Functional differential operators and equations 0792356241, 9780792356240

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CONTENTS

Chapter I. Functional analysis preliminaries 1.1. Banach spaces and bounded linear operators 1.1.1. Topological spaces 1.1.2. Absolutely convergent series in a normed space 1.1.3. The completeness of a quotient space 1.1.4. The completeness of the space of bounded operators 1.1.5. The Banach–Steinhaus theorem 1.1.6. The Banach theorem 1.1.7. The conjugate space 1.1.8. The Hanh–Banach theorem 1.1.9. The second conjugate space 1.1.10. The conjugate operator 1.1.11. Compact operators 1.2. Diagrams 1.2.1. Commutative diagrams 1.2.2. Operators on subspaces and quotient spaces 1.2.3. Short exact sequences 1.2.4. The continuation of a diagram 1.2.5. The existence of a restriction and a quotient operator 1.2.6. The conjugate sequence 1.2.7. The exactness of the conjugate of a short exact sequence 1.2.8. The isomorphism of the simplest rows 1.2.9. The conjugate of a subspace and a quotient space 1.2.10. The closedness of the image of the conjugate operator 1.2.11. The kernel and the image of the conjugate operator 1.3. Lower norms 1.3.1. The definition of the lower norms 1.3.2. The main properties of the lower norms 1.3.3. An equivalent definition of | · |− 1.3.4. The lower norms of the conjugate operator 1.3.5. The continuity of the lower norms 1.3.6. The lower norms as bounds of invertibility 1.3.7. Multiplicative properties of the lower norms 1.3.8. Isometric injections and isometric surjections

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1 1 1 3 4 5 5 5 6 6 6 7 7 7 7 8 9 10 11 11 11 12 12 13 13 14 14 14 15 16 18 18 18 18

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CONTENTS

1.3.9. 1.3.10. 1.3.11. 1.3.12.

Isometric isomophisms of diagrams The closedness of the image of the pre-conjugate operator Fredholm operators The index of a Fredholm operator

19 19 19 20

1.4. Banach algebras 1.4.1. Algebras 1.4.2. The inverse element 1.4.3. The spectrum 1.4.4. Full subalgebras 1.4.5. The spectrum in a subalgebra 1.4.6. Morphisms of algebras 1.4.7. Ideals and quotient algebras 1.4.8. The radical 1.4.9. Holomorphic vector-valued functions 1.4.10. The spectral mapping theorem 1.4.11. The Gel0 fand transform 1.4.12. The Gel0 fand–Naimark theorem

20 20 21 22 23 23 24 24 25 26 28 29 31

1.5. Spaces 1.5.1. 1.5.2. 1.5.3. 1.5.4. 1.5.5. 1.5.6. 1.5.7. 1.5.8. 1.5.9. 1.5.10. 1.5.11. 1.5.12.

31 32 32 33 33 34 37 40 40 42 42 43 43

of continuous and measurable functions The Urysohn theorem The partition of unity Finite-dimensional functions The Stone–Weierstrass theorem The spaces Lp (T ), p < ∞ The spaces L∞ (T ) and L0 (T ) The spaces lq (I) The product of measures ¡ ¢ The isomorphism L1 (T × S, E) ' L1 T, L1 (S, E) The Fubini theorem The Riesz–Thorin theorem The norm of an integral operator

1.6. Infinite matrices 1.6.1. Locally compact abelian groups 1.6.2. The Haar measure 1.6.3. The spaces Lpq and Cq 1.6.4. The matrix representation of operators 1.6.5. The norm of a matrix 1.6.6. Operators with local memory 1.6.7. Operators with uniform memory 1.6.8. Operators with summable memory 1.6.9. Operators with exponential memory 1.6.10. The representation of Cq as a space of sequences 1.6.11. The cut in Cq (R) 1.6.12. The shift operators

44 45 46 48 50 51 52 53 54 55 56 59 60

vii

CONTENTS

1.7. Tensor products 1.7.1. Dual families 1.7.2. The definition of an algebraic tensor product 1.7.3. The definition of a topological tensor product 1.7.4. Examples of topological tensor products 1.7.5. The conjugate cross-norm 1.7.6. The Schatten theorem 1.7.7. The tensor product of operators 1.7.8. The spectral mapping theorem (operator-valued variant) 1.7.9. Irreducible representations of a Banach algebra 1.7.10. The Bochner–Phillips theorem 1.7.11. The tensor product of a direct sum 1.7.12. A remark on the complexification 1.8. Conjugate spaces 1.8.1. The conjugate of Lpq 1.8.2. C00 is ∗-weak dense in L∞ 0 1.8.3. The decomposition l∞ = l00 ⊕ l0⊥ 1.8.4. The modulus of a measure 1.8.5. Measures with a compact support 1.8.6. The decomposition C 0 = C00 ⊕ C0⊥ 1.8.7. Absolutely continuous measures 1.8.8. The Lebesgue–Radon–Nikodym theorem 1.8.9. Mutually singular measures 1.8.10. The Lebesgue decomposition M = Ms ⊕ Mac 1.8.11. The Yosida–Hewitt theorem 1.8.12. The space Mq0 Chapter II. The initial value problem 2.1. The causal invertibility 2.1.1. The definition of a causal operator 2.1.2. The definition of causal invertibility 2.1.3. The monotonicity of causal invertibility 2.1.4. The invertibility of a triangular matrix 2.1.5. The invertibility on all segments implies causal invertibility 2.1.6. Causal invertibility on semi-axes 2.1.7. The additivity of causal invertibility 2.1.8. Local causal invertibility 2.1.9. Local ordinary invertibility 2.2. The causal spectrum 2.2.1. Causal invertibility is a spectral property 2.2.2. The definition of the causal spectrum 2.2.3. The causal spectrum as the union of ordinary spectra 2.2.4. The local causal spectrum 2.2.5. The local ordinary spectrum 2.2.6. The semi-norm G

61 61 61 62 64 66 67 69 71 74 75 78 79 81 81 83 84 85 87 88 89 90 90 92 93 95 97 98 98 101 102 102 104 104 105 105 106 106 106 106 107 107 107 108

viii

CONTENTS

2.3.

2.4.

2.5.

2.6.

2.2.7. An estimation of the spectral radius 2.2.8. The calculation of G(N ) for an integral operator N 2.2.9. The radical in the algebra of causal operators 2.2.10. Compact operators lie in the radical 2.2.11. Compact perturbations of the causal spectrum 2.2.12. Small perturbations preserve causal invertibility Spaces of smooth functions and distributions 2.3.1. The classical and Lebesgue derivatives 1 2.3.2. The spaces Cq1 and Wpq 2.3.3. The derivative of a product in Wp1 loc 1 2.3.4. The isomorphism U between Cq1 and Cq , and Wpq and Lpq 2.3.5. The distribution derivative −1 2.3.6. The spaces Cq−1 and Wpq −1 2.3.7. The equality to zero on (a, b) in Cq−1 and Wpq 2.3.8. The cut in Cq−1 −1 2.3.9. The isomorphism U between Cq−1 and Cq , and Wpq and Lpq −1 2.3.10. Atoms in Wpq −1 2.3.11. The cut in Wpq 1 2.3.12. The duality between Wpq and Wp−1 0 q0 The unique solubility 2.4.1. Initial value problem for an abstract Y 2.4.2. Initial value problem for Y = Lp 2.4.3. Initial value problem for Y = C 2.4.4. Initial value problem for Y = C −1 2.4.5. Initial value problem for Y = Wp−1 2.4.6. The additivity of unique solubility The evolutionary solubility 2.5.1. The definition of evolutionary solubility 2.5.2. Evolutionary solubility as complete unique solubility 2.5.3. The additivity of evolutionary solubility 2.5.4. The decreasing of a segment 2.5.5. Local evolutionary solubility 2.5.6. Local unique solubility 2.5.7. Small perturbations preserve the evolutionary solubility 2.5.8. The operator U as a causal isomorphism 2.5.9. Equations with internal differentiation 2.5.10. Equations with external differentiation Criteria for evolutionary solubility 2.6.1. Multiplication operators 2.6.2. Difference operators 2.6.3. An application to an equation of neutral type 2.6.4. Varying retardation 2.6.5. The spectral mapping theorem for the causal spectrum 2.6.6. The freezing of coefficients

109 109 110 110 111 112 112 112 113 114 115 116 119 120 120 122 123 123 125 127 127 129 130 131 132 135 136 136 137 138 138 138 138 138 138 139 140 140 141 142 144 144 147 148

ix

CONTENTS

2.6.7. An application to an equation of neutral type Chapter III. Stability 3.1. Algebraic preliminaries 3.1.1. The projectors Pa 3.1.2. The representation of Xq as lq (Z, X♥i ) −1 3.1.3. The space C♥i 3.1.4. The operators Qa , Ia , and Ra 3.1.5. A representation of solutions of the initial value problem 3.2. Input–output stability: discrete time 3.2.1. The definition of local solubility 3.2.2. The extended space 3.2.3. The definition of input–output stability 3.2.4. Stability on semi-axes 3.2.5. Input–output stability and causal invertibility 3.2.6. Invertibility in (l∞ , l∞ ) implies that in (l0 , l0 ) 3.2.7. Input–output stability and causal invertibility in (l0 , l0 ) 3.3. Input–output stability: continuous time 3.3.1. The definition of local solubility 3.3.2. Local spaces 3.3.3. The extended space 3.3.4. The definition of input–output stability 3.3.5. Stability on semi-axes 3.3.6. Input–output stability and causal invertibility 3.3.7. Invertibility in (X∞ , Y∞ ) implies that in (X0 , Y0 ) 3.3.8. Input–output stability and causal invertibility in (X0 , Y0 ) 3.3.9. Small perturbations preserve input–output stability 3.3.10. Initial value problem on semi-axis 3.4. Exponential stability: discrete time 3.4.1. Exponential weights and the class e 3.4.2. The class e is full 3.4.3. The independence of invertibility from q for T ∈ e 3.4.4. The case of semi-axes 3.4.5. The definition of uniform solubility 3.4.6. The definition of exponential stability 3.4.7. Exponential stability and causal invertibility 3.4.8. The independence of stability from q for T ∈ e 3.5. Exponential stability: continuous time 3.5.1. Exponential weights on R 3.5.2. The class e (continuous time) 3.5.3. The operators Qa , Ia , and Ra on weighted spaces 3.5.4. The independence of invertibility from q for L ∈ e 3.5.5. The definition of uniform solubility 3.5.6. Weighted spaces on semi-axes 3.5.7. Operators on semi-axes

149 150 150 150 152 153 154 155 157 157 158 159 160 160 161 162 162 163 163 164 165 166 166 168 168 168 170 171 171 174 175 175 176 177 177 179 179 179 182 186 187 187 188 189

x

CONTENTS

3.5.8. The definition of exponential stability 3.5.9. Exponential stability and causal invertibility 3.5.10. The independence of stability from q for L ∈ e 3.5.11. Small perturbations preserve exponential stability 3.5.12. Initial value problem on semi-axis 3.6. Exponential dichotomy: continuous time 3.6.1. The definitions of instability 3.6.2. Small perturbations preserve rough instability 3.6.3. The definition of exponential dichotomy 3.6.4. Dichotomy and invertibility 3.6.5. Dichotomy, stability, and instability

192 192 195 195 195 196 196 197 198 201 207

Chapter IV. Shift invariant operators and equations 4.1. Algebras of bounded measures 4.1.1. Convolution operators on C0 4.1.2. Shift invariant operators 4.1.3. The convolution of measures 4.1.4. The ideal Mac of absolutely continuous measures 4.1.5. The push transform 4.1.6. Convolution operators on L1 4.1.7. The ideal Mc of continuous measures 4.1.8. The subalgebra Md of discrete measures 4.1.9. The main decomposition 4.1.10. Full subalgebras of M 4.1.11. The algebra M+ 4.1.12. Full subalgebras of M+ 4.2. The Fourier transform 4.2.1. The dual group 4.2.2. The character space of Mac 4.2.3. The character space of Md 4.2.4. The topology on X(G) 4.2.5. The Pontrjagin theorem 4.2.6. The invertibility in Md ] 4.2.7. The invertibility in M ac 4.2.8. The character space of Md⊕ac 4.2.9. The invertibility in Md⊕ac 4.2.10. The Kronecker theorem 4.2.11. A generalization of the Kronecker theorem g 4.2.12. A remark on the character space of M sc 4.3. The Laplace transform 4.3.1. The character space of a semi-group 4.3.2. The character space of M+ ac + ] 4.3.3. The invertibility in M

209 209 209 210 211 212 213 214 216 217 218 219 220 221 221 221 225 229 229 230 230 230 231 231 232 232 235 239 239 241

ac

4.3.4. The 4.3.5. The

character space of M+ d invertibility in M+ d

242 243 244

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CONTENTS

4.3.6. 4.3.7. 4.3.8. 4.3.9. 4.3.10.

The character space of M+ (Z) The invertibility in M+ (Z) The character space of M+ d⊕ac The invertibility in M+ d⊕ac + X+ is dense in X+ and in X+ b b tX

244 245 245 245 246

g + 4.3.11. A remark on the character space of M sc 4.4. Convolution operators 4.4.1. Convolution operators on Lp , Lpq , and Cq (scalar case) 4.4.2. The ordering in the space of measures 4.4.3. Operator-valued measures 4.4.4. Convolution operators on Lp , Lpq , and Cq (vector case) 4.4.5. The conjugate of a convolution operator 4.4.6. The norm of Tµ on L1 (G, C) and L∞ (G, C) 4.4.7. The preservation of multiplication (scalar case) 4.4.8. The preservation of multiplication (operator-valued case) 4.4.9. The main decomposition (operator-valued case) 4.4.10. Difference operators 4.4.11. Integral operators 4.4.12. The algebra M+ (R, B) 4.5. Invertibility and causal invertibility 4.5.1. The invertibility in a subalgebra of operator-valued measures 4.5.2. The invertibility of measures 4.5.3. The causal invertibility of measures 4.5.4. A general sufficient criterion 4.5.5. The invertibility of operators 4.5.6. The causal invertibility of operators 4.5.7. The spectrum of a difference operator 4.5.8. The causal spectrum of a difference operator 4.5.9. Differential operators 4.5.10. Sufficient conditions for causal invertibility 4.5.11. Periodic functional operators 4.5.12. Periodic differential operators

247 247 247 250 251 258 260 261 261 262 263 264 264 265 267 267 268 271 273 273 276 277 279 281 283 283 284

Chapter V. Operators with varying coefficients 5.1. Multiplication operators 5.1.1. Multiplication operators 5.1.2. Operators with zero memory 5.1.3. A characterization of operators with zero memory 5.1.4. The equality of X(lq ) and Xz (lq ) for a discrete group 5.1.5. Operators with zero memory on a non-discrete group 5.1.6. A representation of operators A ∈ Xz 5.1.7. The Bohr compactification 5.1.8. The group of isometries of a compact set 5.1.9. Almost periodic functions 5.1.10. Oscillation invariant operators

285 285 286 286 287 287 287 288 289 289 290 291

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CONTENTS

5.2.

5.3.

5.4.

5.5.

5.1.11. The equality of XΨ and Xz 5.1.12. The algebras x, xz , and xψ Difference operators 5.2.1. Difference operators with summable memory 5.2.2. The equality of S, Sz , and SΨ 5.2.3. The mean value of a character 5.2.4. The crossed product S(G, L) 5.2.5. The subalgebra SΨ is full 5.2.6. The subalgebra s is full 5.2.7. The invertibility of D ∈ s is independent from q 5.2.8. The subalgebras Sz (Lpq ) and S(Lpq ) are full 5.2.9. The algebras D, Dz , and DΨ 5.2.10. The invertibility of D ∈ S(Lpq ) is independent from p and q 5.2.11. The subgroup generated by the set of shifts 5.2.12. Difference operators with exponential memory Smoothing operators 5.3.1. Universal operators with summable memory 5.3.2. Smoothing operators with summable memory 5.3.3. Integral operators with bounded kernels 5.3.4. An integral representation for N ∈ N∞ g 5.3.5. The invertibility of N ∈ N ∞ is independent from p and q g 5.3.6. The subalgebra N∞ is full 5.3.7. Universal operators with exponential memory 5.3.8. Smoothing operators with exponential memory 5.3.9. The integral representation for M ∈ M∞ ] 5.3.10. The subalgebra M ∞ is full Integral operators 5.4.1. The class N1 of kernels 5.4.2. The multiplication in N1 5.4.3. Operators of the class N1 5.4.4. The algebra N1 of operators 5.4.5. N∞ is dense in N1 5.4.6. N∞ is an ideal in N1 f1 is full 5.4.7. The subalgebra N f1 is independent from p and q 5.4.8. The invertibility of N ∈ N 5.4.9. The algebra M1 of kernels 5.4.10. M∞ is dense in M1 5.4.11. M∞ is an ideal in M1 g1 is full 5.4.12. The subalgebra M Operators with locally fading memory 5.5.1. Directions on a Banach space 5.5.2. The classes tf and t 5.5.3. A two-diagonal representation for T ∈ tf (lq ) 5.5.4. The subalgebra t(lq ) is full

292 295 296 296 297 298 298 299 301 301 302 302 303 303 304 307 307 309 309 310 311 312 312 313 314 314 315 315 316 317 318 318 319 319 320 320 321 321 322 322 322 324 326 327

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CONTENTS

5.5.5. The isomorphism t(l∞ ) ' t(l0 ) 5.5.6. The equivalence of the invertibility in t(l0 ) and t(l∞ ) 5.5.7. Consistent directions 5.5.8. The Fourier transform maps L1 into C0 5.5.9. Compactly supported sets 5.5.10. Ψ-bounded sets 5.5.11. Differential operators 5.6. Operators with continuous coefficients 5.6.1. Operators with uniformly continuous coefficients 5.6.2. Operators with continuous coefficients 5.6.3. The subalgebra C is full 5.6.4. The conjugate of an operator of the class C 5.6.5. The isomorphisms C(Lp∞ ) ' C(Lp0 ) and C(C∞ ) ' C(C0 ) 5.6.6. An equivalent representation of a convolution 5.6.7. The invariant subspaces 5.6.8. The main example: S(Cq ) ⊆ C(Cq ) 5.6.9. The isomorphism C(L∞q ) ' C(Cq ) 5.6.10. The invertibility of D ∈ S(Cq ) 5.6.11. The equivalence of the lower norms on L∞q and Cq

328 330 330 330 330 331 332 333 333 334 335 335 336 336 336 339 340 340 341

Chapter VI. Differential difference equations 6.1. The local Fredholm property 6.1.1. Locally compact operators 6.1.2. The ideal k 6.1.3. Locally Fredholm operators 6.1.4. The operator T # 6.1.5. The conjugate of a locally compact operator 6.1.6. The conjugate of a locally Fredholm operator 6.1.7. The lower norms of T # 6.1.8. The lower norm on a subspace 6.1.9. Rapidly oscillating characters 6.1.10. An estimate of |D# |+ 6.1.11. Difference operators can not be locally Fredholm 6.2. The solubility for derivatives 6.2.1. Functional equations 6.2.2. The operator U −1 belongs to k 6.2.3. Bounded solution problem 6.2.4. Periodic solution problem 6.2.5. Elliptic equations with shifts 6.2.6. The representation of D ∈ D as a series 6.2.7. Input–output stability and causal invertibility for L ∈ t 6.2.8. Stability 6.3. The independence from the choice of a norm 6.3.1. The case of invertibility with D, B ∈ s 6.3.2. The case of stability with D, B ∈ s

343 343 344 345 346 347 347 347 347 347 348 349 353 355 355 355 356 358 359 361 361 362 366 366 367

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CONTENTS

6.3.3. The case of invertibility with D, B ∈ C 6.3.4. The case of stability with D, B ∈ C 6.3.5. The case of invertibility with D ∈ S and B ∈ V 6.3.6. The case of stability with D ∈ S and B ∈ V 6.3.7. The case of invertibility with D, B ∈ S(Cq ) 6.3.8. The case of stability with D, B ∈ S(Cq ) 6.3.9. The case of invertibility with D ∈ S1 and B ∈ t 6.3.10. The case of stability with D ∈ S1 and B ∈ t 6.3.11. The case of invertibility with D ∈ S1 (C) and B ∈ S(C) 6.3.12. The case of stability with D ∈ S1 (C) and B ∈ S(C) 6.4. Green’s function 6.4.1. The integral representation for solutions 6.4.2. The integral representation for derivative 6.4.3. Distributions of two variables ∂G 6.4.4. The partial derivatives ∂G ∂t and ∂s 6.4.5. The conjugate equation 6.4.6. The fundamental solution 6.4.7. The case of D ∈ S1 (C) and B ∈ S(C) 6.5. Almost periodic operators 6.5.1. Almost periodic operators 6.5.2. The subalgebra BAP is full 6.5.3. More about the algebra DΨ 6.5.4. More about the algebra dψ 6.5.5. The equality tAP = dAP 6.5.6. The ideal h ¡ ¢ 6.5.7. The case of hAP lq (Zn ) 6.5.8. The case of hAP (L) 6.5.9. Almost periodic difference operators 6.5.10. The case of DAP 6.5.11. The case of DAP + hAP 6.5.12. The case of a differential-difference operator Comments Bibliography Index Notation Index

367 368 368 368 369 369 369 371 371 371 372 372 373 375 376 378 380 381 382 382 385 385 386 386 387 390 394 396 397 400 402 403 411 429 433

PREFACE

This is a book on the methods of operator theory in linear functional differential equations. I imagine the reader as a mathematician who is working in functional differential equations and interested in new tools, or as a specialist in operator theory who is interested in new applications. The level of the book is not homogeneous. As far as possible the parts of the book devoted to operator theory and the applications to functional differential equations are written in such a way that they can be read independently. When we deal mostly with functional differential equations, the exposition is quite simple and detailed. We do not spare space for giving several similar definitions and statements. Here we try to avoid long proofs. The objective of this part is to demonstrate how the operator methods discussed in the book arise in the theory of functional differential equations. Against this, the discussion of operator theory is based on the Lebesgue integral, Banach algebras, abstract harmonic analysis, and other usual tools of functional analysis. Formally, necessary prerequisites are collected in chapter 1. But chapter 1 can not be used as a textbook for beginners. Mostly we only recall here known definitions and theorems in a common terminology and notation. We prove only facts which are not well known or are not often found in a relevant account (in our opinion). So we assume that the reader has a certain experience in functional analysis. Undoubtedly, it is not so much important to know a big number of facts but to have the general mathematical culture. Except for the knowledge of the usual university course based on any of the texts [DS], [Edw], [Hil], [Kir], or [Rud2 ] the book is practically self-contained. There are many different books on functional differential equations, for example, [Ant1,2 ], [Bel1,2 ], [Burt], [Dri2 ], [ElN], [Ger], [Gri], [Hal], [Hale2,6 ], [Hin], [Kam3 ], [Kap1 ], [Ko3 ], [Kol1,2,3 ], [Kr], [Mys], [Pin], and [Raz]. Most of them appeal to the ordinary differential equations intuition of the reader. Apparently the simplest introduction to the subject is [ElN]. Many close ideas can be found in the theory of pseudodifferential operators and singular integral equations as well, see, e.g., [Kra2 ], [Lit], [Ste1 ], and [Tayl1 ]. Finally, we mention books on control [Cor1,2 ], [FeS], [Ben], [Deso], and [Wil2 ] as having points of contact with this book. It is impossible to say definitely what is a functional differential equation and what is not. Different authors define this notion in somewhat different ways. Of course, this is natural and usual for such a kind of notion. In this book functional

xv

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PREFACE

differential equations are interpreted in a very wide sense. The simplest examples of functional differential equations are x(t) ˙ + a(t)x(t) + b(t)x(t − h) = f (t) and

Z x(t) ˙ +

n(t, s)x(s) ds = f (t).

We regard all equations which possess similar properties or permit the same methods of investigation as being functional differential equations, too. Thus we consider the properties and methods as the most important features for classification. The traditional definition is as follows. A functional differential equation is an equation which links the values of an unknown function x (and perhaps the values of its derivatives) at the present moment with values of x (and its derivatives) in the past (and sometimes in the future). Here are the main examples which are to be considered as functional differential equations: x(t) ˙ + a(t)x(t) = f (t), ∞ X

am (t)x(t ˙ − hm ) +

m=1 ∞ X

∞ X

bm (t)x(t − hm ) = f (t),

m=1 ∞ X

d am (t)x(t − hm ) + bm (t)x(t − hm ) = f (t), dt m=1 m=1 ∞ X

∞ ¡ ¢ X ¡ ¢ am (t)x˙ τ (hm ) + bm (t)x τ (hm ) = f (t),

m=1 ∞ X

m=1 ∞ X

¡ ¢ d am (t)x τ (hm ) + bm (t)x(τ (hm )) = f (t), dt m=1 m=1 ∞ X m=1 ∞ X m=1

(1) (2i) (2e) (3i) (3e)

am (t)x(t − hm ) = f (t),

(4)

¡ ¢ am (t)x τ (hm ) = f (t),

(5)

Z

t

x(t) + Z

n(t, s)x(s) ds = f (t),

(6)

n(t, s)x(s) ds = f (t).

(7)

−∞ +∞

x(t) + −∞

All sums and integrals are assumed to be convergent in a suitable sense. The usual terminology is the following. Equation (1) is an ordinary differential equation. Equations (2) are called differential difference equations. Equations (3) are usually called equations with deviating argument. Both equations (2) and (3) are examples of functional differential equations. Of course, (1) is a special case of a functional

PREFACE

xvii

differential equation as well. To distinguish features of (2)–(7) from (1) one usually uses the words memory and retardation. Equation (4) is a difference equation; equation (5) is a functional equation (these explain the terms differential difference equation and functional differential equation). Equations (6) and (7) are Volterra and Fredholm integral equations, respectively. Equations (1), (2), and (3) are of the first order. It is convenient to consider equations (4), (5), (6), (7), and their combinations, e.g., the difference integral equation Z +∞ ∞ X am (t)x(t − hm ) + n(t, s)x(s) ds = f (t), m=1

−∞

as functional differential equations of the zero order. In this book we do not consider equations of the second and higher orders because for the problems under discussion there is no essential difference between equations of different order. Moreover, often equations of the zero order will play a more important role in our exposition than equations of the first order. Many results will first be obtained for equations of the zero order and then carried over to actual differential equations. There is no equivalent of the notion of a zero order equation for ordinary differential equations as natural as the notion of a zero order equation for functional differential ones. This fact deserves attention because it is closely connected with the reason why functional differential equations occur. Fundamental laws of physics have the form of differential equations without memory or retardation. In the light of this, how can functional differential equations arise in the description of any physical process? The answer is simple. The general solution of an ordinary differential equation is an integral relation (with a memory), and not a differential one. If an object is described by a system of differential equations and one solves all or some of them then he obtains something like equations (2)–(7). (In practice, some relations can be obtained not by solving equations but from an experiment. Of course, the result is the same). It is interesting to note that in solving hyperbolic partial differential equations one obtains terms of the kind am (t)x(t ˙ − hm ), see, e.g., [Kol1 ]. Thus, using mathematical language the class of differential operators is not closed with respect to the operation of inversion. If one has some initial class of differential operators and forms their sums, products, and inverses in various combinations, one obtains all operators which may occur within the framework of the given problem. Classes of operators closed under algebraic operations are called full subalgebras (of the algebra of all linear operators). It is convenient to work from the outset in a class closed in this sense. This is one of the principal ideas of this book. It is notable that a similar idea provides the basis of the theory of pseudodifferential operators. It can be easily shown that any non-trivial full class contains operators with unbounded memory (which, however, usually can be approximated in norm by operators with bounded memory). Thus, from our point of view, a complete theory of functional differential equations must be a theory of equations with unbounded delay.

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PREFACE

There are some additional types of equations which have properties similar to those of (2)–(7). For example, there are the difference equation with discrete time ∞ X

amn xn = fm ,

(8)

n=−∞

and the partial differential difference equation ∞ X m=1

am (x)∆u(x − hm ) +

∞ X

bm (x)u(x − hm ) = f (x).

(9)

m=1

Although we do not discuss the corresponding applications in detail, many ideas and results of this book can be employed for the investigation of such equations too. The main difference of these equations from (1)–(7) is the following: equations (1)–(7) are equations on R, but (8) and (9) are equations on Z and Rn , respectively. In order to embrace the possibility of applications to equations on Z and Rn we prove general technical results on an arbitrary locally compact abelian group G. Groups Rn , Zn , and T = R/Z are examples of a locally compact abelian group; moreover, roughly speaking, the group Ra × Zb × Tc is an almost general locally compact abelian group. Of course, this generality slightly complicates the book for those who are interested only in equations on R. When it is not difficult to do so, we consider the case where the space E, in which the solution x takes its values, is infinite-dimensional. We proceed to a brief account of the major results of the book. Sections 1.1, 1.2, 1.4, and 1.5 contain more or less standard preliminaries of functional analysis. In §1.3 we discuss lower norms — a convenient tool for the investigation of one-sided invertibility. In §1.6 we introduce four classes of infinite matrices which correspond to the four types of unbounded memory considered throughout the book. In §1.7 we prove the Bochner–Phillips theorem — a generalization of Gelfand’s theory of commutative algebras to a special non-commutative case. In chapter 4 we use this theorem as the main tool for the investigation of shift invariant equations in the spaces of vector functions. In §1.8 we present the Yosida–Hewitt theorem — an analogue of the Lebesgue decomposition of measures for elements of the conjugate of L∞ . This theorem helps one to treat L1 as a pre-conjugate of L∞ . In applications, equations (2), (3), (4), and (5) usually possess the additional properties hm ≥ 0 and τm (t) ≤ t. It is convenient to reformulate these properties in an abstract way. Namely, a linear operator L acting between two spaces of functions defined on R is called causal if for all x and t ¡ ¢ x(s) = 0, s < t, ⇒ Lx (s) = 0, s < t. Equations with causal operators usually describe processes in time. Therefore one can naturally talk about the initial value problem (chapter 2) and stability (chapter 3) for such equations.

PREFACE

xix

We say that the equation Lx = f is evolutionarily soluble if it is locally soluble and the solution x depends on the forcing function f in a causal way. Chapter 2 is devoted to evolutionary solubility. If the first sum in (2) and (3) contains more than one non-zero summand, one says that equations (2) and (3) are of neutral type. If the derivative occurs in equation only at the present moment, i.e., only as x(t), ˙ one says that equations (2) and (3) are of retarded type. In this terminology it is convenient to classify equation (6) as an equation of retarded type, and equations (4) and (5) as equations of neutral type. For equations of retarded type the evolutionary solubility coincides with the ordinary solubility. But for equations of neutral type it is not the case. We say that a causal operator L is causally invertible if the inverse operator L−1 exists and is causal too. It is important that in general the existence of L−1 does not imply that L−1 is causal. Causal operators form a Banach algebra. Thus the causal invertibility is a spectral property. We would like to call the attention of those working in operator theory to the notion of causal invertibility. The main result of chapter 2 states that the equation Lx = f is evolutionarily soluble if and only if the operator L is locally causally invertible. The main result of chapter 3 states that the equation Lx = f is input–output stable if and only if the operator L is causally invertible on the whole of R. (The existence of the ordinary inverse on R is equivalent to the dichotomy of solutions. This illustrates the fact the causal invertibility does not coincide with the ordinary invertibility.) We also note that only equations with an exponential kind of memory can be exponentially stable; for them exponential stability is equivalent to input–output stability. Chapter 4 is concerned with the invertibility and causal invertibility of shift invariant operators and their applications to shift invariant equations. Chapter 5 is the most abstract. We prove that some classes of operators connected with functional differential equations form full subalgebras, see above for the definition. In terms of equations these results mean that the dependence f 7→ x of the solution x of the equation Lx = f upon the forcing function f inherit such properties of an initial operator L as its type (e.g., difference or integral), the kind of its memory (e.g., exponential or summable), the smoothness of its coefficients (e.g., continuous or measurable), and so on. In §6.2 we prove that a differential difference equation of neutral type usually can be reduced to an equation of retarded type (§6.1 contains auxiliary technical preliminaries). For example, if equation stable then: the corresponding ¡ ¢(2) is P ∞ equation (4) is stable too; the operator Dx (t) = m=1 am (t)x(t−hm ) is causally invertible; and if one multiplies (2i) by D−1 or, respectively, carries out the change of variables z = D−1 x in (2e), one obtains the retarded type equation. In §6.3 we show that under natural assumptions the general properties of an equation (such as stability) do not depend on functional spaces in which the equation is investigated. For example, consider equations (2i) and (3i), on the one hand, and equations (2e) and (3e), on the other. ¡ These are two forms or two ways of writing of functional differential equations if am and τm are smooth one can ¢ differentiate (e) and obtain (i) . We call (i) and (e) equations with interior and

xx

PREFACE

exterior differentiation, respectively. Nevertheless, the forms (i) and (e) presuppose different functional spaces for solutions. We prove that equations (2i) and (2e) with relevant coefficients am are stable simultaneously. In §6.4 we study Green’s function, i.e., the properties of the integral representation of the solution. In §6.5 we prove that one-sided invertibility of a differential difference operator with almost periodic coefficients (in particular, with periodic coefficients) implies its two-sided invertibility. This fact can be considered as an analogue of the Fredholm alternative. The text is divided into chapters, sections, and subsections, see Contents. Each subsection contains one statement (theorem, proposition, lemma, or corollary) or none. As a rule a reference of the kind ‘see 1.2.3’ means the reference to such a statement. Some theorems are named after their author(s); all of them are included in the Index. The numbering of formulae starts afresh in each section.

CHAPTER I

FUNCTIONAL ANALYSIS PRELIMINARIES We use functional analysis as a language in our discussion. We assume that the reader is familiar with some conventional university handbooks on functional analysis, e.g., [DS], [Edw], [Hil], [Kir], or [Rud2 ]. In this chapter we recall notation, objects, and statements we use in different places in the book. The exposition is not uniform. For facts which have the same treatment in most of the usual manuals we restrict ourselves to explicit formulation and the elimination of possible ambiguity of terminology. Conversely, if a statement is more special we give it with a proof. Of course, not all the material of this chapter will be needed at once. So, there is no need to keep it in mind constantly: the Index and the Notation index may help to find the necessary reference. Nevertheless, we advise the reader to examine the chapter cursorily and then use it as required.

1.1. Banach spaces and bounded linear operators 1.1.1. Topological spaces. Here we recall some terminology and facts from general topology. Detailed exposition can be found, e.g., in [Bou1 ] or [Kel]. Let T be a set. A family O of subsets of T is called a topology if the union of an arbitrary subfamily of O and the intersection of any finite subfamily of O belong to O. We define the union of the empty family to be ∅ and the intersection of the empty family to be T . Thus ∅ and T belong to O by definition. The pair (T, O) is called a topological space. If no confusion can arise, one says that the set T itself is a topological space. A subfamily O0 ⊆ O is called a base of O if every set O ∈ O can be represented as a union of members of O0 . Members of O are called open sets and complements of them are called closed sets. The largest open set contained in E ⊆ T is called the interior of E; the smallest closed set containing E is called the closure of E and is denoted E. The set E ∩ (T \ E) is called the boundary of E. The set E ⊆ T is called dense in T if E = T. A set U ⊆ T is called a neighbourhood of t ∈ T if there exists an open set O such that t ∈ O ⊆ U . A neighbourhood need not itself be open. A family {Uα } is called a neighbourhood base at t if all Uα are neighbourhoods of t and any neighbourhood of t contains a set Uα . A set U ⊆ T is called a neighbourhood of a set A ⊆ T if there exists an open set O such that A ⊆ O ⊆ U . A topology on T induces the relative topology O0 on any subset T0 ⊆ T . The topology O0 on T0 is the family of all sets O ∩ T0 , O ∈ O.

1

2

I. FUNCTIONAL ANALYSIS PRELIMINARIES

A function f : T → S from a topological spaces T to a topological space S is called continuous if the pre-image f −1 (O) = { t ∈ T : f (t) ∈ O } of any open set O ⊆ S is open in T . A function f is called continuous at a point t if the pre-image of any neighbourhood of f (t) is a neighbourhood of t. A function f is continuous if and only if it is continuous at all points. Let {Uα } be a neighbourhood base at f (t) and {Vβ } be a neighbourhood base at t. A function f is continuous at the point t if the pre-image of any Uα contains some Vβ . A topological space is called Hausdorff if every pair of distinct points of it have disjoint neighbourhoods. Usually we assume that topological spaces under consideration are Hausdorff. A topological space is called normal if every pair of disjoint closed sets in it have disjoint neighbourhoods. A topology O and the topological space T are called discrete if O consists of all subsets of T . In other words, a topological space T is discrete if the set {t} is a neighbourhood of the point t for each t ∈ T . S A collection {Vα } is called a covering of K ⊆ T if K ⊆ α Vα . A topological space T is called compact if it is Hausdorff and every covering {Vα } of T consisting of open sets Vα contains a finite sub-collection which also covers T . A subset K ⊆ T is called compact if it is a compact topological space with respect to the relative topology. A closed subset of a compact set is evidently compact. A compact topological space is normal. A subset N ⊆ T is called conditionally compact if its closure is compact. Let T be a set. A function % : T × T → [0, +∞) is called a metric if it possesses the properties: (i) %(t, s) = %(s, t), (ii) %(t, r) ≤ %(t, s) + %(s, r), (iii) %(t, s) = 0 if t = s, (iv) t = s if %(t, s) = 0 for all t, s, r ∈ T . Sometimes assumption (iv) is omitted; such a metric is called degenerate. The pair (T, %), or simply the set T itself, is called a metric space. The family of all sets Ut,ε = { s ∈ T : %(t, s) < ε }, t ∈ T and ε > 0, is a base of the topology. This topology is called the topology induced by the metric. The topology induced by a metric is Hausdorff if and only if the metric is non-degenerate. Let N be a subset of a metric space. A subset S is called an ε-net for N if the collection of all open balls with centres in S and radius ε covers N . The set N is conditionally compact if and only if it possesses a finite ε-net for any ε > 0. It is easy to show that the set N is conditionally compact if it possesses a conditionally compact ε-net for any ε > 0. Let T be a metric space. A sequence tn ∈ T is called a Cauchy sequence if ∀ε > 0 ∃N

∀n, m > N

%(tn , tm ) < ε.

A metric space T is called complete if any Cauchy sequence in it has a limit. A metric space is compact if and only if any sequence in it contains a Cauchy subsequence. A topological space T is called locally compact if each of its points has a compact neighbourhood. A locally compact topological space T can be embedded in a compact space Te by adjoining one additional point. We recall this construction. Let ∞ be an arbitrary point not in T . We denote by Te the union of T and ∞. We define the topology on Te as follows. The set U ⊆ Te is open if either ∞ ∈ /U

1.1. BANACH SPACES AND BOUNDED LINEAR OPERATORS

3

and U is open in T , or ∞ ∈ U and T \ U is compact in T . It is easy to see that Te is a compact topological space, and the topology on T induced by the embedding in Te coincides with the initial topology. The topological space Te is called the one point compactification of T . Let T be a topological space, H be a set, and F be a collection of functions f : H → T . It is easy to see that the family O of all finite intersections of the sets of the form f −1 (V ), where f ∈ F and V ⊆ T is open, forms a base for a topology. Clearly, it is the weakest topology in which all the functions f ∈ F are continuous. Let Tα , α ∈ A, be a family Q of topological spaces. The product topology on the Cartesian product T = α∈A Tα is the weakest topology in which all natural projections qα : T → Tα , α ∈ A, are continuous. Proposition ([Bou1 , ch. 1, §9], [Kel, ch. 5]). (a) Let T and S be Hausdorff topological spaces, and let f : T → S be a continuous function. If T is compact then the image f (T ) is also compact. If, in addition, the function f has the inverse f −1 : S → T then f −1 is continuous, too. Q (b) Let Tα , α ∈ A, be compact. Then α∈A Tα is also compact. 1.1.2. Absolutely convergent series in a normed space. To simplify our exposition we assume that all linear spaces are considered over the field C of complex numbers unless otherwise stated. The real case usually can be reduced to a complex case by means of complexification, see also the discussion in 1.7.12. A linear space X is called normed if there is fixed a function k · k : X → R, called a norm, satisfying the following properties kx + yk ≤ kxk + kyk, kλxk = |λ| · kxk, kxk ≥ 0, kxk 6= 0

if x 6= 0

for all x, y ∈ X and λ ∈ C. Sometimes the last assumption is omitted; such a norm is called a degenerate norm or a semi-norm. Evidently, the kernel { x : kxk = 0 } of a degenerate norm is a subspace. Clearly, the function %(x, y) = kx − yk is a metric. A normed space is called Banach if it is complete. It is well known that any non-complete normed space can be completed, i.e., embedded isometrically in a Banach space as a dense subspace. A space with a degenerate norm may also be complete or be completed (the standard example is the definition of the Lebesgue spaces Lp , see 1.5.5 below). We say that the norms k·k and k·k∗ on X are equivalent if there exist constants c, C ∈ (0, ∞) such that ckxk ≤ kxk∗ ≤ Ckxk

for all x ∈ X.

Clearly, equivalent norms induce the same topology. Moreover, they generate the same class of Cauchy sequences, and the completions with respect to equivalent norms are the same.

4

I. FUNCTIONAL ANALYSIS PRELIMINARIES

P∞ P∞ A series n=1 xn , xn ∈ X, is called absolutely convergent if n=1 kxn k < ∞. It is often convenient to check the completeness by means of the following statement. Proposition. (a) Any Cauchy sequence {xn } in a normed space contains a subsequence {xnk } such that the series ∞ X xn1 + (xnk+1 − xnk ) k=1

converges absolutely. (Clearly, the partial sums of this series are {xnk }.) (b) A normed space X is complete if and only if any absolutely convergent series in it has a limit. (c) A subspace of a Banach space is closed if and only if any absolutely convergent series has a limit in it. Proof. (a) It is enough to extract the subsequence {xnk } so that kxnk+1 − xnk k ≤ 2−k . (b) follows from (a). (c) follows from (b). ¤ 1.1.3. The completeness of a quotient space. Let X be a linear space and X0 be a subspace of it. The set { x + X0 } = { x + x0 : x0 ∈ X0 } is called an equivalence class modulo X0 . The family of all equivalence classes is a linear space with respect to the operations { x + X0 } + { y + X0 } = { (x + y) + X0 } and λ{ x + X0 } = { λx + X0 }. It is called a quotient space and is denoted X/X0 . We denote by dim X0 the dimension of X0 , and by codim X0 the codimension of X0 , i.e., the dimension of the quotient space X/X0 . We denote by j : X0 → X the natural embedding, and by q : X → X/X0 the natural projection. If X is a normed space we define the norm on X/X0 by the formula ke xk = inf{ kxk : x ∈ x e },

x e ∈ X/X0 .

The norm on X/X0 can be degenerate. Clearly, kqk ≤ 1. Proposition. Let X be a Banach space. Then the quotient space X/X0 is complete. If X0 is closed the norm on X/X0 is non-degenerate. P∞ en , x e ∈ X/X0 , be Proof. First, we show that X/X0 is complete. Let n=1 x an absolutely convergent series in X/X0 . By the definition of the norm on X/X0 we can choose xn ∈ x en such that kxn k ≤ ke xn k + 2−n .

P∞ Then the series complete n=1 xn is also absolutely convergent. Since X Pis P ∞ ∞ en conn=1 x n=1 xn converges to some x ∈ X. And since kqk ≤ 1 the series verges to x e = qx. Now we show that the norm on X/X0 is non-degenerate. Let ke xk = 0 for some x e ∈ X/X0 . This means that there exists a sequence xn ∈ x e which converges to 0. We note that since X0 is closed the set x e = { x + X0 } is also closed. Therefore x e contains zero. Thus x e coincides with the zero class { 0 + X0 }. ¤

1.1. BANACH SPACES AND BOUNDED LINEAR OPERATORS

5

A subspace X 0 of X is called a complement of X0 if X = X0 ⊕ X 0 . If a complement exists, X0 is called complemented. If X is a Banach space and X0 is closed, it is usually implied that X 0 must be closed, too. Unfortunately, a closed subspace of a Banach space can have no a closed complement. It is easy to show that a complement X 0 is (topologically) isomorphic to the quotient space X/X0 . 1.1.4. The completeness of the space of bounded operators. We denote by BX the closed unit ball { x ∈ X : kxk ≤ 1 } in a normed space X. Let X and Y be normed spaces, and let A : X → Y be a linear operator. In such a case we say that the operator A acts from X (in)to Y , and that X is the domain and Y is the range of A; if X = Y we say that the operator A acts on X. As usual by the norm of A we mean the number kAk = kA : X → Y k = sup{ kAxk : x ∈ BX } or, equivalently (see §1.3 below for discussion), kAk = inf{ l ≥ 0 : ABX ⊂ lBY }. An operator A is called bounded if kAk < ∞. An operator A is continuous if and only if it is bounded. We denote by B = B(X, Y ) the set of all bounded linear operators A acting from a Banach space X into a Banach space Y . If X = Y we employ the brief notation B = B(X). The symbols 1 and 1X denote the identity operator on X. Clearly, B(X, Y ) is a normed space. The topology and the convergence on B(X, Y ) induced by the norm are called uniform. Proposition. The uniform topology on the space B(X, Y ) is complete. Proof. The proof follows easily from 1.1.2. ¤ 1.1.5. The Banach–Steinhaus theorem. The weakest topology on B(X, Y ) in which all functions A 7→ Ax, x ∈ X, are continuous (see 1.1.1) is called the strong topology. The following theorem is useful for working with the strong topology. Theorem ([DS, ch. 2, §1, theorem 11], [Rud2 , theorem 2.6]). Let a family { Aα ∈ B(X, Y ) : α ∈ Λ } satisfy the following property: for any x ∈ X the set { kAα xk : α ∈ Λ } is bounded. Then there exists a constant K such that kAα k ≤ K

for all α ∈ Λ.

1.1.6. The Banach theorem. We recall that we denote by 1 = 1X the identity operator from X to X. An operator A : X → Y is called invertible if there exists an operator B : Y → X such that AB = 1Y and BA = 1X . In such a case the operator B is called the inverse of A and is denoted by A−1 . Let A : X → Y be a linear operator. We denote by Ker A the kernel of A, i.e., the subspace Ker A = { x ∈ X : Ax = 0 }. And we denote by Im A the image of A, i.e., the subspace Im A = { y ∈ Y : y = Ax for some x ∈ X }. We call the subset A−1 F = { x ∈ X : Ax ∈ F } the pre-image of a subset F ⊆ Y . If Im A = Y we say that the operator A is surjective or acts onto Y . If Ker A = {0}, the operator A is called injective. Clearly, a linear operator A : X → Y is invertible if and only if Ker A = {0} and Im A = Y , i.e., it is both surjective and injective. An invertible operator is also called bijective.

6

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Theorem ([DS, ch. 2, §2, theorem 2], [Rud2 , corollary 2.12(c)]). If the inverse of A ∈ B(X, Y ) exists then it is also linear and bounded. An invertible operator A ∈ B(X, Y ) can be interpreted as a (topological ) isomorphism from X onto Y . We say that X and Y are (topologically) isomorphic and write X ' Y if there exists an isomorphism A ∈ B(X, Y ). An isomorphism A is called isometric if kAxk = kxk for all x ∈ X, or, equivalently, kAk = 1 and kA−1 k = 1, or kAk ≤ 1 and kA−1 k ≤ 1. 1.1.7. The conjugate space. The conjugate space of a Banach space X is the space X 0 consisting of all bounded linear functionals x0 : X → C. We denote the value of the functional x0 ∈ X 0 at x ∈ X by x0 (x) or mainly by hx, x0 i. We consider on X 0 the usual norm kx0 k = sup{ |hx, x0 i| : x ∈ BX }, where BX = { x ∈ X : kxk ≤ 1 }. By 1.1.4 X 0 is complete. Let F be a subspace of X 0 . The weakest topology (see 1.1.1) on X in which all functions x 7→ hx, x0 i, x0 ∈ F , are continuous is called the hX, F i-topology on X. If F = X 0 it is called the weak topology on X. Clearly, any x ∈ X induces a functional on X 0 , i.e., an element of X 00 (see also 1.1.9 below). The hX 0 , Xi-topology on X 0 is called the ∗-weak topology. Proposition ([DS, ch. 5, §4, theorem 2], [Rud2 , theorem 4.3]). The closed unit ball BX 0 is compact in the ∗-weak topology. 1.1.8. The Hanh–Banach theorem. Theorem ([Rud2 , theorems 3.3 and 3.7]). (a) Let X0 be a subspace of a Banach space X, and let f0 : X0 → C be a bounded linear functional. Then there exists a bounded linear functional f : X → C such that f coincides with f0 on X0 and kf k = kf0 k. (b) Let B be a convex balanced (i.e., αx ∈ B for all x ∈ B and α ∈ C, |α| ≤ 1) closed subset of a Banach space X and x0 ∈ / B. Then there exists a functional f ∈ X 0 such that hx0 , f i > 1,

and

|hx, f i| ≤ 1

for all x ∈ B.

1.1.9. The second conjugate space. The second conjugate space of a Banach space X is the conjugate of X 0 , i.e., the space of all bounded linear functionals x00 : X 0 → C. We denote it by X 00 . We denote by x00 (x0 ), or by hx0 , x00 i, or (mainly) by hx00 , x0 i the value of x00 ∈ X 00 at x0 ∈ X 0 . For any x ∈ X we denote by κx the functional x0 7→ hx, x0 i on X 0 . Thus we obtain the operator κ : X → X 00 . We call it the natural embedding. Clearly, kκk ≤ 1. From the Hanh–Banach theorem it follows that for any x ∈ X there exists x0 ∈ X 0 , kx0 k = 1, such that hx, x0 i = kxk. Hence kκxk = kxk for all x ∈ X, i.e., the embedding κ is isometric. Therefore one can identify X with the closed subspace κX of X 00 . If Im κ = X 00 the space X is called reflexive. Proposition ([DS, ch. 5, §4, theorem 5], [Rud2 , theorem 3.15]). The set κBX is ∗-weak dense in BX 00 (see 1.1.7 for the definition of the ∗-weak topology).

1.2. DIAGRAMS

7

1.1.10. The conjugate operator. The conjugate operator of A ∈ B(X, Y ) is an operator A0 : Y 0 → X 0 possessing the property hAx, gi = hx, A0 gi

for all x ∈ X and g ∈ Y 0 .

For any A ∈ B(X, Y ) the conjugate operator exists and is unique. From the Hanh–Banach theorem it follows that kA0 k = kAk. An operator A ∈ B(X, Y ) is called the pre-conjugate operator of B ∈ B(Y 0 , X 0 ) if B = A0 . Not every operator B ∈ B(Y 0 , X 0 ) has a pre-conjugate operator. Proposition. Let F be a subspace of X 0 . An operator A ∈ B(X) is continuous with respect to the hX, F i-topology on X if and only if the subspace F is invariant under the operator A0 : X 0 → X 0 . Proof. The proof follows directly from the definition of the hX, F i-topology, see 1.1.7. ¤ 1.1.11. Compact operators. An operator K ∈ B(X, Y ) is called compact if the set KBX = { Kx : x ∈ BX } is conditionally compact. Here BX is the unit ball. We denote the set of all compact operators K ∈ B(X, Y ) by K = K(X, Y ). Proposition ([DS, ch. 6, §5], [Kir, ch. 3, §2, 2, theorem 15], [Rud2 , theorems 4.18 and 4.19]). (a) The set K(X, Y ) is a closed linear subspace of B(X, Y ). (b) An operator K ∈ B(X, Y ) is compact if and only if its conjugate K 0 is compact. (c) If K ∈ K and A, B ∈ B then AK, KB ∈ K.

1.2. Diagrams We use diagrams as a language for the discussion of operators on subspaces and quotient spaces. For example, they allow us to explain in a rigorous way what we mean in saying that we do not distinguish some naturally isomorphic spaces. We consider only the simplest properties of diagrams and do not immerse ourselves deeply in category theory. We restrict ourselves to the category of linear spaces and linear operators, and the category of Banach spaces and bounded linear operators. 1.2.1. Commutative diagrams. By a diagram we mean a scheme consisting of linear (or Banach) spaces, which are denoted by letters, and linear (respectively, linear bounded) operators, which are denoted by arrows marked by letters. Diagram is a convenient tool for describing in a simple graphic way of a family of operators and connections between their domains and ranges. Because of their special form, some diagrams are called squares, sequences, etc.. The meaning of these terms will be clear from the context. A diagram is called commutative if compositions of operators along any pair of paths with the same origin and the same end do not depend on the path. For

8

I. FUNCTIONAL ANALYSIS PRELIMINARIES

example, let us consider the diagram A

B

1 1 X1 −−−− → Y1 −−−− → Z1       TX y TY y TZ y

A

(1)

B

2 2 X2 −−−− → Y2 −−−− → Z2

consisting of linear spaces X1 , Y1 , Z1 , X2 , Y2 , Z2 and linear operators A1 , B1 , A2 , B2 . It has two paths from X1 to Y2 ; they are X1 − → Y1 − → Y2 and X1 − → X2 − → Y2 . The commutativity of the diagram implies that TY A1 = A2 TX . To verify that the whole diagram (1) is commutative it suffices to show that both of its squares are commutative, i.e., TY A1 = A2 TX and TZ B1 = B2 TY . Indeed, the equalities TZ B1 A1 = B2 TY A1 = B2 A2 TX (associated with the paths from X1 to Z2 ) follow from them. We say that the rows of the diagram (1) are isomorphic if there exist invertible operators TX , TY , and TZ such that the whole diagram is commutative. If the spaces are Banach and the operators TX , TY , and TZ are bounded we say that the isomorphism is topological. If, in addition, TX , TY , and TZ are isometric we say that the isomorphism is isometric. 1.2.2. Operators on subspaces and quotient spaces. Let X be a linear (Banach) space, and X0 be a (closed) subspace of it, and X/X0 be the quotient space, see 1.1.3. We usually denote by j : X0 → X and q : X → X/X0 the natural embedding and the natural projection, respectively. Let X and Y be linear spaces, and X0 and Y0 be subspaces of them, respectively, and A : X → Y be a linear operator. If there exists an operator A0 : X0 → Y0 such that the diagram jX X0 −−−−→ X     A0 y Ay jY

Y0 −−−−→ Y is commutative we call A0 the restriction of the operator A. If there exists an e : X/X0 → Y /Y0 such that the diagram operator A qX

X −−−−→ X/X0     e Ay Ay

(2)

qY

Y −−−−→ Y /Y0 e the quotient operator induced by A. is commutative we call A Proposition. Let X and Y be linear spaces, and X0 and Y0 be subspaces of them, respectively, and A : X → Y be a linear operator. (a) If the restriction A0 exists then A0 is unique. e exists then A e is unique. (b) If the quotient operator A e exist if AX0 ⊆ Y0 . (c) The operators A0 and A

9

1.2. DIAGRAMS

Assume, in addition, that X and Y are Banach spaces, and X0 and Y0 are closed e ≤ kAk. subspaces, and A ∈ B(X, Y ). Then kA0 k ≤ kAk and kAk Proof. (a) and (c) are evident. e is unique because in (2) Im qX = X/X0 . (b) is almost evident. The operator A e ≤ kAk. Let us assume Clearly, kA0 k ≤ kAk. We prove the inequality kAk that x e ∈ X/X0 . We take an arbitrary ε > 0 and choose x ∈ x e such that kxk ≤ e (1 + ε)ke xk. Then kAe xk = kqY Axk ≤ kqY k · kAk · kxk ≤ 1 · kAk(1 + ε)ke xk. The e arbitrariness of ε > 0 and x e shows that kAk ≤ kAk. ¤ We shall use this proposition in various situations. The extreme cases are the e : X/ Ker A → Y . In all cases we shall call A0 a following: A0 : X → Im A and A e a quotient operator. restriction and A A

B

1.2.3. Short exact sequences. A diagram X −→ Y −→ Z is called exact at Y if Im A = Ker B. A more complicated diagram is exact if it is exact at all its terms. An important example of an exact diagram is the canonical sequence j

0

q

0

0− → X0 − →X− → X/X0 − → 0.

(3)

Here and below zeros denote zero spaces and operators. An exact diagram of the form 0

A

B

0

0− → X −→ Y −→ Z − →0

(4)

is called a short exact sequence. Proposition. Any short exact sequence (4) is isomorphic to a diagram of the form (3). The isomorphism is topological if the spaces are Banach and the operators are bounded. Proof. We consider the Banach case; the linear case is handled in a similar way with evident simplifications. The exactness at X means that Ker A = {0}. Thus the restriction A0 : X → Im A is bijective. The exactness at Y implies that Im A is closed. Therefore by the Banach theorem A0 : X → Im A has a bounded inverse. e : Y / Ker B → Z. Clearly, By 1.2.2(c) the operator B induces the operator B e = Z and Ker B e = {0}, so that B e is invertible. Thus B e is a topological Im B isomorphism from Y / Ker B to Z. We represent the proved facts by the commutative diagram j

0

0 −−−−→ Im A −−−−→ x  A0  0

0 −−−−→

X

A

Y ° ° ° B

−−−−→ Y −−−−→ ° ° ° q

Z   −1 e y B

0

−−−−→ 0

0

Y −−−−→ Y / Ker B −−−−→ 0,

10

I. FUNCTIONAL ANALYSIS PRELIMINARIES

and then rewrite it as 0

0 −−−−→

X   A0 y

0

A

B

j

q

−−−−→ Y −−−−→ ° ° °

0

Z   e −1 y B

−−−−→ 0

0

0 −−−−→ Im A −−−−→ Y −−−−→ Y / Ker B −−−−→ 0. The commutativity of the latter diagram means that diagram (4) is topologically isomorphic to the diagram (3). ¤ 1.2.4. The continuation of a diagram Proposition. In the diagram J

Q

J

Q1

0

X −−−−→ Y −−−−→ Z −−−−→ 0       Ay By Cy 0

0 −−−−→ X1 −−−1−→ Y1 −−−−→ Z1 let the spaces X, Y , Z, X1 , Y1 , and Z1 be linear (Banach) and the linear (bounded) operators J, Q, J1 , Q1 , and B be given. (a) Assume the diagram is exact at Y , Y1 , and X1 . If there exists a linear (bounded) operator C : Z → Z1 such that the right square is commutative then there exists a unique linear (bounded) operator A : X → X1 such that the left square is commutative. (b) Assume the diagram is exact at Y , Y1 , and Z. If there exists a linear (bounded) operator A : X → X1 such that the left square is commutative then there exists a unique linear (bounded) operator C : Z → Z1 such that the right square is commutative. Proof. We restrict ourselves to the consideration of the Banach case. (a) The commutativity of the right square means that Q1 B = CQ; the exactness at Y implies that QJ = 0; finally, the exactness at Y1 means that Ker Q1 = Im J1 . Consequently Q1 BJ = CQJ = 0. Hence Im BJ ⊆ Ker Q1 = Im J1 . We consider the restriction J01 : X1 → Im J1 of the operator J1 . The exactness at X1 implies that Ker J1 = {0}, and the exactness at Y1 implies that Im J1 is closed. Consequently by the Banach theorem the inverse of the operator J01 is bounded. −1 As we have seen above, Im BJ ⊆ Im J1 . Therefore we can set A = J01 BJ. (b) The commutativity of the left square and the exactness at Y and Y1 imply B(Ker Q) = B(Im J) = Im BJ = Im J1 A ⊆ Im J1 = Ker Q1 ; e which makes the thus B(Ker Q) ⊆ Ker Q1 . By 1.2.2(c) there exists the operator B left square of the diagram q

e Q

q1

e1 Q

0

Y −−−−→ Y / Ker Q −−−−→ Z −−−−→ 0     e By By Y1 −−−−→ Y1 / Ker Q1 −−−−→ Z1 e is an isomorphism. Thus we can set C = Q e1 B eQ e −1 . commutative. Here Q

¤

11

1.2. DIAGRAMS

1.2.5. The existence of a restriction and a quotient operator Corollary. Let X and Y be linear spaces, and let X0 and Y0 be subspaces of them, respectively. Then the restriction A0 : X0 → Y0 of a linear operator e : X/X0 → Y /Y0 exists. A : X → Y exists if and only if the quotient operator A Proof. The proof follows from 1.2.4.

¤

1.2.6. The conjugate sequence. Below in this section we assume that all diagrams consist of Banach spaces and linear bounded operators. Proposition ([Kir, ch. 3, §2, 3, theorem 17]). Assume the diagram A

B

C

X −→ Y −→ Z −→ is exact at Y and Z. Then the conjugate diagram A0

B0

C0

X 0 ←− Y 0 ←− Z 0 ←− is exact at Y 0 ; here X 0 , Y 0 , Z 0 ; A0 , B 0 , and C 0 are conjugate spaces and operators, respectively; see 1.1.7 and 1.1.10 for the definitions. Proof. Since Im A = Ker B it follows that BA = 0. Consequently A0 B 0 = 0. Therefore Im B 0 ⊆ Ker A0 . Thus it suffices to show that Im B 0 ⊇ Ker A0 . Let g ∈ Ker A0 . Then hx, A0 gi = 0 for all x ∈ X or, equivalently, hAx, gi = 0 for all x ∈ X. This means that g is equal to zero on Im A = Ker B. Therefore by 1.2.2(c) g induces the functional ge ∈ (Y / Ker B)0 . On the other hand, by the exactness at Z the image of B is closed. Therefore by e : Y / Ker B → Im B is bounded. the Banach theorem the inverse of the operator B e is defined according to 1.2.2(c).) We define h : Im B → C by the rule (Here B e −1 . Let H : Z → C be a bounded continuation of h to Z (see the Hanh– h = geB Banach theorem). We represent the situation by the diagram e B

q

j

Y −−−−→ Y / Ker B −−−−→ Im B −−−−→ Z         gy g ey Hy hy C C C C. e We note that j Bq e = B. Hence g = HB. Then by the definition Thus g = Hj Bq. 0 of the conjugate operator, g = B H. So that g ∈ Im B 0 . ¤ 1.2.7. The exactness of the conjugate of a short exact sequence Corollary. (a) The diagram 0

B0

A0

0

−0 0← − X 0 ←− Y 0 ←− Z 0 ← conjugate of the short exact sequence (4) is exact. (b) The diagram 0

j0

q0

0

−0 0← − (X0 )0 ←− X 0 ←− (X/X0 )0 ← conjugate of the canonical sequence (3) is exact. Proof. The proof follows immediately from 1.2.6.

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12

I. FUNCTIONAL ANALYSIS PRELIMINARIES

1.2.8. The isomorphism of the simplest rows Lemma. Assume the diagram 0

J

0

J

0 −−−−→ X −−−−→ Y ° ° ° 0 −−−−→ X1 −−−1−→ Y is exact at X and X1 ; and the images of J and J1 are the same closed subspace Y0 of Y . Then the rows are (topologically) isomorphic. This isomorphism is unique. Proof. We complete the preceding diagram to make the diagram 0

J

q

0

0

J

q

0

0 −−−−→ X −−−−→ Y −−−−→ Y /Y0 −−−−→ 0 ° ° ° ° ° ° 0 −−−−→ X1 −−−1−→ Y −−−−→ Y /Y0 −−−−→ 0 with exact rows. By 1.2.4 there exists an isomorphism between X and X1 .

¤

1.2.9. The conjugate of a subspace and a quotient space. Let X be a Banach space and X0 be a closed subspace of it. The subspace X0⊥ of X 0 consisting of all f ∈ X 0 which are equal to zero on X0 , is called the annihilator of X0 . Theorem. Let X be a Banach space, and let X0 be a closed subspace of it. Then the exact rows of the diagram 0

0 ←−−−− (X0 )0

0

j0

q0

q1

j1

0

←−−−− X 0 ←−−−− (X/X0 )0 ←−−−− 0 ° ° °

0 ←−−−− X 0 /X0⊥ ←−−−− X 0 ←−−−−

X0⊥

0

←−−−− 0

are isomorphic. These isomorphisms are unique. Consequently (X0 )0 ' X 0 /X0⊥ ,

(X/X0 )0 ' X0⊥ ,

(X0 )00 ' X0⊥⊥ ,

(X/X0 )00 ' X 00 /X0⊥⊥ .

Remark. (a) In 1.3.9 we shall establish that these isomorphisms are isometric. (b) This theorem states not only that the isomorphisms (X0 )0 ' X 0 /X0⊥ and (X/X0 )0 ' X0⊥ exist. It also claims that these isomorphisms are natural. Proof. The exactness of the rows has been discussed above. By 1.2.8 (and the exactness at X 0 ) it suffices to prove that Ker j 0 = X0⊥ . Indeed, for any f ∈ X 0 we have f ∈ Ker j 0 ⇐⇒ j 0 f = 0 ⇐⇒ ∀x0 ∈ X0 hx0 , j 0 f i = 0 ⇐⇒ ∀x0 ∈ X0 hjx0 , f i = 0 ⇐⇒ f ∈ (Im j)⊥ = X0⊥ . The first and the second formulae follow from the diagram. The others follow from the equalities (X0 )00 ' (X 0 /X0⊥ )0 ' X0⊥⊥ and (X/X0 )00 ' (X0⊥ )0 ' X 00 /X0⊥⊥ . ¤

13

1.2. DIAGRAMS

1.2.10. The closedness of the image of the conjugate operator. Assume the image of A ∈ B(X, Y ) is closed. Then, by definition, the sequence j

0

q

A

0

0− → Ker A − → X −→ Y − → Y / Im A − →0 is exact. Proposition. Assume the image of an operator A ∈ B(X, Y ) is closed. Then the image of A0 is closed, too. Remark. The converse statement will be proved in 1.3.10. Proof. The proof follows from the exactness of the conjugate sequence j0

0

q0

A

0

0← − (Ker A)0 ←− X 0 ←− Y 0 ←− (Y / Im A)0 ← −0 at X 0 (see 1.2.6).

¤

1.2.11. The kernel and the image of the conjugate operator Theorem. Assume the image of an operator A ∈ B(X, Y ) is closed. Then the exact rows of the diagram 0

0 ←−−−−

(Ker A)0

0

j0

A0

Y 0 ←−−−− (Y / Im A)0 ←−−−− 0 ° ° °

q1

A0

Y 0 ←−−−− (Im A)⊥ ° ° °

q2

A0

←−−−− X 0 ←−−−− ° ° °

0 ←−−−− X 0 /(Ker A)⊥ ←−−−− X 0 ←−−−− ° ° ° 0

0 ←−−−−

X 0 / Im A0

q0

0

j1

j2

←−−−− X 0 ←−−−− Y 0 ←−−−−

Ker A0

0

←−−−− 0

0

←−−−− 0

are isomorphic. This isomorphism is unique. Consequently Im A0 = (Ker A)⊥ ' (X/ Ker A)0 , (Ker A)0 ' X 0 / Im A0 = X 0 /(Ker A)⊥ , Im A00 = (Im A)⊥⊥ ' (Im A)00 ,

Ker A0 = (Im A)⊥ ' (Y / Im A)0 , (Im A)0 ' Y 0 / Ker A0 = Y 0 /(Im A)⊥ , Ker A00 = (Ker A)⊥⊥ ' (Ker A)00 ,

(Y / Im A)00 ' Y 00 / Im A00 = Y 00 /(Im A)⊥⊥ , (X/ Ker A)00 ' X 00 / Ker A00 = X 00 /(Ker A)⊥⊥ . Remark. In 1.3.9 we shall establish that these isomorphisms are isometric. Proof. The third row is exact by definition. The first row is exact by 1.2.10. The first and the second rows are isomorphic by 1.2.9. A reference to 1.2.4 completes the consideration of the diagram. The second and the third formulae follow from the diagram. The first and the fourth formulae follow from them and 1.2.9. The other formulae follow from the proved formulae and 1.2.9. ¤

14

I. FUNCTIONAL ANALYSIS PRELIMINARIES

1.3. Lower norms Lower norms are convenient tools for the investigation of invertibility. They allow one to replace some reasonings by calculations. Roughly speaking, the two lower norms are associated with two kinds of one-sided invertibility. Sometimes it is more convenient to study separately left and right invertibility instead of two-sided invertibility at once. 1.3.1. The definition of the lower norms. Let X and Y be Banach spaces, and let A ∈ B(X, Y ). We set |A|+ = |A : X → Y |+ = inf{ kAxk : kxk = 1 }, |A|− = |A : X → Y |− = sup{ l ≥ 0 : lBY ⊆ ABX }, where BX = { x ∈ X : kxk ≤ 1 } and BY = { y ∈ Y : kyk ≤ 1 } are closed unit balls in X and Y , respectively. Clearly, |A|+ is the best constant in the estimate kAxk ≥ |A|+ kxk. Thus, evidently, |A|+ = sup{ l ≥ 0 : kAxk ≥ lkxk for all x ∈ X }. Example. (a) Let j : X0 → X be the natural embedding. Then, evidently, |j|+ = 1. (b) Let q : X → X/X0 be the natural projection. It is easy to see that |q|− = 1. (c) Let κ : X → X 00 be the natural embedding described in 1.1.9. Clearly, |κ|+ = 1. We call |A|+ and |A|− the lower norms of A. In the literature (see, e.g., [GiT], [Wei2 ]) sometimes they are also called the minimum modulus and the surjection modulus of A, respectively. 1.3.2. The main properties of the lower norms Theorem. Let A ∈ B(X, Y ). (a) |A|+ > 0 if and only if Ker A = {0} and Im A is closed. If Ker A = {0} and Im A is closed then 1 |A|+ = , k(A0 )−1 k where A0 : X → Im A is the restriction. (b) |A|− > 0 if and only if Im A = Y . If Im A = Y then |A|− =

1 , e−1 k kA

e : X/ Ker A → Y is the quotient operator. where A (c) The operator A is invertible if and only if |A|+ > 0 and |A|− > 0. If A is invertible then 1 |A|+ = |A|− = . kA−1 k

1.3. LOWER NORMS

15

Proof. (a) Let |A|+ > 0. Then, evidently, Ker A = {0}. We show that Im take an absolutely convergent series P∞A is closed. We make use of 1.1.2(c). We P ∞ −1 n=1 yn , yn ∈ Im A, and consider the series P∞ n=1 xn , where xn = (A0 ) yn . By the estimate |A|+ kxn k ≤ kyn k, P the series n=1 xn is absolutely convergent; we ∞ denote its limit by x∗ . Clearly, n=1 yn converges to Ax∗ ∈ Im A. Thus Im A is closed. Conversely, assume Ker A = {0} and Im A is closed. Then by the Banach theorem the operator A0 : X → Im A has bounded inverse. Clearly, ½ ¾ kAxk |A|+ = inf : x ∈ X, x 6= 0 kxk ½ ¾ kyk = inf : y ∈ Im A, y 6= 0 k(A0 )−1 yk = 1/k(A0 )−1 k > 0. (b) Assume |A|− > 0. Then by the definition of | · |− we have Im A = Y . e : X/ Ker A → Y is invertConversely, assume Im A = Y . Then the operator A e −1 k ≤ |A|− . We take l > 0 such that l < 1/kA e−1 k. ible. First, we show that 1/k(A) e−1 y. Clearly, we have Assume y ∈ lBY . We set x e=A e−1 k · kyk < 1 · l = 1. ke x k ≤ kA l We choose x ∈ x e such that kxk < 1. Evidently, Ax = y. Consequently y ∈ ABX . e−1 k by the choice of l. Thus lBY ⊆ ABX , i.e., |A|− ≥ l. Hence |A|− ≥ 1/kA In particular, we have |A|− > 0. e −1 k. Assume 0 < l < |A|− , i.e., lBY ⊆ ABX . Now we show that |A|− ≤ 1/k(A) Then for a given y ∈ BY there exists x ∈ X such that kxk ≤ 1/l and Ax = y. ex = y Let x e ∈ X/ Ker A be the equivalence class of x. Clearly, ke xk ≤ 1/l and Ae e−1 y = x e−1 k = sup{ kA e−1 yk : y ∈ BY } ≤ 1/l or or, equivalently, A e. Thus kA e−1 k ≥ l. Since l ∈ (0, |A|− ) is arbitrary we arrive at 1/kA e−1 k ≥ |A|− . 1/kA (c) follows from (a) and (b). ¤ 1.3.3. An equivalent definition of | · |− Lemma ([Pie, B.3.5], see also [Rud2 , lemma 4.13(a)]). For any operator A ∈ B(X, Y ) one has |A|− = sup{ l ≥ 0 : lBY ⊆ ABX }, where ABX is the closure of ABX . Proof. Clearly, it suffices to prove that sup{ l ≥ 0 : lBY ⊆ ABX } ≤ sup{ l ≥ 0 : lBY ⊆ ABX } provided that the left hand side is not zero. Without loss of generality we may assume that the left hand side is equal to 1, i.e., BY ⊆ ABX . We take an arbitrary l < 1 and show that lBY ⊆ ABX .

16

I. FUNCTIONAL ANALYSIS PRELIMINARIES

We choose q > 0 such that l/(1 − q) ≤ 1. Since ABX is dense in BY , for any y ∈ BY there exists x ∈ BX such that ky − Axk ≤ q. Consequently for any y ∈ Y there exists x ∈ X such that kxk ≤ kyk

and

ky − Axk ≤ qkyk;

we denote such an x by x(y). We fix y∗ ∈ lBY and consider the sequence y0 = y∗ ,

yk+1 = yk − Ax(yk ).

Clearly, kx(yk )k ≤ kyk k = kyk−1 − Ax(yk−1 )k ≤ qkyk−1 k ≤ q k l. P∞ We set x∗ = k=0 x(yk ). It is easy to see that kx∗ k ≤ l/(1 − q) and Ax∗ = y∗ . Since l/(1−q) ≤ 1 we have y∗ ∈ ABX . Thus we have proved that lBY ⊆ ABX . ¤ 1.3.4. The lower norms of the conjugate operator Proposition ([Pie, B.3.8], see also [Rud2 , lemma 4.13(b)]). For any operator A ∈ B(X, Y ) one has |A0 |− = |A|+

and

|A0 |+ = |A|− .

Example. (a) |j 0 |− = 1 and |q 0 |+ = 1 because (see the example in 1.3.1) |j|+ = 1 and |q|− = 1. (b) Let A ∈ B(X, Y ), and let X0 ⊆ Ker A be a closed subspace. We claim that |A : X/X0 → Y |− = |A : X → Y |− . Indeed, both operators obviously have the same image. If the images are not equal to Y , both lower norms are zero. Suppose the images coincide with Y . Then, in particular, Im A is closed. By 1.2.10 Im A0 is closed, too. By 1.2.11 X0⊥ ⊇ Im A0 . Therefore |A : X/X0 → Y |− = |A0 : Y 0 → X0⊥ |+ = |A0 : Y 0 → X 0 |+ = |A : X → Y |− . (c) Let A ∈ B(X, Y ). Assume that the operator A0 is invertible. Then by 1.3.2(c) the operator A is invertible as well.

17

1.3. LOWER NORMS

Proof of the proposition. We prove the four inequalities: |A|+ ≤ |A0 |− , and |A0 |− ≤ |A|+ , and |A0 |+ ≤ |A|− , and |A|− ≤ |A0 |+ . Without loss of generality we may assume that the left sides of these inequalities are greater than 0. (i) Assume 0 < l < |A|+ . We consider the restriction A0 : X → Im A. By 1.3.2(a) we have kA−1 0 k < 1/l. We take an arbitrary f ∈ lBX 0 and define the functional g : Im A → C by the rule hy, gi = hA−1 0 y, f i

for y ∈ Im A.

Clearly, kgk ≤ kA−1 0 k · kf k < (1/l) · l = 1. Let G : Y → C, kGk < 1, be a continuation of g according to the Hanh–Banach theorem. Then hy, Gi = hA−1 0 y, f i for all y ∈ Im A and consequently hAx, Gi = hx, f i

for all x ∈ X.

Thus A0 G = f . We recall that kGk < 1. Therefore lBX 0 ⊆ A0 BY 0 , i.e., |A|− ≥ l. Hence by the choice of l we obtain |A|+ ≤ |A0 |− . (ii) Assume 0 < l < |A0 |− . Let us take an arbitrary x ∈ X, kxk = 1. We choose f ∈ X 0 , kf k = 1, such that hx, f i = 1. From the assumption lBX 0 ⊆ A0 BY 0 it follows that lf = A0 g for some g ∈ Y 0 , kgk ≤ 1. We have kAxk ≥ |hAx, gi| = |hx, A0 gi| = |hx, lf i| = l. Thus |A|+ ≥ l. Consequently |A0 |− ≤ |A|+ . (iii) Assume |A0 |+ > l for some l > 0. We show that lBY ⊆ ABX , see 1.3.3. We assume the contrary: let there exist y0 ∈ Y , ky0 k = 1, such that ly0 ∈ / ABX . 0 Then by the Hanh–Banach theorem there exists g ∈ Y such that hly0 , gi > 1,

and

|hy, gi| ≤ 1 for all y ∈ ABX .

hly0 , gi > 1,

and

|hAx, gi| ≤ 1

and

|hx, A0 gi ≤ 1 for all x ∈ BX ,

Hence for all x ∈ BX ,

or, equivalently, hly0 , gi > 1, which implies that lkgk > 1

and

kA0 gk ≤ 1.

But the last formulation means that |A0 |+ < l. This is a contradiction. Thus lBY ⊆ ABX , and therefore |A|− ≥ l. Consequently |A0 |+ ≤ |A|− . (iv) Assume |A|− > l, i.e., lBY ⊆ ABX . For any g ∈ Y 0 we have kA0 gk = sup{ |hx, A0 gi| : x ∈ BX } = sup{ |hAx, gi| : x ∈ BX } = sup{ |hy, gi| : y ∈ ABX } ≥ sup{ |hy, gi| : y ∈ lBY } = lkgk. Thus |A0 |+ ≥ l. And consequently |A|− ≤ |A0 |+ .

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18

I. FUNCTIONAL ANALYSIS PRELIMINARIES

1.3.5. The continuity of the lower norms Proposition ([Pie, B.3.11], see also [GiT, lemma 2.2]). For all A, B ∈ B(X, Y ) one has ¯ ¯ ¯ ¯ ¯|A|+ − |B|+ ¯ ≤ kA − Bk ¯|A|− − |B|− ¯ ≤ kA − Bk. and Proof. We have kAxk ≥ kBxk − kAx − Bxk ≥ |B|+ · kxk − kA − Bk · kxk for all x ∈ X. Thus |A|+ ≥ |B|+ − kA − Bk or |B|+ − |A|+ ≤ kA − Bk. The proof of the inequality |A|+¯ − |B|+ ≤ kA ¯ − Bk is analogous. ¯ The inequality |A|− − |B|− ¯ ≤ kA − Bk follows from the proved inequality and 1.3.4. ¤ 1.3.6. The lower norms as bounds of invertibility. The property of an operator to be invertible is preserved under small perturbations. Lower norms allow one to measure the magnitude of the perturbations which do not spoil the invertibility. Corollary. Let A, B ∈ B(X, Y ). (a) If |A|+ > 0 and kA − Bk < |A|+ then |B|+ > 0, too. (b) If |A|− > 0 and kA − Bk < |A|− then |B|− > 0, too. (c) If A is invertible and kA − Bk · kA−1 k < 1 then B is also invertible. Proof. The proof follows from 1.3.5 and 1.3.2.

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1.3.7. Multiplicative properties of the lower norms Proposition. For all B ∈ B(X, Y ) and A ∈ B(Y, Z) one has |A|+ · |B|+ ≤ |AB|+ ≤ kAk · |B|+ ,

|A|+ · kBk ≤ kABk,

|A|− · |B|− ≤ |AB|− ≤ |A|− · kBk,

kAk · |B|− ≤ kABk.

Proof. The first line inequalities follow immediately from the definition of |·|+ . The second line inequalities can be deduced from them and 1.3.4. ¤ 1.3.8. Isometric injections and isometric surjections. We call an operator J ∈ B(X, Y ) an isometric injection if kJk = 1 and |J|+ = 1. By 1.3.2(a) this means that J0 : X → Im J is an isometric isomorphism. We call an operator Q ∈ (Y, Z) an isometric surjection if kQk = 1 and |Q|− = 1. e : Y / Ker Q → Z is an isometric isomorphism. By 1.3.2(b) this means that Q Proposition. J

J

1 (a) Assume that in the diagram X − → Y ←− X1 the operators J and J1 are isometric injections with the same images. Then X and X1 are isometrically isomorphic. Q Q1 (b) Assume that in the diagram Z ←− Y −−→ Z1 the operators Q and Q1 are isometric surjections with the same kernels. Then Z and Z1 are isometrically isomorphic.

1.3. LOWER NORMS

19

Proof. (a) By 1.3.2(a) both X and X1 are isometrically isomorphic to Im J = Im J1 . (b) By 1.3.2(b) both Z and Z1 are isometrically isomorphic to Y / Ker Z = Y / Ker Z1 . ¤ Example (cf. the examples in 1.3.1 and 1.3.4). It is easy to see that the natural embeddings j : X0 → X and κ : X → X 00 are isometric injections, and the natural projection q : X → X/X0 is an isometric surjection. 1.3.9. Isometric isomophisms of diagrams Corollary. The isomorphisms in theorems 1.2.9 and 1.2.11 are isometric. Proof. The proof follows from 1.3.8, the example in 1.3.8, and 1.3.4. ¤ 1.3.10. The closedness of the image of the pre-conjugate operator Theorem ([Rud2 , theorem 4.14]). The image of an operator A ∈ B(X, Y ) is closed if and only if the image of its conjugate A0 is closed. Proof. By virtue of 1.2.10 it is sufficient to prove that Im A is closed provided that Im A0 is closed. So let Im A0 be closed. We show that Im A is closed, too. We denote by Z ⊆ Y the closure of the image of the operator A and consider the restriction A0 : X → Z of A. Clearly, Im A0 = Im A. Let us consider the diagram X X     A0 y Ay j

Z −−−−→ Y and its conjugate

X0 x  A0 

X0 x  (A0 )0  q

Y 0 /Z ⊥ ←−−−− Y 0 (see 1.2.9 for the isomorphism Z 0 = Y 0 /Z ⊥ ). Since the diagram is commutative Im(A0 )0 = Im A0 . In particular, it follows that Im(A0 )0 is closed. We show that Ker(A0 )0 = {0}. Assume (A0 )0 h = 0 for some h ∈ Z 0 . Then hA0 x, hi = hx, (A0 )0 hi = 0

for all x ∈ X.

Hence hy, hi = 0 for all y ∈ Im A. So h is equal to zero on a dense subspace Im A of Z. Therefore h = 0. We have proved that Ker(A0 )0 = {0} and Im(A0 )0 is closed. Applying 1.3.2 we obtain that |(A0 )0 |+ > 0. Then from 1.3.4 it follows |A0 |− > 0. Applying 1.3.2 once more we obtain Im A0 = Z. Consequently Im A is closed. ¤ 1.3.11. Fredholm operators. An operator A ∈ B(X, Y ) is called Fredholm if its image is closed and the spaces Ker A and Coker A = Y / Im A are finitedimensional.

20

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Proposition. An operator A ∈ B(X, Y ) is a Fredholm operator if and only if A is a Fredholm operator. Moreover, 0

dim Ker A0 = codim Im A

codim Im A0 = dim Ker A.

and

Proof. The proof follows from 1.3.10 and 1.2.11.

¤

1.3.12. The index of a Fredholm operator. Let A ∈ B(X, Y ) be a Fredholm operator. The number ind A = dim Ker A − codim Im A is called the index of A. Proposition ([Kir, ch. 3, §2, 3, theorem 22], [Rud2 , theorem 4.25]). Let K ∈ B(X, Y ) be a compact operator (see 1.1.11 for the definition). Then the operator 1 + K is Fredholm and ind(1 + K) = 0.

1.4. Banach algebras 1.4.1. Algebras. A set A is called an algebra if: there are defined three operations — αA (the multiplication by scalars), A + B (the addition) and AB (the multiplication or composition) — for all A, B ∈ A and α ∈ C; A is a linear space with respect to the operations of multiplication by scalars and addition; and A(BC) = (AB)C, (A + B)C = AC + BC,

A(B + C) = AB + AC,

α(AB) = (αA)B = A(αB) for all A, B, C ∈ A and α ∈ C. An element 1 = 1A ∈ A is called a unit if A1 = 1A = A

for all A ∈ A.

If A has (no) unit, it is called an algebra with(out) a unit. An algebra with a unit is also called unital. It is easy to show (see, e.g., [Rud2 , 10.1]) that the unit is unique provided it exists. In the sequel, unless otherwise specified, we shall usually assume that all algebras under consideration have a unit. We denote the zero element of A by the symbols 0 and 0A . If, additionally, A is a normed linear space and k1A k = 1, kABk ≤ kAk · kBk

for all A, B ∈ A,

A is called a normed algebra. If A has no unit the first condition is omitted. A normed algebra is called a Banach algebra if it is complete as a normed space. The completion of a normed algebra (interpreted as a normed space) can be considered naturally as a Banach algebra.

1.4. BANACH ALGEBRAS

21

e the algebra consisting of all pairs (α, A) Let A be an algebra. We denote by A and α ∈ C, A ∈ A, with the operations (α, A) + (β, B) = (α + β, A + B), λ(α, A) = (λα, A), (α, A)(β, B) = (αβ, αB + βA + AB). If A is normed we set k(α, A)k = |α| + kAk. e and A e is actually a normed algebra It is easy to verify that (1, 0) is the unit of A; e provided that so is A. Thus any algebra A can be embedded in the algebra A e is called the algebra with an adjoint unit associated with a unit. The algebra A with A. We note that in this construction A may have a unit itself. Example. (a) The simplest examples of Banach algebras are the zero algebra {0} and the field C. Note that the zero algebra is a special case. According to our definition it has a unit (it is 0), and its unique element 0 is invertible and has an empty spectrum (see below for the definition). Unfortunately, it cannot satisfy the assumption k0k = 1 in the definition of a normed algebra. Nevertheless, sometimes it is convenient to consider this algebra as a Banach algebra with k0k = 0. (b) The most important example of a Banach algebra is the algebra B = B(X) of all bounded linear operators acting on a Banach space X. (c) Let T be a locally compact topological space. We consider the Banach space C = C(T ) of all bounded continuous functions x : T → C with the norm kxk = supt∈T |x(t)|. Clearly, C is a Banach algebra with respect to pointwise multiplication. This is an algebra with a unit. (d) Let T be a locally compact non-compact topological space. Let C0 = C0 (T ) denote the subspace of C consisting of all functions x vanishing at infinity, i.e., satisfying the following condition: for any ε > 0 there exists a compact subset K ⊆ T such that |x(t)| < ε for all t ∈ / K. Clearly, C0 is a Banach algebra without f0 with an adjoint unit associated with C0 . a unit. Let us consider the algebra C f0 consists of pairs (α, x). It is easy to see that We recall that, by definition, C f0 is isomorphic to the subspace Clim = Clim (T ) of C consisting of all functions C which have a limit at infinity (in a sense similar to the above). This isomorphism f0 and Clim are equivalent. Clearly, Clim is is not isomorphic, but the norms on C isometrically isomorphic to C(Te), where Te is the one point compactification (see 1.1.1 for the definition) of T . If T is compact it is natural to define C0 (T ) to be C(T ). 1.4.2. The inverse element. An element A of an algebra A with a unit is called invertible (in A) if there exists an element B ∈ A such that AB = BA = 1. The element B is called the inverse of A. We denote it by the symbol A−1 . It is easy to show (see, e.g., [Rud2 , 10.5]) that the inverse element is unique provided it exists.

22

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Proposition (cf. 1.3.6). Let A be a Banach algebra, and let A, B ∈ A. If the element A is invertible and kBk · kA−1 k < 1 then the element A − B is also invertible. Moreover, k(A − B)−1 k ≤

kA−1 k . 1 − kBk · kA−1 k

Proof. We consider the series A−1 + A−1 BA−1 + A−1 BA−1 BA−1 + . . . . By assumption this series converges absolutely. A straightforward calculation shows that the result of the multiplication of this series by A − B is the unit element. The estimate is an estimate of the sum of the series. ¤ 1.4.3. The spectrum. Let A be an algebra with a unit, and let A ∈ A. The set of all λ ∈ C such that the element λ1 − A has no inverse is called the spectrum of the element A (in A). We denote it by σ(A) or σA (A). The complement %(A) = %A (A) = C \ σ(A) is called the resolvent set of A. The function Rλ = (λ1 − A)−1 is called the resolvent of A. It is defined for λ ∈ %(A). The number R(A) = RA (A) = sup{ |λ| : λ ∈ σ(A) } is called the spectral radius of A. The element of the form λ1, λ ∈ C, is called a scalar element. Usually we denote it simply by λ. Proposition. Let A be a Banach algebra with a unit, and let A ∈ A. Then (a) σ(A) is a compact subset of C, and R(A) ≤ kAk. (b) More precisely, there exists limn R(A) = inf n

p n

p n kAn k and

kAn k = lim n

p n kAn k.

Proof. (a) Let λ ∈ %(A). By 1.4.2 all elements B in the (1/k(λ − A)−1 k)neighbourhood of λ − A are invertible. Hence %(A) is an open subset of C. By 1.4.2 the element λ − A is invertible provided kAk · |λ−1 | < 1. Therefore σ(A) is contained in the circle in C of the radius kAk centred at zero. This implies that R(A) ≤ kAk. (b) The proof uses holomorphic vector-valued functions (see below 1.4.9) and can be found in many books on functional analysis; see, e.g., [Hil, theorem 4.7.3], [Nai, ch. 2, §9, 5], [Rud2 , theorem 10.13]. ¤

23

1.4. BANACH ALGEBRAS

1.4.4. Full subalgebras. Let A be an algebra with a unit. A subset B ⊂ A is called a subalgebra if αA, A + B, AB ∈ B for all A, B ∈ B and α ∈ C; and 1A ∈ B. If A has no unit the assumption 1A ∈ B is omitted. If A has a unit, but 1A ∈ / B, we say that B is a subalgebra without a unit. Evidently, in a Banach algebra the closure of a subalgebra is a subalgebra, too. A subalgebra B of A is called full if any element B ∈ B, which is invertible in A, is also invertible in B; in other words, for any B ∈ B, if there exists B −1 ∈ A such that BB −1 = B −1 B = 1 then B −1 ∈ B. Let Q be a subset of an algebra A. The set Γ(Q) = { A ∈ A : AQ = QA for all Q ∈ Q } is called the centralizer of Q. Proposition. Let A be an algebra with a unit. (a) A subalgebra B of A is full if and only if σB (B) = σA (B)

for all B ∈ B.

(b) The centralizer Γ(Q) of any subset Q ⊆ A is a full subalgebra of A. The centralizer Γ(Q) is closed if A is Banach. Proof. (a) is evident. (b) Clearly, Γ(Q) is a closed subalgebra. We verify that Γ(Q) is full. Let A ∈ Γ(Q) be invertible. We show that A−1 ∈ Γ(Q). Multiplying the identity AQ = QA by A−1 both from the left and from the right we obtain A−1 AQA−1 = A−1 QAA−1 or QA−1 = A−1 Q, which means that A−1 ∈ Γ(Q). ¤ Example. (a) Let X be a Banach space. We denote by L(X) the algebra of all (not obligatorily bounded) linear operators A : X → X. By the Banach theorem B(X) is a full subalgebra of L(X). Next, we denote by Bc (X 0 ) the algebra of all operators A ∈ B(X 0 ) having a pre-conjugate operator. By the example (c) in 1.3.4 Bc (X 0 ) is a full subalgebra of B(X 0 ). (b) Let lp = lp (Z, C), 1 ≤ p ≤ ∞, be the space of all sequences x = { xk ∈ C : k ∈ Z } bounded by the usual norms (see 1.5.7 below). An operator A ∈ B(lp ) is called causal if for all x ∈ lp and n ∈ Z xk = 0,

k < n,

⇒

(Ax)k = 0,

k < n.

Clearly, the set B+ (lp ) of all causal operators on lp forms a subalgebra of B(lp ). We claim that this subalgebra is not full. Indeed, let us consider the shift operator (Sx)k = xk−1 . Obviously, S is causal. But the inverse operator (S −1 x)k = xk+1 is not causal. We shall discuss the algebra of causal operators in ch. 2 in detail. 1.4.5. The spectrum in a subalgebra. A component of a topological space σ is the smallest member in the family of all both closed and open subsets of σ. Proposition. Let A be a Banach algebra with a unit, and let B be a closed subalgebra of it containing the unit. (a) ([Bou4 , ch. 1, §2, 6], [Rud2 , theorem 10.18(b)]). For any B ∈ B the set σB (B) is the union of σA (B) and (possibly empty) collection of bounded components of %A (B). In particular, the boundary of σA (B) lies in σB (B). (b) For every B ∈ B one has RB (B) = RA (B).

24

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Example. We consider the operator S ∈ B+ (l∞ ) ⊂ B(l∞ ) from the example (b) in 1.4.4. Let σ and σ + denote the spectrum of S in B(l∞ ) and B+ (l∞ ), respectively. Evidently, kSk = 1 and kS −1 k = 1. By 1.4.2 these implies that σ ⊆ U, where U = { λ ∈ C : |λ| = 1 }. Clearly, the sequence x = { xk : xk = λk }, λ ∈ U, is an eigen-sequence of S with eigenvalue λ−1 , i.e., Sx = λ−1 x. Therefore σ ⊇ U as well. So σ = U. Thus %(S) consists of two components: { λ ∈ C : |λ > 1 } and { λ ∈ C : |λ| < 1 }. We have 0 ∈ σ + because S −1 is not causal. The proposition then states that the whole component { λ ∈ C : |λ| < 1 } lies in σ + . There are no other bounded components of %(S). Hence σ + (S) = { λ ∈ C : |λ| ≤ 1 }. Proof of the proposition. (a) Clearly, σA (B) ⊆ σB (B). Let Λ be a bounded component of %A (B). Suppose Λ contains points both of σB = σB (B) and %B = %B (B). Then Λ must contain a point λ0 of the boundary of σB . (We recall that σB is closed. Thus λ0 ∈ σB .) We take a sequence λn ∈ %B which converges to λ0 . The elements λn −B (including λ0 −B) are invertible in A. By 1.4.2 (λn − B)−1 converges to (λ0 − B)−1 in norm. But (λn − B)−1 belongs to B for n 6= 0. Consequently (λ0 − B)−1 also belongs to B, i.e., λ0 ∈ %B (B). This is a contradiction. It shows that Λ is contained entirely either in σB or in %B . (b) is an immediate corollary of (a). ¤ 1.4.6. Morphisms of algebras. Let A and B be algebras. We say that a mapping ϕ : A → B is a morphism (of algebras) if ϕ(A + B) = ϕ(A) + ϕ(B), ϕ(αA) = αϕ(A), ϕ(AB) = ϕ(A)ϕ(B) for all A, B ∈ A and α ∈ C. If A and B have units and, in addition, ϕ(1A ) = 1B , we say that ϕ is a morphism of algebras with a unit. If A and B are Banach algebras and ϕ is continuous, we say that ϕ is a morphism of Banach algebras. We say that ϕ is an isomorphism, and that A and B are isomorphic if ϕ is invertible. A morphism ϕ : A → B(X) is usually called a representation. Proposition. Let A and B be algebras with a unit, and let ϕ : A → B be a morphism of algebras with a unit. If A ∈ A is invertible then ϕ(A) is invertible, too. Consequently ¡ ¢ σB ϕ(A) ⊆ σA (A). Proof. Clearly, ϕ(A−1 ) is the inverse of ϕ(A). ¤ 1.4.7. Ideals and quotient algebras. Let A be an algebra. A subspace J ⊂ A is called a (two-sided ) ideal of A if AJ, JA ∈ J for all J ∈ J and A ∈ A. The closure of an ideal in a Banach algebra is an ideal, too. An ideal J is called proper if J 6= A. Clearly, a proper ideal of an algebra with a unit cannot contain invertible elements. Therefore by 1.4.2 the closure of a proper ideal in a Banach algebra is a proper ideal, too.

1.4. BANACH ALGEBRAS

25

Example. The kernel of a morphism ϕ : A → B is an ideal. Proposition. Let A be a Banach algebra, and let J be a closed two-sided ideal of it. Then the quotient space A/J is also a Banach algebra with respect to the eBk e ≤ kAk e · kBk e for all A, e B e ∈ A/J. natural multiplication. In particular, kA Proof. The proof is evident, cf. 1.1.3. ¤ Example. Let X be a Banach space. Then by 1.1.11 the set K(X) of all compact operators K ∈ B(X) is a closed two-sided ideal in B(X). It is known (see, e.g., [Pie, theorem 26.3.4]) that an operator A ∈ B(X) is Fredholm (see 1.3.11) if and only if its natural projection into the Calkin algebra (see [Cal]) B(X)/K(X) is invertible. 1.4.8. The radical. Let A be a Banach algebra with a unit. The set of all R ∈ A such that the element 1 + AR is invertible for all A ∈ A is called the radical of A. We denote it by the symbol Rad A. A non-trivial example of a radical will arise in 2.2.9. The following proposition shows that the asymmetry in this definition is apparent. Proposition. Let A be a Banach algebra with a unit. (a) For any A, R ∈ A the elements 1 + AR and 1 + RA are invertible simultaneously. Consequently σ(AR) coincides with σ(RA) except zero. (b) Rad A is a closed two-sided ideal. For all R ∈ Rad A one has σ(R) = {0}. (c) For all A ∈ A and R ∈ Rad A one has σ(A + R) = σ(A). (d) Assume a two-sided ideal R of A possesses the following property: for any R ∈ R the element 1 + R is invertible. Then R ⊆ Rad A. Proof. (a) Clearly, it suffices to prove that the invertibility of 1 + AR implies the invertibility of 1 + RA. We show that (1 + RA)−1 = 1 − R(1 + AR)−1 A. Indeed, ¡ ¢ 1 − R(1 + AR)−1 A (1 + RA) =1 − R(1 + AR)−1 A + RA − R(1 + AR)−1 ARA ¡ ¢ =1 + RA − R (1 + AR)−1 (1 + AR) A =1. In a similar way one can show that ¡ ¢ (1 + RA) 1 − R(1 + AR)−1 A = 1.

26

I. FUNCTIONAL ANALYSIS PRELIMINARIES

(b) We show that Rad A is a subspace of A. Assume R1 , R2 ∈ Rad A and A ∈ A. Then 1 + AR1 is invertible. Hence 1 + A(R1 + R2 ) can be represented as ¡ ¢ 1 + A(R1 + R2 ) = (1 + AR1 ) 1 + (1 + AR1 )−1 AR2 which is invertible by the definition of the radical. Clearly, 1 + A(αR1 ) = 1 + (αA)R1 , α ∈ C, is invertible, too. From the definition of the radical and (a) it follows immediately that AR, RA ∈ Rad A provided A ∈ A and R ∈ Rad A. Thus Rad A is an ideal. e belongs to the closure of Rad A. We show that Rad A is closed. Assume R e − Rk < 1/kAk. Then We take an A ∈ A and choose R ∈ Rad A such that kR e − R) is invertible by 1.4.2. Therefore we can represent the the element 1 + A(R e as element 1 + AR ¡ ¢ e = (1 + A(R e − R)) 1 + (1 + A(R e − R))−1 AR 1 + AR which is evidently invertible. (c) Assume A is invertible. Then by virtue of (b) the element 1 + RA−1 is also invertible. But ¡ ¢−1 A−1 (1 + RA−1 )−1 = (1 + RA−1 )A = (A + R)−1 . Hence the element A + R is invertible, too. In a similar way one can prove that the invertibility of A + R implies the invertibility of A. The substitution of λ − A and λ − (A + R) instead of A and A + R, respectively, completes the prove. (d) is evident. ¤ Example. The following example will not be used in this book, nevertheless, it is curious. Let X be a Banach space, and let K(X) be the closed two-sided ideal of all compact operators K ∈ B(X). Let F(X) denote the set of all operators F ∈ B(X) with finite-dimensional image, and let denote G(X) its closure. The operators G ∈ G(X) are called approximable. It is easy to see that F(X) and G(X) are two-sided ideals in B(X); and F(X) ⊆ K(X), and consequently G(X) ⊆ K(X). It is known (see, e.g., [Pie, the proof of theorem 10.4.6]) that there exist Banach spaces X for which G(X) 6= K(X). On the other hand, an operator A ∈ B(X) is Fredholm if and only if its natural projection into B(X)/G(X) is invertible; also an operator A ∈ B(X) is Fredholm if and only if its natural projection into B(X)/K(X) is invertible, see, e.g., [Pie, §26.3]. According to the previous proposition this implies that the natural projection of K(X) into B(X)/G(X) is included in the radical of B(X)/G(X). 1.4.9. Holomorphic vector-valued functions. We say that a function z : [a, b] → C is a smooth path if z is continuously differentiable and z 0 (t) 6= 0 for all t ∈ [a, b]. Two smooth paths z : [a, b] → C and z1 : [a1 , b1 ] → C are called equivalent if there exists a continuously differentiable function ϕ : [a1 , b1 ] → [a, b] such that ϕ(a1 ) = ϕ(a), ϕ(b1 ) = ϕ(b), ϕ0 (s) > 0, and z1 (s) = z(ϕ(s)) for all s. An equivalence class Γ of smooth paths is called a smooth curve. Any path from

1.4. BANACH ALGEBRAS

27

this equivalence class is called a parametrization of the curve. Clearly, a smooth curve possesses the natural orientation. For a continuous function f from Γ to a Banach space X we set Z Z b f (λ) dλ = f (z(t)) z 0 (t) dt. Γ

a

Clearly, the integral does not depend on the choice of the parameterization z. Remark. This definition can be generalized simply to piecewise smooth curves. The integral also makes sense for rectifiable curves, see, e.g., [Hil, §3.11]. But these generalizations are not necessary for the integration of holomorphic functions since any continuous path can be approximated by a smooth path, and, by the following theorem, the integrals over neighboring paths are equal. Therefore by paths and curves one may mean only smooth ones. A path (a curve) is called closed if z(a) = z(b). A contour is a finite family of closed curves without self-intersections forming the oriented boundary of a subset U of C. (Whilst saying that a curve has no self-intersections, nevertheless, we allow z(a) = z(b).) Let X be a Banach space, let U be an open subset of C∪{∞}, and let f : U → X be a function. The function f is called differentiable at a finite point λ0 ∈ U if it has the complex derivative f 0 (λ0 ) = lim

λ→λ0

f (λ) − f (λ0 ) . λ − λ0

The function f is called differentiable at infinity if the function z 7→ f (1/z) is differentiable at zero. The function f is called holomorphic on U if it is differentiable at all points of U . Theorem (see, e.g., [Hil, §3.11], [Nai, §4.7]). Assume that a function f takes its values in a Banach space X. (a) Let a contour Γ be the boundary of an open set U ⊆ C. Assume f is holomorphic on U and continuous on the closure of U . Then Z f (λ) dλ = 0. Γ

(b) Let a contour Γ be the oriented boundary of an open set U ⊆ C. Assume f is holomorphic on U and continuous on the closure of U . Then Z 1 f (λ) dλ = f (λ0 ) for all λ0 ∈ U . 2πi Γ λ − λ0 (c) Assume f is holomorphic on the open circle { λ : |λ − λ0 | < r } and continuous in the closed circle { λ : |λ − λ0 | ≤ r }. Then f can be represented in the open circle as the sum of the absolutely convergent series Z ∞ X 1 f (λ) k dλ. f (λ) = ck (λ − λ0 ) with ck = 2πi |λ−λ0 |=r (λ − λ0 )k+1 k=0

28

I. FUNCTIONAL ANALYSIS PRELIMINARIES

In particular, kck k ≤ max{ kf (λ)k : |λ − λ0 | = r }/rk . (d) If f is holomorphic and bounded on the whole of C then f is equal to a constant. Example. (a) Let A be a Banach algebra, and assume that T ∈ A. The function λ 7→ Rλ = (λ − T )−1 is called the resolvent of T . It is defined on the resolvent set %(T ) and takes its values in A. By 1.4.2 (cf. 1.4.3) if |λ| > R(T ) then 1 T T2 + 2 + 3 + ... . λ λ λ The following property of the resolvent is called the Hilbert identity: Rλ − Rµ Rλ Rµ = − . λ−µ The proof of this identity is plain: ¡ ¢ Rλ (λ − µ)Rµ = Rλ (λ − T ) − (µ − T ) Rµ ¡ ¢ = (λ − T )−1 (λ − T ) − (µ − T ) (µ − T )−1 Rλ =

= Rµ − Rλ . From the Hilbert identity it is evident that the resolvent is a holomorphic function. (b) We show that the spectrum of an element T of a non-zero algebra A with a unit is a non-empty set. If it were empty the resolvent Rλ = (λ − T )−1 would be holomorphic on the whole of C. On the other hand, clearly Rλ → 0 as λ → ∞. By assertion (d) of the theorem these imply that Rλ is a constant. Then λ − T = (Rλ )−1 does not depend on λ, too. This is a contradiction. (c) Assume that a function f taking its values in a Banach algebra is holomor¡ ¢−1 phic and the element f (λ) is invertible for all λ. Then the function λ 7→ f (λ) is holomorphic as well. Indeed, the expression ¡ ¢−1 ¡ ¢−1 ¡ ¢ ¡ ¢−1 f (λ0 ) − f (λ) ¡ ¢−1 f (λ) − f (λ0 ) = f (λ) f (λ0 ) λ − λ0 λ − λ0 evidently has a limit as λ → λ0 . 1.4.10. The spectral mapping theorem. Let A be a Banach algebra with a unit, and let ¡T be a fixed element of it. ¢ Let O = O σ(T ), C denote the set of all holomorphic functions f : U → C, where U is a neighbourhood of σ(T ). Of course, U = U (f ) depends of f . We regard two functions f, g ∈ O as equivalent if they coincide on a neighbourhood of σ(T ). Thus, to be precise, O consists of equivalence classes. Clearly, O is an algebra with respect to pointwise operations. There is no natural norm on O (but there is a natural topology, see, e.g., [Bou4 , ch. 1, §4]). For any f ∈ O we define the element f (T ) ∈ A by the formula Z 1 f (T ) = f (λ) (λ − T )−1 dλ, 2πi Γ where Γ is a contour in U (f ) surrounding σ(T ). (More precisely, Γ is the oriented boundary of a neighbourhood of σ(T ) being contained in U .) It is easy to show that the integral does not depend on the choice of Γ.

1.4. BANACH ALGEBRAS

29

Theorem ([Bou4 , ch. 1, §4], [Hil, §5.2–3], [Rud2 , theorems 10.27 and 10.28]). (a) The mapping ϕ : f 7→ f (T ) is a morphism from the algebra O to the algebra A. Moreover, ϕ takes the function u(λ) = 1 to the unit 1, and the function v(λ) = λ to the element T . Consequently p(T ) = a0 1A + a1 T + a2 T 2 + · · · + an T n n for a polynomial p(λ) =¡a0 + a¢1 λ +¡a2 λ2 + ¢ · · · + an λ . (b) For all f ∈ O one has σ f (T ) = f σ(T ) , or in detailed form,

¡ ¢ σ f (T ) = { f (λ) : λ ∈ σ(T ) }. 1.4.11. The Gel0 fand transform. An algebra A is called commutative if AB = BA

for all A, B ∈ A.

Let A be a commutative Banach algebra with a unit. Any morphism ζ : A → C of algebras with a unit is called a character of A. Thus ζ : A → C is a character if ζ(A + B) = ζ(A) + ζ(B), ζ(αA) = αζ(A), ζ(AB) = ζ(A)ζ(B), ζ(1A ) = 1C for all A, B ∈ A and α ∈ C. We denote the set of all characters of A by X = X(A) and call it the character space or space of characters of A. Let A be an algebra without a unit. In this case any morphism of algebras ζ : A → C is called a character of A. We denote the zero character of A by ζ∞ ; and we denote the set of all non-zero characters of A by X = X(A) and call it the e be the algebra with an adjoint unit associated with character space of A. Let A e then the restriction of ζe to A is a character A (see 1.4.1). If ζe is a character of A of A. And conversely, if ζ is a character of A then the mapping ¡ ¢ ζe (α, A) = α + ζ(A) e moreover, ζe is a unique character of A e which coincides with ζ is a character of A; ¡ ¢ on A. Clearly, the zero character of A corresponds to the character ζe∞ (α, A) = α e Thus X(A) can be identified with the set X(A) e lacking one point which of A. corresponds to the zero character ζ∞ of A. For any character ζ one has kζk ≤ 1; moreover, kζk = 1 if A has a unit. Indeed, without loss of generality we may assume that A has a unit. By 1.4.6 the value ζ(A) belongs to σ(A) for all A ∈ A. Therefore |ζ(A)| ≤ R(A) ≤ kAk, A ∈ A. Thus kζk ≤ 1. On the other hand, the equality ζ(1A ) = 1 implies kζk ≥ 1.

30

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Let A be a commutative Banach algebra and A ∈ A. The function ϑA : X → C defined by the rule ϑA (ζ) = ζ(A) is called the Gel0 fand transform of A. The mapping ϑ : A 7→ ϑA is called the Gel0 fand transform on A. We define the topology on X = X(A) to be the weakest topology in which all functions ϑA : X → C, A ∈ A, are continuous (see 1.1.1). Thus the family of all sets VA1 ,A2 ,...,An ,ε (ζ0 ) = { ζ ∈ X(A) : |ζ(Ak )−ζ0 (Ak )| < ε, k = 1, 2, . . . , n }, where A1 , A2 , . . . , An ∈ A and ε > 0 are arbitrary, forms a neighbourhood base (see 1.1.1) at ζ0 ∈ X(A). By virtue of the definition of a character, X is a Hausdorff space. If A has no unit the topology on X = X(A) coincides with the topology induced ¡ ¢ ¡ ¢ e since ζe (α, A) − ζe0 (α, A) = ζ(A) − ζ0 (A). by the embedding X(A) ⊂ X(A) Theorem. Let A be a commutative Banach algebra. (a) If A has a unit, the space X(A) is compact. The Gel0 fand transform ϑ is a morphism from the algebra A to the algebra C(X). In this case kϑk = 1. e is the (b) If A has no unit, the space X(A) is locally compact and X(A) 0 one point compactification of X(A). The Gel fand transform on A is a morphism from the algebra A to the algebra C0 (X) with kϑk ≤ 1. (c) Assume A has a unit. Then the Gel0 fand transform ϑ : A → C(X) preserves the spectrum, i.e., σ(A) = σ(ϑA ) = { ϑA (ζ) : ζ ∈ X } = { ζ(A) : ζ ∈ X } for all A ∈ A. In particular, an element A ∈ A is invertible if and only if ϑA (ζ) 6= 0 for all ζ ∈ X. Proof. (a) First we show that the space X = X(A) is compact. Actually, a character is a special case of a linear functional. Thus X can be regarded as a part of the conjugate space A0 of A, considered as a Banach space. Moreover, X is a subset of the unit sphere of A0 since kζk = 1. We observe that the topology on X coincides with the ∗-weak topology on A0 . We recall from 1.1.7 that the unit ball of A0 is ∗-weakly compact. It is straightforward to verify that X is closed in A0 . Clearly, ϑ is a morphism of algebras with a unit. The identity kϑk = 1 follows from kζk = 1. (b) follows immediately from (a). Indeed, a compact space lacking one point is locally compact. We only note that the Gel0 fand transform ϑeA of A is equal to zero at the point ζ∞ for all A ∈ A. (c) The inclusion σ(A) ⊆ σ(ϑA ) follows from 1.4.6. The opposite statement is less trivial. Its proof can be found in [Bou4 , ch. 1, §3, 3, proposition 3], [Hew, theorem C.20], [Nai, ch. 3, §11, 2], or [Rud2 , theorem 11.9]. See also 1.7.9 below. ¤

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

31

1.4.12. The Gel0 fand–Naimark theorem. Let A be an algebra. An operation A 7→ A∗ on A is called an involution if (A + B)∗ = A∗ + B ∗ , (αA)∗ = αA∗ , (AB)∗ = B ∗ A∗ , A∗∗ = A for all A, B ∈ A and α ∈ C. Here α is the complex conjugate of α. A subalgebra B of A is called a ∗-subalgebra if B ∈ B implies B ∗ ∈ B. A Banach algebra A with an involution is called a C ∗ -algebra if kAA∗ k = kAk2

for all A ∈ A.

Example. (a) The algebra B(H) of all bounded linear operators acting on a Hilbert space H is a C ∗ -algebra. (b) Let T be a locally compact topological space. Then C(T, C) is a C ∗ -algebra with respect to the natural involution x∗ (t) = x(t). (c) Let T be a locally compact topological space with a positive measure λ. Then L∞ (T, C) is a C ∗ -algebra with respect to the natural involution x∗ (t) = x(t). (d) Let G be a locally compact abelian group (see 1.6.1 below). Then the space AP (G, C) of all almost periodic (see 5.1.9 below for the definition) functions x : G → C forms a C ∗ -algebra. Theorem ([Bou4 , ch. 1, §6, 4, theorem 1], [Nai, ch. 3, §16, 2, theorem 1], [Rud2 , theorem 11.18]). Let A be a commutative C ∗ -algebra with a unit. Then the Gel0 fand transform ϑ : A → C(X, C) is an isometric isomorphism of C ∗ -algebras. Example. (a) Let T be a locally compact topological space with a positive measure λ. Then by the theorem there exists a compact topological space Tb such that L∞ (T, C) is isometrically isomorphic to C(Tb, C). This statement becomes false if one replaces C by an arbitrary Banach space E. Namely, it can be shown that C(Tb, E) is isomorphic only to the subspace E ⊗ε L∞ (T, C) of L∞ (T, E) consisting of all functions with compact image in a locally essential sense. (For the definition of the tensor product and the cross-norm ε see §1.7 below.) See also the remark in 1.7.4. (b) It can be shown that the character space of the C ∗ -algebra AP (G, C) of almost periodic functions (see 5.1.9 below for the definition) is Gb (see 5.1.7 and 4.2.1 below for the definition of Gb ). Hence by the theorem AP (G, C) is isometrically isomorphic to C(Gb , C), cf. 5.1.9.

1.5. Spaces of continuous and measurable functions The greater part of this section is devoted to the theory of the Lebesgue integral. We assume the reader is familiar with it. We continue this topic in §1.8, where we discuss absolutely continuous measures. We do not dare to say that the knowledge of any university course of the Lebesgue integral is completely enough for the reading of this book, because various expositions of this theory differ to a greater or less extent. Unfortunately, the

32

I. FUNCTIONAL ANALYSIS PRELIMINARIES

author has failed to base his exposition solely on the final results of the theory (such as the Fubini and Lebesgue–Radon–Nikodym theorems), which formulations are the same in most variants of the theory. Sometimes we are forced to use initial notions (such as measurable and integrable functions) which are equivalent to, but not the same as those in various expositions. We use the variant of the Lebesgue theory (more correctly, the Lebesgue– Bochner theory) of Bourbaki [Bou3 ] because it is the one most adapted to the language of functional analysis. Essentially the same theory can be found in [Edw]; the advantage of the latter book is its self-contained and detailed exposition. A self-contained comprehensive exposition of almost the same theory can also be found in [Nai], and one very close to it in [Hew]. We note that [Bou4 ], [Nai], and [Hew] are our main reference sources on spectral theory and harmonic analysis. Giving a complete theory of the Lebesgue integral would mean turning this book into a textbook on the foundations of functional analysis. Therefore we only recall the main steps of the construction of the Lebesgue integral from [Bou3 ]. We hope that it will be enough for the reader (accustomed to another variant of the theory) to read over this point of view and understand our proofs. 1.5.1. The Urysohn theorem. Let E be a fixed (complex) Banach space. We usually denote the norm on E by | · |. Let T be a locally compact topological space. The most important examples of T are R, [a, b], [a, +∞), and (−∞, a]. For any function x : T → E we call the closure of the set { t : x(t) 6= 0 } the support of x. We denote the support of x by the symbol supp x. We say that x is supported in K if supp x ⊆ K. We denote by C = C(T ) = C(T, E) the Banach space of all continuous bounded functions x : T → E with the supremum norm kxk = sup |x(t)|, t∈T

cf. the example in 1.4.1. We denote by C0 = C0 (T ) = C0 (T, E) its closed subspace that consists of all functions x vanishing at infinity, i.e., satisfies the following assumption: for any ε > 0 there exists a compact subset K ⊆ T such that |x(t)| < ε for all t ∈ / K. Theorem ([Bou1 , ch. 9, §4, 1–2], [Kel, ch. 4, lemma 4]). Let T be a locally compact topological space and K ⊆ T be a compact subset of it, or let T be a normal topological space and K ⊆ T be a closed subset of it. Then any continuous function α : K → [0, 1] has a continuous extension α e : T → [0, 1]. 1.5.2. The partition of unity Corollary ([Bou1 , ch. 9, §4, proposition 3], [Nai, ch. 3, §15, 2]). Let K be a compact topological space. For any finite covering U1 , U2 , . . . , Un of K by open sets, there exist Pn continuous functions α1 , α2 , . . . , αn : K → [0, 1] such that supp αk ⊆ Uk and k=1 αk (t) = 1 for all t ∈ K. The family { α1 , α2 , . . . , αn } is called the partition of unity subordinated to the covering { U1 , U2 , . . . , Un }.

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

33

Proof. First we construct open sets Vk ⊆ Uk such that V1 , V2 , . . . , Vn cover K and the closure Vk of each Vk is contained in Uk . Indeed, since a compact Sn space is normal Sn we can take disjoint open neighbourhoods V1 and W1 of U1 \ m=2 Um = K \ m=2 Um and K \ U1 , respectively. Assume Vk−1 is constructed. Then we ¡Sk−1 ¢ Sn take disjoint open neighbourhoods Vk and Wk of Uk \ m=1 Vm ∪ m=k+1 Um and K \ Uk . Clearly, on each step, V1 , . . . , Vk , Uk+1 , . . . , Un is a covering of K and Vk ⊆ K \ Wk ⊆ Uk . By Urysohn’s theorem there exist functions βk : K → [0, 1]³ such that β´ k (t) = 1, Pn t ∈ Vk , and βk (t) = 0, t ∈ / Uk . Finally, we set αk (t) = βk (t)/ k=1 βk (t) . ¤ 1.5.3. Finite-dimensional functions. We say that a function x : T → E is finite-dimensional if its image is contained in a finite-dimensional subspace of E. Proposition. (a) Let K be a compact topological space. Then the subspace of all finitedimensional functions is dense in C(K, E). (b) Let T be a locally compact topological space. Then the subspace of all finite-dimensional functions is dense in C0 (T, E). Proof. (a) We fix arbitrary x ∈ C(K, E) and ε > 0. For any t ∈ K we take an open neighbourhood UtSof t such that |x(s) − x(t)| < ε for all s ∈ Ut . Next, we n choose t1 , . . . , tn so that k=1 Utk = K. By 1.5.2 there existP continuous functions αk : K → [0, 1] such that Pnαk is equal to n = 1 for all t. Obviously, x(t) = k=1 αk (t)x(t). zero outside Utk and k=1 αk (t)P n We consider the function z(t) = k=1 αk (t)x(tk ). Clearly, it is finite-dimensional. It remains to observe that n n ¯X ¯ X ¯ ¯ |x(t) − z(t)| = ¯ αk (t)x(t) − αk (t)x(tk )¯ k=1

≤ ≤

n X k=1 n X

k=1

¯ ¯ αk (t)¯x(t) − x(tk )¯ αk (t)ε = ε.

k=1

(b) Let Te be the one point compactification of T . We represent C0 (T, E) as the subspace of C(Te, E) consisting of all functions vanishing at ∞. Under this interpretation (b) follows from (a). ¤ 1.5.4. The Stone–Weierstrass theorem. Let T be a locally compact topological space. We say that a subset H ⊆ C(T, C) separates points of T if for any t, s ∈ T there exists x ∈ H such that x(t) 6= x(s). We say that a subset H ⊆ C(T, C) does not vanish at t ∈ T if there exists x ∈ H such that x(t) 6= 0. We recall from 1.4.12 that a subalgebra H ⊆ C(T, C) is called a ∗-subalgebra if x ∈ H implies x ∈ H, where x(t) = x(t) is the complex conjugate function. Let C(T, R) denote the set of all continuous bounded functions x : T → R. Clearly, the set C(T, R) can be considered as a Banach algebra over the field R.

34

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Theorem. (a) ([Bou1 , ch. 10, §4, 2, theorem 3], [Nai, ch. 1, §2, 10, theorem 1]). Let K be a compact topological space. Let H be a subalgebra, with a unit, of the algebra C(K, R), and let H separate points. Then H is dense in C(K, R). (b) Let K be a compact topological space. Let H be a ∗-subalgebra, with a unit, of the C ∗ -algebra C(K, C), and let H separate points. Then H is dense in C(K, C). (c) Let T be a locally compact topological space. Let H be a ∗-subalgebra of the C ∗ -algebra C0 (T, C), and let H separate points and not vanish at all points t ∈ T . Then H is dense in C0 (T, C). Proof. (b) It is easy to see that in this case the subset HR of all real-valued functions belonging to H separates points, too. By (a) HR is dense in C(K, R). Consequently H is dense in C(K, C). e and C f0 be algebras with adjoint units associated with H and C0 (T, C), (c) Let H f0 as C(Te, C), where Te is the one point respectively, see 1.4.1. We represent C e is dense in C(Te, C). Clearly, H consists compactification of T , see 1.1.1. By (b) H e vanishing at ∞. Therefore H is dense in C0 (T, C). ¤ of all x ∈ H 1.5.5. The spaces Lp (T ), p < ∞. Let T be a locally compact topological space, and let E be a Banach space with the norm | · |. For any compact subset K ⊆ T we denote by CK (T ) = CK (T, E) the closed subspace of C consisting of all functions supported in K. By C00 = C00 (T, E) we denote the union of CK (T, E) over all compact K ⊆ T . From the Urysohn theorem it follows that C00 is dense + in C0 . Finally, we denote by C00 the subset of all functions x ∈ C00 (T, C) taking their values in [0, +∞). A (positive) measure or a (positive) integral on T is an arbitrary linear functional λ : C00 (T, C) → C possessing the following property: λ(x) ≥ 0 provided + x ∈ C00 . It can be shown that each positive measure λ is continuous in the natural topology of C00 , namely, the restriction of λ to CK (T, C) is continuous in the norm of C for all compact K ⊆ T . A measure λ on T is called bounded or finite if it is continuous in the norm of C on the whole of C00 (T, C). In such a case we set kλk = sup{ |λ(x)| : x ∈ C00 }. R We R usually denote the value of λ on x ∈ C00 by x dλ or, in more detailed form, x(t) dλ(t). Below in this subsection we assume that λ is a positive measure on T . A function z : T → [0, +∞] is called lower semi-continuous if for every t0 ∈ T and h ∈ R such that h < z(t0 ) there exists a neighbourhood U of t0 such that h < z(t)

for all t ∈ U .

For example, the characteristic function of an open set is lower semi-continuous. We denote by M + = M + (T ) the set of all lower semi-continuous functions z : T → [0, +∞]. It can be shown that for any z ∈ M + and t ∈ T one has z(t) = sup { y(t) : y ∈ C00 , 0 ≤ y(t) ≤ z(t) } .

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

35

Let λ be a positive measure on T . The upper integral of z ∈ M + is the quantity ½Z ¾ Z ∗ z dλ = sup y dλ : y ∈ C00 , 0 ≤ y(t) ≤ z(t) . Now let x : T → [0, +∞] be an arbitrary function. The upper integral of x is Z

½Z

∗

x dλ = inf

¾

∗

+

z dλ : z ∈ M , x(t) ≤ z(t)

.

It is easy to show that for a sequence xn : T → [0, +∞] one has Z ∗ ³X ∞

´ xn dλ ≤

n=1

∞ Z X

∗

xn dλ.

n=1

The upper measure λ∗ (E) of a set E ⊆ T is the upper integral of its characteristic function 1E ; we recall that by definition 1E (t) = 1 if t ∈ E, and 1E (t) = 0 if t∈ / E. A set E ⊆ T is called null , or λ-null, or negligible, or a set of measure zero if λ∗ (E) = 0. We say that a property holds almost everywhere (briefly, a.e.) if it is violated only on a null set. Let F = F(T, E) denote the space of all x : T → E. Two functions x, y ∈ F are said to be equivalent if x(t) = y(t) a.e.. A function x is called null , or λ-null, or negligible if it is equivalent to the zero function. For every 1 ≤ p < ∞ and x ∈ F(T, E) we set µZ Np (x) =

∗

¶1/p |x(t)| dλ(t) . p

By the Minkowski inequality Np possesses the properties of a semi-norm. Hence the set Fp = Fp (T, E) of all x ∈ F(T, E) with a finite Np (x), and the kernel Kp = Kp (T, E) of Np form subspaces. It can be shown that Kp does not depend on p and consists of all null functions. We denote by Fp the quotient space Fp /Kp . Sometimes it is convenient to consider functions x : T → E defined only a.e.; in this case the upper integral of |x| is the upper integral of |¯ x|, where x ¯ is an arbitrary function defined everywhere and equivalent to x. One can modify the definition of F and take for F the set of all functions x : T → E defined a.e.. This leads to an equivalent definition of Fp . But in this case F and Fp are not linear spaces, they are solely semi-groups with respect to addition, see the discussion in [Bou3 , ch. 4, §2, 5]. Therefore Fp should not be defined to be the quotient space, but to be directly the space of equivalence classes. We also note that it is convenient to imagine the undefined values of a function defined a.e. as unknown or defined in an arbitrary way, but not as non-existing. Proposition ([Bou3 , ch. 4, §3, 3, proposition 6]). Let 1 ≤ P∞ Pp∞< ∞. Assume the series n=1 xn , xn ∈ Fp , converges absolutely in Fp , i.e., n=1 Np (xn ) < +∞. P∞ Then for almost all t ∈ T the series n=1 xn (t) converges absolutely in E. Let

36

I. FUNCTIONAL ANALYSIS PRELIMINARIES

P∞ P∞ x(t) denote the sum n=1 xn (t) (we define x(t) in an arbitrary way if n=1 P∞xn (t) does not converge). Then the function x belongs to Fp and the series n=1 xn converges to x in norm, more precisely, ³ Np x −

n X

´ xk ≤

k=1

∞ X

Np (xk ).

k=n+1

Consequently the space Fp is complete. It is also convenient to regard Fp as a complete, but not Hausdorff, space. Let Lp = Lp (T, E) denote the closure of C00 (T, E) in Fp . The space Lp = Lp (T ) = Lp (T, E) = Lp (T, E, λ), 1 ≤ p < ∞, is defined to be the subspace of Fp induced by Lp . Equivalently, Lp is the closure in Fp of the natural image of C00 . Clearly, Lp is a Banach (i.e., Hausdorff and complete) space. Functions x ∈ L1 are called integrable. Functions x ∈ Lp are called integrable to the power p or pintegrable. A function x belongs to Lp if and only if the function t 7→ |x(t)|p−1 x(t) belongs to L1 . Let x ∈ C00 (T, C) take its values in R. Then representing x as the sum of + x = (x + |x|)/2 and x− = (x − |x|)/2 we see that: ¯Z ¯ Z ¯ ¯ ¯ x dλ¯ ≤ |x| dλ = N1 (x). If x ∈ C00 (T, C) takes its values in C the same estimate also holds.¯R Indeed, ¯ R ¯ x dλ¯ = without loss of generality we may assume that x dλ ∈ [0, +∞). Then R R R Re x dλ ≤ |x| dλ. Since C00 is dense in L1 (T, C) the functional x 7→ x dλ can be extended to the R R whole of L1 by continuity. We denote this extension by the symbol x dλ or x(t) dλ(t) and call it the integral of x with respect to λ. Clearly, ¯Z ¯ Z ¯ ¯ for all x ∈ L1 (T, C). ¯ x dλ¯ ≤ |x| dλ = N1 (x) x Ras x = Pn PnLet x ∈ C00 (T, E) be a finite-dimensional function.R We represent C). We set x dλ = k=1 ek xk dλ. k=1 ek xk , where ek ∈ E and xk ∈ C00 P(T, m If one has another representation x = i=1 e˜i x ˜i then, clearly, Z Z n m DX E Z DX E 0 0 ek xk dλ, e = hx(t), e i dλ(t) = e˜i x ˜i dλ, e0 k=1

i=1

for all e0 ∈ E0 . Thus this definition Pn does not depend on the choice of the represen0 0 tation of x in the form R x =0 k=1 ek xk . Clearly, for such Ran x and any e ∈ E , 0 ke k ≤R1, we have |h x dλ, e i| ≤ N1 (x) and consequently | x dλ| ≤ N1 (x). Thus x 7→ x dλ is a vector-valued functional taking its values in E, continuous in the semi-norm N1 , and defined on a dense subspace of C00 (see 1.5.3). Hence it possesses the extension by R continuity to L1 (T, E) and L1 (T, E). We denote this extension by the symbol x dλ and call the integral of x with respect to λ. Clearly, ¯Z ¯ Z ¯ ¯ for all x ∈ L1 (T, E). ¯ x dλ¯ ≤ |x| dλ = N1 (x)

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

For x ∈ L1 the function |x|(t) = |x(t)| is integrable and

R∗

|x| dλ =

R

37

|x| dλ.

Remark. In terms of §1.7 the vector-valued integral is the tensor product of 1 ∈ B(E) and the scalar integral on L1 (T, C), Note that by 1.7.4 L1 (T, E) ' E ⊗π L1 (T, C) and the cross-norm π is uniform. A set E ⊆ T is called summable R or integrable if its characteristic function 1E is integrable. The number λ(E) = 1E dλ is called the measure of E. Any compact set is summable; and an open set O is summable provided that λ∗ (O) < ∞. A set E is summable if and only if for any ε > 0 there exist an open set O and a compact set K such that K ⊆ E ⊆ O and λ(O \ K) < ε. The family of all summable sets is a σ-ring and λ is σ-additive on it. 1.5.6. The spaces L∞ (T ) and L0 (T ). Let T be a locally compact topological space, and let λ be a positive measure on T . A function x : T → E is called measurable if for every compact set K ⊆ T and ε > 0 there exists a compact set K1 ⊆ K such that λ(K \ K1 ) ≤ ε and the restriction of x to K1 is continuous. A set E ⊆ T is called measurable if its characteristic function 1E is measurable. Closed and open sets are measurable. The family of all measurable sets forms a σ-algebra. A real-valued function x is measurable if and only if the set { t : x(t) ≤ a } is measurable for all a ∈ R. A function x : T → E belongs to Lp , 1 ≤ p < ∞, if and only if xP is measurable and n Np (x) < +∞. The subspace of all functions x of the form x = k=1 ek 1Ek , where ek ∈ E and Ek ⊆ T are summable and pairwise disjoint, is dense in Lp , p < ∞. R Let E ⊆ T Rbe a measurable set, and let x : T → E be a function. We define x dλ to be 1E x dλ. E A set E ⊆ T is called locally null or locally negligible if its intersection with any compact set K ⊆ T is null. We say that a property holds locally almost everywhere (briefly, l.a.e.) if it is violated only on a locally null set. A function is called locally null or locally negligible if it is equal to the zero function l.a.e.. Clearly, if x is measurable and y(t) = x(t) l.a.e. then y is also measurable. A function x : T → E is called universally measurable if it is measurable with respect to any positive measure on T . A set E ⊆ T is called universally measurable if its characteristic function is universally measurable. In other words, a set is universally measurable if it is measurable with respect to any positive measure on T . Clearly, the family of all universally measurable sets is a σ-algebra. For any λ-measurable function x : T → E there exists a universally measurable function y : T → E which coincides with x λ-locally almost everywhere, see [Bou3 , ch. 5, §3, 4, proposition 7]. Consequently for any λ-measurable set E ⊆ T there exists a universally measurable set F ⊆ T such that both F \ E and E \ F are λ-locally null. Example. Clearly, closed and open subsets of T are universally measurable. We show that any locally compact (with respect to the relative topology) subset T0 ⊆ T is universally measurable. We denote by T0 the closure of T0 in T . Let t0 ∈ T0 , and let K ⊆ T0 be a compact neighbourhood of t0 in T0 . By the definition of a relative topology there exists an open neighbourhood U = U (t0 ) of t0 in T such that U ∩ T0 ⊆ K. We observe that the set U \ T0 = U \ K is open in T . Therefore

38

I. FUNCTIONAL ANALYSIS PRELIMINARIES

the intersection of U \ T0 with T0 is empty. Consequently U ∩ T0 = U ∩ T0 . Let O = ∪t0 ∈T0 U (t0 ). Evidently, O is open. By what has been proved, O ∩ T0 = T0 . Thus T0 is the intersection of an open set and a closed set. A measurable set E is called σ-finite (it would be more natural to call it σsummable) if it is contained in a countable union of summable sets. Clearly, for all x ∈ Lp , 1 ≤ p < ∞, the set of all t such that x(t) 6= 0 is σ-finite. A measure λ is called σ-finite if the whole space T is σ-finite. If λ is σ-finite every locally null set is null. Example. Let Rd denote the group R considered with the discrete topology. (For instance, the group Rd will arise in the definition of the Bohr compactification Rb of R, which in turn is closely connected with almost periodic functions, see 5.1.7.) In this case the space C00 (Rd , C) consists of all functions x which are equal Rto zero outside a finite set. The Haar measure on Rd is the functional P x 7→ x dλ = t∈R x(t) on C00 . Clearly, the measure λ is not σ-finite. A less trivial example of a space not σ-finite is the group Rd × R. In the latter case there exist locally null subsets of Rd × R which are not null. See also the example in 1.5.12. A measurable set E is called σ-compact if it is contained in a countable union of compact sets. A measure λ is called σ-compact if the whole space T is σ-compact. Since any compact subset has a finite measure, any σ-compact set (measure) is σ-finite. (From 1.6.2(e) below it follows easily that if λ is a Haar measure then, conversely, any σ-finite set is σ-compact.) In our most important examples the space T will be σ-compact. By 1.5.5 and 1.1.2 any function x ∈ Lp , 1 ≤ p < ∞, is equal to zero outside the union of a null set and a sequence of compact sets. Hence any σ-finite set is the union of a σ-compact set and a null set. Let x : T → E be a function. We say that e ∈ E is a (locally) essential value of x if for any neighbourhood U of e the set x−1 (U ) is not (locally) null. Clearly, if x and y coincide (locally) almost everywhere then they have the same (locally) essential values. Assume x is measurable. It is easy to see that the set of all (locally) essential values of x is closed and non-empty. If T is compact, σ-compact, or σ-finite then by the definition of a measurable function the set of all essential values of x is σ-compact. In particular, the set of all essential values of x ∈ Lp , 1 ≤ p < ∞, is σ-compact. For every x : T → E we set ess sup |x(t)| = inf sup{ |y(t)| : y(t) = x(t) a.e. }, y

t

t

N∞ (x) = loc ess sup |x(t)| = inf sup{ |y(t)| : y(t) = x(t) l.a.e. }. t

y

t

It is easy to see that the infima in these formulae can be replaced by minima. We also note that these quantities are the maximum (locally) essential values of |x|. If N∞ (x) < ∞, the function x is called locally essentially bounded. Let L∞ = L∞ (T, E) denote the space of all measurable functions x : T → E with N∞ (x)

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

39

being finite. Clearly, N∞ is a semi-norm, and its kernel K∞ consists of all locally null functions. The space L∞ = L∞ (T ) = L∞ (T, E) = L∞ (T, E, λ) is defined to be L∞ /K∞ . The space L∞ is Banach. We note that elements of L∞ can be enlarged easily to classes of functions that are defined l.a.e.. This leads to an equivalent definition of L∞ . We denote by L0 = L0 (T, E) the subspace of L∞ that consists of all classes x e ∈ L∞ vanishing at infinity in the following sense: for any ε > 0 there exists a compact subset K ⊆ T such that loc ess supt∈K / |x(t)| < ε. Clearly L0 is closed and therefore is a Banach space. The space L0 ⊆ L∞ is defined in a similar way. Sometimes it is convenient to extend the set Lp , 1 ≤ p < ∞, in the following way, see [Bou3 , ch. 5, §1] and [Edw, 4.14] for details. For an arbitrary function x : T → [0, +∞] we call Z

½Z

∗

x dλ = sup

¾

∗

1K x dλ : K ⊆ T is compact

,

where 1K is the characteristic function of a set K, the essential upper integral of R∗ R∗ the function x. It is easy to see that x dλ ≤ x dλ. The equality holds if + x ∈ M or if x is equal to zero outside a σ-finite set. Clearly, E is locally null if R∗ and only if 1E dλ = 0. For 1 ≤ p < ∞ and x : T → R we set ÃZ Np (x) =

∗

!1/p |x(t)|p dλ(t)

.

Let Fp = Fp (T, E) denote the space of all x with finite Np (x); and let Kp = Kp (T, E) denote the kernel of Np . Clearly, Kp = K∞ . It can be proved that the quotient space Fp = Fp /Kp is complete. Clearly, one can enlarge elements of Fp to classes of functions defined l.a.e.. Let Lp denote the closure of C00 (T, E) in Fp . The space Lp , 1 ≤ p < ∞, is defined to be the subspace of Fp = Fp /Kp induced by Lp . The functions x ∈ Lp are called essentially integrable to the power p or essentially p-integrable. A function x : T → E belongs to Lp , 1 ≤ p < ∞, if and only if x coincides with a function y ∈ Lp l.a.e.; in this case Np (x) = Np (y). Thus the spaces Lp and Lp , 1 ≤ p < ∞, are naturally isometrically R isomorphic. By virtue of this isomorphism the extension of the functional x 7→ x dλ to L1 is well defined. We denote this extension by R R R the symbol x dλ and call it the essential integral of x. Clearly, x dλ = x dλ R R for x ∈ L1 . It is easy to verify R that x dλ = limK K x dλ for x ∈ L1 . Here limK means the following:R limK K x dλ = a if for any ε > 0 there exists a compact set K0 ⊆ T such that | K x dλ − a| < ε for all compact sets K ⊇ K0 . Remark. One may also consider the space L∞ arising from the identification a.e.. If T is σ-finite both definitions obviously coincide; but in general this is not the case. The conjugate space of L1 is L∞ with identification l.a.e., see 1.8.1. This is the main advantage of the chosen variant of the definition of L∞ .

40

I. FUNCTIONAL ANALYSIS PRELIMINARIES

If no confusion can arise we do not distinguish carefully between Lp and Lp (and Lp and Lp , respectively), i.e., we identify the equivalence class x e ∈ Lp with any of its representatives x ∈ Lp . Let E ⊂ T be a measurable subset. For any function x : E → E we consider the continuation x ˜ : T → E defined by the rule x(t) = 0, t ∈ T \ E. We say that a function x : E → E is measurable (integrable, etc.) if so is x ˜. On the basis of this idea we define the spaces Lp (E, E) and Lp (E, E). We denote the norm on Lp induced by Np by the symbols k · k, k · kp , and k · kLp . 1.5.7. The spaces lq (I). Let I be a set. We consider I with the discrete topology and assume that the measure of each point i ∈ I is equal to unity. Let 1 ≤ q ≤ ∞ or q = 0. We denote the space Lq (I, E) by the symbol lq = lq (I) = lq (I, E). Sometimes it is convenient to consider the more general spaces lq (I, Ei ) defined as follows. Assume we have a collection of Banach spaces {Ei : i ∈ I} with the norms | · |. We denote by lq = lq (I) = lq (I, Ei ) the spaces of all families x = { xi ∈ Ei : i ∈ I } bounded in the usual norms: kxk =

³X

q

|xi |

´1/q

,

q < ∞,

i∈I

kxk = sup |xi |,

q = ∞.

i∈I

By l0 we denote the closed subspace of l∞ consisting of all families {xi } vanishing at infinity, i.e., such that for any ε > 0 there exists a finite subset K ⊆ I such that |xi | < ε for all i ∈ / K. Finally, we denote by l00 ⊆ lq the subspace of all families x = {xi } having only a finite number of non-zero members. Clearly, l00 is dense in lq for q 6= ∞. 1.5.8. The product of measures. Let T and S be locally compact topological spaces. We define a topology on T × S in the usual way, see 1.1.1. Let K ⊆ T and N ⊆ S be compact subsets. We denote by CK (T, C) ⊗ CN (S, E) (note the asymmetry between C and E) the subspace of CK×N (T ×S, E) consisting of all functions z : T × S → E which can be represented as a finite sum z(t, s) =

n X

xi (t)yi (s)

i=1

with xi ∈ CK (T, C) and yi ∈ CN (S, E). Cf. §1.7. Theorem. Let T and S be locally compact topological spaces, and let K ⊆ T and N ⊆ S be compact subsets. (a) The space CK (T, C) ⊗ CN (S, E) is dense in the space CK×N (T × S, E). (b) The space¡ CK×N (T ×¢ S, E) is naturally isometrically isomorphic to the space CK T, CN (S, E) . Consequently the ¡ space C0 (T ¢ × S, E) is naturally isometrically isomorphic to the space C0 T, C0 (S, E) .

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

41

(c) Assume that λ and µ are measures on T and S, respectively. Then for any z ∈ C00 (T × S, E) the formula (the right hand sides make sense by (b)) ¶ ¶ Z Z µZ Z µZ z d(λ ⊗ µ) = z(t, s) dµ(s) dλ(t) = z(t, s) dλ(t) dµ(s) defines the linear continuous functional λ⊗µ on C00 (T ×S, E). Particularly, λ ⊗ µ is a measure on T × S. If λ and ν are bounded, kλ ⊗ µk ≤ kλk · kµk. The measure λ ⊗ µ is called the productRR of λ and µ. The integral with respect to λ ⊗ µ is usually denoted by the symbol z d(λ ⊗ µ). Remark. The ¡ ¢ space C00 (T × S, E) is also naturally isomorphic to the space C00 T, C00 (S, E) . But the meaning of this statement is less simple. The point is that the natural topology of the space C00 (S, E) is the inductive limit of the topologies of the spaces CN (S, E), where N runs over all compact subsets of S. This topology is defined to be the strongest locally convex topology on C00 (S, E) in which all natural embeddings CN (S, E) → C00 (S, E) are continuous, see [Bou2 ], [Edw, §6.3] or [Scha, ch. 2, §6] for details. It is easy to show (see, e.g., [Scha, ch. 2, 6.1]) that a linear mapping from C00 (S, E) to a locally convex space is continuous if and only if its restriction to each CN (S, E) is continuous, cf. the ¡ ¢ definition of a measure. The topologies on C00 (T × S, E) and C00 T, C00 (S, E) are defined in a similar way, but one should remember that here C00 (S, E) is a locally convex space itself. In order to simplify our exposition we do not use these facts explicitly. Proof. (a) The case E = C follows from the Stone–Weierstrass theorem. Hence for an arbitrary E the subspace CK (T, C) ⊗ CN (S, E) contains all functions z whose image lies in a (one-dimensional and consequently in a) finite-dimensional subspace of E. By 1.5.3 the set of all such functions is dense in CK×N (T × S, E). (b) We denote by F (T, E) the linear space of all functions from the set T to a linear space E. First we observe that the natural mapping ω that takes a function z :¡T × S → E¢ to the function t 7→ z(t, ·) is an isomorphism from F (T × S, E) onto F T, F (S, E) . ¡ ¢ It is easy to see that ω −1 maps C T, C (S, E) into CK×N (T × S, E). What K N ¡ ¢ ω maps CK×N (T × S, E) into CK T, CN (S, E) follows from the fact any compact space is a uniform space and any continuous function on a compact space is uniformly continuous (see [Bou1 , ch. 2, §4, 1, theorems 1 and 2]). The equality kωzk = kzk for z ∈ CK×N is evident. (c) The linearity and continuity of the functional ¶ Z Z µZ z 7→ z d(λ ⊗ µ) = z(t, s) dµ(s) dλ(t) follow from (b). The identity ¶ ¶ Z µZ Z µZ z(t, s) dµ(s) dλ(t) = z(t, s) dλ(t) dµ(s) for z of the form z(t, s) = x(t)y(s) is trivial. For other z it follows from (a).

¤

42

I. FUNCTIONAL ANALYSIS PRELIMINARIES

¡ ¢ 1.5.9. The isomorphism L1 (T × S, E) ' L1 T, L1 (S, E) . Let λ and µ be positive measures on T and S, respectively. We denote by L1 (T × S, E) the space induced by the measure λ ⊗ µ. Corollary. The space L1 (T ×S, E) is naturally isometrically isomorphic to the ¡ ¢ space L1 T, L1 (S, E) . Proof. Assume K ⊆ T and N ⊆ S are compact sets. ¢Let ω denote the ¡ natural isomorphism from CK×N (T ×S, E) onto CK T, CN (S, E) taking a function z : T × S → E to the function t 7→ z(t, ·). By the definition of λ ⊗ µ, for any z ∈ CK×N (T × S, E) we have ¶ ZZ Z µZ |z(t, s)| d(λ ⊗ µ)(t, s) = |z(t, s)| dµ(s) dλ(t). This equality means that the norm of z in ¢L1 (T × S, E) is equal to the norm ¡ of the function t 7→ z(t, .) in L1 T, L1 (S, E) . Since the union of the subspaces CK×N (T × S, E) (that is equal to C00 ) is dense in L1 (T × S, E) the extension of ¡the isomorphism ω by continuity is an isomorphism from L1 (T × S, E) into ¢ L1 T, L1 (S, E)¡ . It remains¢ to observe that ¡ ω acts onto ¢ because the union of the subspaces CK T, CN (S, E) is dense in L1 T, L1 (S, E) . ¤ 1.5.10. The Fubini theorem Theorem ([Bou3 , ch. 5, §8, 4]). Let T and S be locally compact topological spaces with positive measures λ and µ, respectively. (a) For any z ∈ L1 (T × S, E, λ ⊗ µ) the function Z t 7→ z(t, s) dµ(s) is defined for almost all t ∈ T and belongs to L1 (T, E, λ), and ¶ ZZ Z µZ z d(λ ⊗ µ) = z(t, s) dµ(s) dλ(t). (b) Let z : T × S → E be a λ ⊗ µ-measurable function, and let z be equal to zero outside a set which is σ-finite with respect to the measure λ ⊗ µ, and ¶ Z ∗ µZ ∗ |z(t, s)| dµ(s) dλ(t) < ∞. Then z ∈ L1 (T × S, E, λ ⊗ µ). Remark. Running ahead, we note that by 1.8.4(c) the theorem remains valid for complex measures. In such a case the condition in (b) must be written as follows ¶ Z ∗ µZ ∗ |z(t, s)| d|µ|(s) d|λ|(t) < ∞.

1.5. SPACES OF CONTINUOUS AND MEASURABLE FUNCTIONS

43

1.5.11. The Riesz–Thorin theorem Theorem ([BerL, theorem 5.1.2]). Let T be a locally compact topological space with a positive measure λ. Let 1 ≤ α < β < γ ≤ ∞, and let a linear operator A act continuously both on Lα = Lα (T, E) and Lγ = Lγ (T, E). Then A acts continuously on Lβ = Lβ (T, E) and ¡ ¢1−ϑ ¡ ¢ϑ kA : Lβ → Lβ k ≤ kA : Lα → Lα k kA : Lγ → Lγ k , where ϑ is determined from the condition 1−ϑ ϑ 1 = + . β α γ The statement remains valid if one interprets L∞ as L0 . By saying here that the operator A acts simultaneously on several spaces we mean that A is the same on their common part. We note that since β < ∞ the set C00 — the common part of Lα and Lγ — is dense in Lβ . Therefore the operator A : Lβ → Lβ is determined uniquely. 1.5.12. The norm of an integral operator. In this subsection we recall the simplest criterion of the action of integral operators on Lp . Let T be a locally compact topological space with a positive measure λ and E be a Banach space. Let n : T × T → B(E) be a measurable function. We consider the integral operator Z ¡ ¢ N x (t) = n(t, s)x(s) dλ(s). T

Theorem. Assume T is σ-finite. (a) If Z K1 = ess sup kn(t, s)k dλ(t) < ∞ s∈T

T

then the operator N acts on L1 (T, E) and kN : L1 → L1 k ≤ K1 . (b) If Z K∞ = ess sup kn(t, s)k dλ(s) < ∞ t∈T

T

then the operator N acts on L∞ (T, E) and kN : L∞ → L∞ k ≤ K∞ . (c) If both K1 and K∞ are finite, the operator N acts on Lp (T, E) for all ϑ 1 ≤ p ≤ ∞, and kN : Lp → Lp k ≤ K11−ϑ K∞ ≤ max(K1 , K∞ ), where ϑ is defined as in 1.5.11. Proof. (a) Let x ∈ L1 . Clearly, the function (t, s) 7→ n(t, s) x(s) is measurable. Furthermore, we have ¶ Z ∗ µZ kn(t, s)k dλ(t) kx(s)k dλ(s) ≤ K1 kxkL1 . T

T

44

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Therefore by Fubini’s theorem N x ∈ L1 and kN xkL1 ≤ K1 kxkL1 . (b) Let x ∈ L∞ . We consider an arbitrary compact subset U ⊆ T . We show that the restriction of the function (t, s) 7→ n(t, s) x(s) to U × T belongs to L1 (U × T ). Actually, ¶ Z µZ ∗

U

T

kn(t, s) x(s)k dλ(s) dλ(t) ≤ λ(U )K∞ kxkL∞ .

Therefore the restriction of the function N x to U belongs to L1 (U ) for any compact set U . Hence N x is measurable on the whole of T . Evidently, we have kN xkL∞ ≤ K∞ kxkL∞ . (c) follows from (a) and (b), and the Riesz–Thorin theorem. ¤ Example. Some problems arise in the case in which T is not σ-finite. T = [0, 1] × [0, 1]d , where [0, 1] is considered with the usual topology Lebesgue measure, and [0, 1]d is considered with the discrete topology discrete measure, i.e., λ({t}) = 1 for all t ∈ [0, 1]d . Clearly, T is not cf. the example in 1.5.6. (a) We consider the kernel ½ n(t1 , t2 ; s1 , s2 ) =

1

for s1 = t2 ,

0

for s1 6= t2 ,

Assume and the and the σ-finite,

where t1 , s1 ∈ [0,RR1] and t2 , s2 ∈ [0, 1]d . It is easy to see that n is measurable and that sups1 ,s2 T n(t1 , t2 ; s1 , s2 ) dt1 dt2 = 1. Straightforward calculations show that N x = 0 for all x ∈ L1 . Thus the estimate kN : L1 → L1 k ≤ K1 is very rough. (b) We consider the kernel ½ n(t1 , t2 ; s1 , s2 ) =

1 0

for t1 = s2 , for t1 = 6 s2 ,

where t1 , s1 ∈ [0,RR 1] and t2 , s2 ∈ [0, 1]d . It is easy to see that n is measurable and that supt1 ,t2 T n(t1 , t2 ; s1 , s2 ) ds1 ds2 = 1. Let R ⊆ [0, 1] be a Lebesgue non-measurable subset. It is straightforward to verify that the operator N takes the characteristic function of the measurable set [0, 1] × R to the characteristic function of the non-measurable set R × [0, 1]d . It is interesting to note that in both examples n is locally null, but not null. In particular, n’s are not equal to zero outside a σ-finite set. Consequently the applicability of Fubini’s theorem is restricted.

1.6. Infinite matrices This book is aimed at applications to functional differential equations. Therefore the most important functional spaces for our exposition are the spaces of functions on R. But it would be inconvenient to restrict our consideration to the spaces on R and on the parts of R (such as [a, b] and [a, ∞)) only. For example, the periodic value problem can be reduced to an equation on T = R/Z; and equations on Z can be interpreted as functional equations with discrete time; moreover,

1.6. INFINITE MATRICES

45

sometimes it is convenient to reduce the consideration of difference equations with continuous time, such as x(t) = x(t − 1) + 2x(t − 2), which are parts of differential difference equations of neutral type, to the consideration of equations on Z. All three sets mentioned above — R, T, and Z — are locally compact abelian groups. This fact is of great importance. It allows one to apply various methods of harmonic analysis. Therefore we develop a considerable part of our theory for functional spaces on locally compact abelian groups. Of course, there are some additional (less trivial) reasons for the consideration of general locally compact abelian groups. If one represents the space Lp (R, E) ¡ ¢ as lp Z, Lp¡([0, 1], E) one ¢ can assign to each operator A : Lp → Lp the matrix { Aij ∈ B Lp ([0, 1], E) : i, j ∈ Z }. Matrix representations are convenient for the investigation of asymptotic properties of functional and functional differential equations. To be systematic, in this section we gather definitions of various classes of matrices and functional spaces associated with matrix representations. 1.6.1. Locally compact abelian groups. Let G be a group; usually we assume that G is abelian and write the group operation on G as addition. The group G is called a topological group if G is a Hausdorff topological space and the functions (t, s) 7→ t + s and t 7→ −t are continuous. It can be shown ([Hew, 8.13]) that any locally compact abelian group is a normal topological space. We usually call locally compact abelian groups simply groups. A topological group G is called locally compact, compact, or discrete if the topology on G possesses the corresponding property. A group G is called compactly generated if there exists a compact neighbourhood U of 0 ∈ G such that the sets U0 = U − U , Uk+1 = Uk + U0 , k = 1, 2, . . . , cover the whole of G. Let G1 and G2 be (topological) groups. A morphism of (topological ) groups is a (continuous) mapping ϕ : G1 → G2 that preserves the group operation, in other words, ϕ(g + h) = ϕ(g) + ϕ(h). In particular, this implies that ϕ(0G1 ) = 0G2 . A morphism ϕ of (topological) groups is called a (topological ) isomorphism if the inverse mapping exists (and is continuous). Clearly, if ϕ−1 exists it is also a morphism of groups. We always endow the product G1 × G2 of topological groups with the product topology, see 1.1.1. If G ≈ G1 × G2 we say that G contains G1 as a direct factor. Let G be an abelian topological group, and let H be a subgroup of it. We define the topology on the quotient group G/H to be the strongest topology in which the natural projection q : G → G/H is continuous. Thus a subset of G/H is open (closed) if and only if its pre-image under q is an open (closed) subset of G. Examples. (a) The group R of real numbers with the usual topology, the group Z of integers with the discrete topology, and the quotient group T = R/2πZ with the topology of a quotient space, are locally compact abelian groups. These are the main examples we deal with in this book. A more general locally compact abelian group is the group Ra × Zb × Tc , where a, b and c are non-negative integers, with the topology of Cartesian product. This group contains all the preceding groups as special cases. (b) The group U = { z ∈ C : |z| = 1 } with the multiplication as the group operation is a locally compact abelian group. Clearly,

46

I. FUNCTIONAL ANALYSIS PRELIMINARIES

U is isomorphic to T. (c) We denote by Rd the group R with the discrete topology. We recall that Rd is not σ-compact. In this book we use the group Rd as an auxiliary object for the investigation of equations on R. (d) For n ∈ N we denote the quotient group Z/nZ by Zn . For example, the group Z2 can be used as a model for the space of elementary events in the problem of random throwing of a coin. Then the group (Z2 )N is the set of elementary events for a sequence of independent throwings arising in the laws of large numbers. (e) The Hermite poly2 2 nomials Hn (x) = (−1)n ex d/dx (e−x ) satisfy the equation Hn0 (x) = 2nHn−1 (x). Similarly, the Laguerre polynomials Ln (x) = (1/n!) ex dn /dxn (xn e−x ) satisfy the equation xL0n (x) + (n + 1 − x)Ln (x) − (n + 1)Ln+1 (x) = 0, see, e.g., [Tikh]. These are equations on the group R × Z. More examples of equations on R × Z can be found in [Pin, ch. 7]. (f) An important example of a compact group — the Bohr compactification Gb of a locally compact abelian group G — will be defined in 5.1.7. The group Gb has a difficult structure even for our main example G = R. It will be used in §6.5 for the discussion of almost periodic and difference operators. Proposition. (a) ([Hew, theorems 5.22 and 5.21]) Let G be a locally compact abelian group, and let H be a closed subgroup of it. Then G/H is locally compact, too. The topology on G/H is discrete if and only if H is open. (b) ([Bou4 , ch. 2, §2, 1], [Hew, theorem 9.8]) Any locally compact compactly generated abelian group G is topologically isomorphic to the group Ra × Zb × K, where a and b are non-negative integers, and K is a compact abelian group. (c) ([Kuro, §20] Any finitely generated (discrete) abelian group G is isomorphic to Zb × F, where b is a non-negative integer, and F is a finite group. (d) ([Bou4 , ch. 2, §2, 2, proposition 3]), [Hew, theorem 24.30]) Any locally compact abelian group G is topologically isomorphic to the group Rc × V, where c is a non-negative integer, and the locally compact abelian group V contains a subgroup K which is both compact and open. We note that assertion (c) is a special case of (b) for discrete groups. When speaking about locally compact abelian group G we usually assume that the representation G ' Rc × V in (d) and the subgroup K are fixed. We stress that by (a) the quotient group D = V/K in (d) is discrete. 1.6.2. The Haar measure. A positive non-zero measure λ on a locally compact abelian group G is called a Haar measure if it is shift invariant, i.e., for any summable M ⊆ G R and h ∈ G the setRM + h is summable and λ(M + h) = λ(M ) or, equivalently, G x(g − h) dλ(g) = G x(g) dλ(g) for all x ∈ C00 (G, C). Proposition. (a) ([Bou3 , ch. 7, §1, 2], [Hew, theorem 15.5], [Nai, §27]) On any locally compact abelian group G there exists a Haar measure λ. The Haar measure is determined uniquely up to a constant factor. (b) For any compact set K ⊆ G one has λ(K) < ∞. For any non-empty open set O ⊆ G one has λ(O) > 0.

1.6. INFINITE MATRICES

47

(c) ([Bou3 , ch. 7, §1, 3], [Hew, theorem 15.9]) λ(G) < ∞ if and only if G is compact. (d) ([Bou3 , ch. 7, §1, 3]) If G is discrete then λ({t}) 6= 0 for all t ∈ G. If G is non-discrete then for any t ∈ G and ε > 0 there exists an open neighbourhood U of t such that 0 < λ(U ) < ε; in particular, λ({t}) = 0. (e) Any λ-summable set is σ-compact. Consequently for any x ∈ Lp (G), 1 ≤ p < ∞, the set of all t ∈ G such that x(t) 6= 0 or x(t) is undefined is σ-compact. Cf. 1.5.6. Proof. (b) By Urysohn’s theorem thereR exists a non-negative function ψ ∈ C00 which is equal to 1 on K; clearly, λ(K) ≤ ψ dλ < ∞. And if λ(O) were zero for a non-empty open set O then λ(K) would be zero for all compact sets K and consequently λ would be zero on C00 . (c) Since λ(K) < ∞ for all compact sets K it follows that, in particular, λ(G) < ∞ if G is compact. We prove the converse. Let V be a compact neighbourhood of zero such that λ(V ) > 0. (If there were no such a neighbourhood V then λ would be zero on C00 .) We set U = V − V . Since the image of a compact set under a continuous function is compact U is also a compact neighbourhood of zero. Since G is not compact there exists a sequence {tn } such that tn+1 ∈ / ∪nk=1 (tk + U ). In such a case the sets tn + V are pairwise P∞ disjoint. Since λ is shift invariant λ(tn + V ) = λ(V ) for all n. Thus λ(G) ≥ n=1 λ(tn + V ) = ∞. (d) If G is discrete then λ({t}) > 0 for all t ∈ G because the set {t} is open. We prove the second assertion. Since the Haar measure is shift invariant it suffices to consider the case t = 0. By (a) λ(U ) > 0 for all open neighbourhoods of zero. Let U be an open neighbourhood of zero. Since G is a Hausdorff space and G is non-discrete, U contains two non-empty disjoint open sets U1 and U2 . Clearly, either λ(U1 ) ≤ λ(U )/2 or λ(U2 ) ≤ λ(U )/2. We recall that the shifts of U1 and U2 have the same measures. Therefore the infimum of λ(U ) over all open neighbourhoods U of zero is equal to zero. (e) Let λ(E) < ∞. Then there exists an open set O ⊇ E such that λ(O) < ∞. It suffices to show that O is σ-compact. Obviously, the characteristic function 1O of O is lower semi-continuous. ThereR fore λ(O) = sup{ y dλ : y ∈ CR00 , 0 ≤ y ≤ 1O }. For any n ∈ N we choose yn ∈ S C00 , 0 ≤ yn ≤ 1O , so that | yn dλ| > λ(O) − 1/n. We set Kn = supp yn and ∞ K = n=1 Kn . Clearly, λ(Kn ) > λ(O) − 1/n and consequently λ(K) = λ(O). We observe that the set O \ K does not contain non-empty open subsets. Otherwise λ(K) would be less than λ(O), since the Haar measure of a non-empty open set is not zero. Let V be a compact neighbourhood of the zero of G. It is easy to see that the set V + K contains O. S∞ It suffices to observe that V + K = n=1 V + Kn and that by 1.1.1(a) the sets V + Kn are compact. ¤ Example. If G = Rn the Haar measure on G is the usual Lebesgue measure. We recall that in this case λ is usually normalized so that λ([0, 1]n ) = 1.

48

I. FUNCTIONAL ANALYSIS PRELIMINARIES

If G is discrete we normalize the Haar measure λ so that λ({t}) = 1 for all t ∈ G. If G is compact we normalize λ in such a way that λ(G) = 1. 1.6.3. The spaces Lpq and Cq . Let us consider a locally compact abelian group G of the form Rc × V, see 1.6.1(d). We normalize the Haar measures λ on R and V so that the measures of the sets [0, 1] ⊆ R and K ⊆ V be equal to unity. In such a case the measures of the sets Q = [0, 1]c × K

and

e = (0, 1]c × K Q

are equal to unity, too. We also set D = V/K and consider the discrete group I = Zc × D. We write elements of I as i = (k, d), where k ∈ Zc and d ∈ D. We recall that elements of D are (simultaneously compact and open) subsets of V of the form d = f + K, f ∈ V. Thus elements of I can be considered as subsets of G. We observe that the subsets e i = (k + (0, 1]c ) × d, Q

i = (k, d) ∈ I,

are pairwise disjoint and cover the whole of G. The subsets Qi = (k + [0, 1]c ) × d,

i = (k, d) ∈ I,

possess the similar property of also covering G, with the union of their pairwise e i ) = 1. If one intersections being a locally null set. We stress that λ(Qi ) = λ(Q c ¯ ¯ +Q fixes a point d ∈ d for each d ∈ D, one may identify (k + [0, 1] ) × d with (k, d) ¯ + Q. e and identify (k + (0, 1]c ) × d with (k, d) Proposition. Any compact set K ⊆ G can be covered by a finite number of e i or Qi , i ∈ I. the sets Q Proof. We consider the image of K under the natural projection from Rc × V to Rc × D. By 1.1.1 the image is compact. Hence it is contained in the product of a subset of Rc of the form [−m, m]c , m ∈ N, and a finite subset of D. Clearly, this product possesses a desirable covering. ¤ For any function x : G → E we consider the family { xi : i ∈ I }, where xi is the restriction of x to Qi . Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. ¡We define the ¢ space Lpq = Lpq (G) = Lpq (G, E) to be naturally isomorphic to lq I, Lp (Qi , E) . Namely, the space Lpq consists of all classes of measurable functions x : G → E (with the identification l.a.e.) for which each function xi : Q ¡ ¢ i → E, i ∈ I, belongs to Lp (Qi , E) and the family {xi } belongs ¡ to lq I, L¢p (Qi , E) . We define the norm of x ∈ Lpq to be the norm of {xi } in lq I, Lp (Qi , E) . It is easy to see that Lpq are Banach spaces. We denote by Lpq the set of all functions x : G → E (defined l.a.e.) such that the equivalence class of x belongs to Lpq . We do not discuss (and shall not use) the independence of this definition from the choice of the isomorphism e i. G ' Rc × V. We note that in this definition one may change Qi to Q

49

1.6. INFINITE MATRICES

To simplify notation one may imagine xi , i = (k, d) ∈ I,¡ as the restriction ¢ ¯ to Q, and Lpq (G, E) as lq I, Lp (Q, E) . Here of the function t 7→ x(t − (k, d)) ¡ ¢ d¯ ¡∈ d are fixed ¢ points, see above. Note that the isomorphism lq I, Lp (Qi , E) ' lq I, Lp (Q, E) depends on the choice of d¯ ∈ d. We note that for G = R the norms on Lpq are defined by the following formulae: ÃZ kxk = sup k∈Z

à kxk = à kxk =

!1/p

k+1

p

|x(t)| dt

,

p < ∞, q = 0, ∞,

k

∞ X ¡

¢q ess sup |x(t)|

!1/q

k=−∞ t∈[k,k+1] ∞ ³Z X k=−∞

k+1

´q/p |x(t)|p dt

,

p = ∞, q 6= 0, ∞,

!1/q ,

p < ∞, q 6= 0, ∞.

k

We also note that Lpp coincides with the space Lp arising from the identification l.a.e., see 1.5.6. In particular, L∞∞ = L∞ and L∞0 = L0 . In any case Lpp ' Lp . If G is discrete then Lpq = lq . If G is compact then Lpq = Lp . Remark. We consider the spaces Lpq with two indices for two reasons. First, they arise naturally in the classical theory of differential equations, see, e.g., [DaK] and [MaS]. For example, the operator of multiplication ¡ ¢ Ax (t) = a(t)x(t) is bounded as acting from Wp1 (R, E) to Lp (R, E) if a ∈ Lp∞ . Second, the consideration of the spaces Lpq with two indices is necessary for some of our proofs, see, e.g., 5.3.5 and 6.5.8. Now let x : G → E be a continuous function. We consider the restrictions xi , i ∈ I, of x to Qi . Clearly, each xi belongs to C(Qi , E). Let 1 ≤ q ≤ ∞ or q = 0. We define the spaces Cq = Cq (G) = Cq (G, E) as those consisting of all continuous functions x : G →¢ E for which the family of restrictions xi : Qi → E, i ∈ I, belongs ¡ to lq I, C(Qi , E) . Clearly, the space C∞ coincides with C, and the space C0 coincides with the space C0¡ considered¢in 1.5.1. We define the norm of x ∈ Cq to be the norm of {xi } in lq I, C(Qi , E) . Thus the norms on C∞ and C0 are the same as was defined earlier. If G = R the norms on Cq , q 6= 0, ∞, are à ∞ !1/q X ¡ ¢q kxk = sup |x(t)| . k=−∞ t∈[k,k+1]

We note that the natural embedding of Cq in L∞q is isometric and thus Cq can be interpreted as C ∩ L∞q . Indeed, since open subsets of Qi = (k + [0, 1]c ) × d have non-zero measure the value x ∈ C(Qi , E) at any point t ∈ Qi is an essential value of x considered as an element of L∞ (Qi , E), see 1.5.6 for the definition of an essential value. Hence |x(t)| is all the more an essential value of |x|. Consequently kxkC(Qi ,E) ≤ kxkL∞ (Qi ,E) . The opposite inequality is trivial.

50

I. FUNCTIONAL ANALYSIS PRELIMINARIES

1.6.4. The matrix Prepresentation of operators. Let I be a set of indices. We say that a series j∈I yj of elements yj of a Banach space Y converges unconditionally to y ∈ Y if for any ε > 0 there exists a finite subset H ⊆ I such that for any finite subset K ⊇ H one has ° °X ° ° yj − y ° < ε. ° j∈K

P It is easy to show that if the series converges absolutely, i.e., j∈I kyj k < ∞ then it converges unconditionally. See, e.g., [Kad] for more on unconditional convergence. ¯ i : i ∈ I } be collections of Banach spaces with the Let { Ei : i ∈ I } and { E ¯ i ) : i, j ∈ I } a matrix . norms | · |. We call any family {Tij } = { Tij ∈ B(Ej , E For 1 ≤ q ≤ ∞ or q = 0 we consider the spaces lq = lq (I, {Ei }), and ¯ i }). Assume we have a matrix {Tij }. We say that an operator lq = lq (I, {E T ∈ B(lq , lq ) is induced by the matrix {Tij } if X (T x)i = Tij xj (1) j∈I

for all i ∈ I and x = {xj } ∈ lq with the series converges unconditionally. Clearly, there exist (infinite) matrices which do not induce bounded operators. We note the evident inequality kTij k ≤ kT k, i, j ∈ I. In particular, it implies that the mapping {Tij } 7→ T is injective. We define linear operations with matrices in the natural way. Clearly, the correspondence {Tij } 7→ T preserves the linear operations. We define the product of matrices {Tij } and {Bjk } to be the matrix with the entries X Cik = Tij Bjk j∈I

provided the series converges unconditionally. Of course, the product only makes ¯ i } are correlated. sense if the corresponding families {Ei } and {E ¯ i be the natLet Jj : Ej → lq be the natural embedding, and let Qi : lq → E ural projection. By the matrix of an operator T ∈ B(lq , lq ) we mean the family ¯ i ) } defined by the rule { Tij ∈ B(Ej , E Tij = Qi T Jj . Thus we have obtained two operations {Tij } 7→ T and T 7→ {Tij }. We say that an operator T ∈ B(lq , lq ) is restored by its matrix if its matrix induces an operator and this operator coincides with T . Example. (a) Let l∞ = l∞ (Z, C). Let us consider a non-zero functional f : l∞ → C which vanishes on l0 ⊆ l∞ . The existence of such a functional follows from the Hanh–Banach theorem. Next, we consider the operator (T x)i = f (x)

1.6. INFINITE MATRICES

51

as acting from l∞ into l∞ . Clearly, T is equal to zero on l0 . Therefore all its matrix entries Tij are zeros. But T is not a zero operator. Thus T can not be restored by its matrix. (b) Let E = l∞ (Z, C). We denote by Qi : E → C, i ∈ Z, and Jj : C → E, j ∈ Z, the natural projection and the natural embedding, respectively. We note that Pi = Ji Qi : E → E is the projector onto the i-th co-ordinate. We consider the space l∞ = l∞ (Z, E) and the operator T : l∞ (Z, E) → l∞ (Z, E) defined by the rule (T x)i = { Qi xj : j ∈ Z } ∈ E, where x = { xj ∈ E : j ∈ Z }. Clearly, the matrix entries of T are defined by the rule Tij e = Pi e, e ∈ E. We stress that in this case the series (1) does not converge unconditionally. It is also interesting to note that l0 = l0 (Z, E) is not invariant under T . The following proposition shows that if q 6= ∞ the situation is simpler Proposition. (a) Let 1 ≤ q ≤ ∞ or q = 0. Assume that a matrix {Tij } induces an operator T ∈ B(lq , lq ). Then the matrix of T coincides with the initial matrix {Tij }. (b) Let 1 ≤ q < ∞ or q = 0 (but q 6= ∞). For any operator T ∈ B(lq , lq ) its matrix induces the initial operator T , with the unconditional convergence of the series (1). Proof. (a) follows from the definitions. (b) Let l00 ⊆ lq denote the subspace of all families x = {xi } having only a finite number of non-zero members. We recall that the subspace l00 is dense in lq since q 6= ∞. For given x ∈ lq and ε > 0 we pick z ∈ l00 such that kz − xk < ε. We denote by H ⊆ I the set of all i ∈ I such that zi 6= 0. We take a finite K ⊇ H and define y ∈ l00 as follows: yi = xi for i ∈ K, and yi = 0 if i ∈ / K. Clearly, ky−xk < ε too. Hence kT y − T xk ≤P εkT k and consequently k(T y) i − (T x)i k ≤ εkT k ¯for all ¯P ¯ ¯ i ∈ I. We observe that j∈K Tij xj = (T y)i . Thus j∈K Tij xj − (T x)i ≤ ε, which implies the unconditional convergence of (1). ¤ Assume compact abelian group of the form Rc × V. Then since ¡ G is a locally ¢ Lpq ' lq I, Lp (Qi , E) we can use the matrix representation for operators acting ¯ on Lpq . We ¡ recall that ¢ fixing d ∈ d for each d ∈ D we can also use the isomorphism Lpq ' lq I, Lp (Q, E) and hence corresponding matrices. These will be our main application of matrix representation. 1.6.5. The norm of a matrix ¯ i ) }. Theorem (cf. 1.5.12). We consider a matrix {Tij } = { Tij ∈ B(Ej , E (a) If X K1 = sup kTij k < ∞ j∈I

i∈I

then the matrix {Tij } induces a bounded operator T acting from l1 to l1 and kT : l1 → l1 k ≤ K1 .

52

I. FUNCTIONAL ANALYSIS PRELIMINARIES

(b) If K∞ = sup i∈I

X

kTij k < ∞

j∈I

then the matrix {Tij } induces a bounded operator T acting from l∞ to l∞ and kT : l∞ → l∞ k ≤ K∞ . (c) If both K1 and K∞ are finite then the matrix {Tij } induces a bounded ϑ operator T acting from lq to lq for all 1 ≤ q ≤ ∞ and kT k ≤ K11−ϑ K∞ ≤ max (K1 , K∞ ), where ϑ is defined as in 1.5.11. Proof. The proof is evident. We note that the only locally null subset of a discrete space I is the empty set. ¤ ¡ ¢ ¯ i the 1.6.6. Operators with local memory. We denote by tf = tf I, Ej , E set of all matrices {Tij } such that any row Ti = {Tij : j ∈ I} and any column Tj = {Tij : i ∈ I} of {Tij } contain only a finite number of non-zero entries. We call a matrix belonging to the class tf a matrix with locally bounded memory. We note that some matrices belonging to the class¢tf do not induce bounded ¡ ¯ i ) the set of all (bounded) operators. We ¡denote by tf = ¢tf lq (I, Ej ), lq (I, E ¯ i ) induced by matrices of the class tf . We denote operators¡T ∈ B lq (I, Ej ), lq¢(I, E ¡ ¢ ¯ i ) the closure of tf lq (I, Ej ), lq (I, E ¯ i ) in norm. Note by t = t lq (I, Ej ), lq (I, E that the class t of operators depends on q. c Let G by ¡ be a locally ¢ compact abelian group of the form R ¡× V. We denote ¢ tf = tf Lpq (G, E) the set of all (bounded) operators T ∈ B L (G, E) induced pq © ¡ ¢ ª by matrices T ∈ B L (Q , E), L (Q , E) : i, j ∈ I of the class tf . We denote ij ¢ p j p ¡i ¡ ¢ by t = t Lpq (G, E) the closure of tf Lpq (G, E) in norm. Note that the class t of operators depends on p and q. We call an operator T ∈ tf an operator with locally bounded memory. We call an operator T ∈ t an operator with locally fading memory. Proposition. (a) The set of all matrices of the class tf is a linear space with respect to the natural operations. The product of matrices of the class tf is well defined ¯ i for all i then tf is an and is also a matrix of the class tf . If Ei = E algebra. (b) The set of all operators of the class tf is a linear space. The set of all operators of the class t is a Banach space. The product of operators of the class tf (respectively, t) is also an operator of the class tf (respectively, t). ¯ i for all i then tf is an algebra and t is a Banach algebra. If Ei = E (c) An operator T ∈ t(lq ) is restored by its matrix. Proof. (a) and (b) are plain. (c) Assume Tn ∈ tf converges to T in norm. For a fixed ε > 0 we choose n so that kTn − T k < ε. Then, in particular, k(Tn x)i − (T x)i k < εkxk.

53

1.6. INFINITE MATRICES

We take an arbitrary i ∈ I. Let H = H(i) be the set of all indices j such that (Tn )ij 6= 0. By the definition of tf , H is finite. Assume K ⊇ H is an arbitrary finite set. For any x ∈ lq we denote by xK the family {xK j } defined as follows: K K xj = xj for j ∈ K, and xj = 0 for j ∈ / K. Then we have °X ° ° ° ° ° Tij xj − (Tn x)i ° = °(T xK )i − (Tn xK )i ° ° j∈K

≤ kT − Tn k · kxK k ≤ εkxk. Hence ° ° °X °X ° ° ° ° Tij xj − (Tn x)i ° + k(Tn x)i − (T x)i k Tij xj − (T x)i ° ≤ ° ° j∈K

j∈K

≤ 2εkxk. P Therefore the series j∈I Tij xj converges unconditionally to (T x)i . We shall discuss the classes tf and t more detailed in §5.5.

¤

1.6.7. Operators with uniform memory. Let I be an abelian group with ¯ i ) : i, j ∈ I } be a matrix. We call a discrete topology. Let {Tij } = { Tij ∈ B(Ej , E the set { Tij : i − j = h } the h-th diagonal of the matrix {Tij }. We call the 0-th diagonal the main diagonal. We say that a matrix { Tij : i, j ∈ I } is main diagonal if Tij = 0 for all i 6= j. ¡ ¢ ¯ i the set of all matrices { Tij ∈ B(Ej , E ¯ i) : We denote by df = df I, Ej , E i, j ∈ I } with a finite number of non-zero diagonals and uniformly bounded entries Tij . By the uniform boundedness we mean that there exists K such that kTij k ≤ K for all i and j. Clearly, df ¡⊆ tf . ¢ ¯ i ) the set of all operators induced by We denote by df = df lq (I, Ej ), lq (I, E ¡ ¢ ¯ i ) the closure of matrices of the class d . We denote by d = d l (I, E ), l (I, E f q j q ¡ ¢ ¯ i ) in norm. Note that the class d of operators depends on q. df lq (I, Ej ), lq (I, E c Let G be ¡a locally compact abelian group of the form ¢ ¡ R × V.¢ We denote by df = ©df Lpq (G, ¡ E) the set of all ¢operatorsªT ∈ B Lpq (G, E) induced by matrices Tij ∈ B ¡ ¢ Lp (Qj , E), Lp (Qi¡, E) : i, j ∈ ¢ I of the class df . We denote by d = d Lpq (G, E) the closure of df Lpq (G, E) in norm. Note that the class d of operators depends on p and q. We call an operator T ∈ df an operator with uniformly bounded memory. We call an operator T ∈ d an operator with uniformly fading memory. Proposition. (a) The set of all matrices of the class df is a linear space with respect to the natural operations. The product of matrices of the class df is also a ¯ i for all i then df is an algebra. matrix of the class df . If Ei = E ¡ ¢ ¯ i) , (b) Any matrix {Tij } ∈ df induces an operator T ∈ B lq (I, Ej ), lq (I, E 1 ≤ q ≤ ∞ or q = 0. The mapping {Tij } 7→ T preserves linear operations

54

I. FUNCTIONAL ANALYSIS PRELIMINARIES

¯ i for all i then the mapping {Tij } 7→ T is a and multiplication; if Ei = E morphism from the algebra df into the algebra ¡ B. ¢ ¯ i ) is a linear (c) The set of all operators of the class df = df lq (I, Ej ), l¡q (I, E ¢ ¯ i) space, and the set of all operators of the class d = d lq (I, Ej ), lq (I, E is a Banach space; the product of operators of the class df (respectively, d) is also an operator of the class df (respectively, d). In particular, if ¯ i for all i then df is an algebra and d is a Banach algebra. Ei = E Proof. (a) is evident. (b) It suffices to observe that if {Tij } has only one non-zero diagonal then it induces a bounded linear operator T , and kT k ≤ sup kTij k. (c) follows from (a) and (b). ¤ 1.6.8. Operators with summable memory. For any matrix {Tij } we set αh = sup kTij k. i−j=h

Clearly, this is the supremum norm of the h-th diagonal. ¡ ¢ ¯ i the set of all matrices for which the number We denote by s = s I, Ej , E X {Tij } = αh h∈I

is finite. We call a matrix¡ of the class s a matrix with summable memory. ¢ ¯ We denote by s = s lq (I, Ej ), lq (I, Ei ) the set of all operators induced by matrices of the class s. Since df ⊆ s ⊆ d the closure of s in norm coincides with d. In 5.2.7 we shall prove that s(lq ) is a full subalgebra of the algebra B(lq ). This will imply that d is full as well, see 5.2.9. Let¡ G be a locally compact abelian group of ¡the form R¢c × V. We denote by ¢ s© = s Lpq¡(G, E) the set of all¢ operators ª T ∈ B Lpq (G, E) induced by matrices Tij ∈ B Lp (Qj , E), Lp (Qi , E) : i, j ∈ I of the class s. We call the operator T ∈ s an operator with summable memory. Proposition. (a) The set of all matrices of the class s is a Banach space with respect to the natural operations and the norm · . (b) The product of matrices of the class s is also a matrix of the class s and ¯ i for all i then s is {Cik } ≤ {Tij } · {Bjk } . In particular, if Ei = E a Banach algebra. ¡ ¢ ¯ i ) for all (c) Any matrix {Tij } ∈ s induces an operator T ∈ B lq (I, Ej ), lq (I, E 1 ≤ q ≤ ∞ or q = 0; and kT : lq → lq k ≤ {Tij } . The mapping {Tij } 7→ T preserves linear operations and multiplication; if ¯ i for all i it is a morphism from the algebra s into the algebra B. Ei = E ¡ ¢ ¯ i ) is a linear (d) The set of all operators of the class s = s lq (I, Ej ), lq (I, E space; the product of operators of the class s is also an operator of the ¯ i for all i then s is an algebra. class s. In particular, if Ei = E

55

1.6. INFINITE MATRICES

Proof. (a) is evident (see 1.1.2). (b) Let kTij k ≤ αi−j and kBjk k ≤ βj−k . The estimate X X X X sup kTij k · kBjk k ≤ sup αi−j βj−k h∈I

i−k=h

j∈I

h∈I

≤

i−k=h

XX

αm βh−m

h∈I m∈I

=

µX

m∈I

j∈I

αm

¶µX

¶ βh−m

h∈I

= {Tij } · {Bjk } P implies that the series Cik = j∈I Tij Bjk converges absolutely and {Cik } ≤ {Tij } · {Bjk } . (c) Let {Tij } ∈ s. For any h ∈ I we consider the matrix {Tijh : i, j ∈ I} defined as follows: Tijh = Tij if i − j = h and Tijh = 0 if i − j 6= h. Clearly, {Tijh } induces P an operator T h and kT h k ≤ αh . We observe that the series h∈I T h converges in the norm of B(lq , lq ). It is easy to verify that the matrix {Tij } defines the same P operator T as h∈I T h and kT k ≤ {Tij } . Assume {Tij }, {Bjk } ∈ s induce the operators T and B. As above we consider g the one-diagonal matrices { Tijh }, h ∈ G, and {Bjk }, g ∈ G, and the corresponding g h g operators T and B . It is straightforward to verify that {Tijh } · {Bjk } induces the ´ ³P ´ ³P g h operator T h B g . Consequently g∈G {Bjk } induces the operator h∈H {Tij } · ´ ´ ³P ³P g h B for any finite subsets H, G ∈ I. Since the mapping T · g∈G h∈H {Tij } 7→ T is continuous this implies that {Tij } · {Bjk } induces T B. (d) follows from (a), (b), and (c). ¤ 1.6.9. Operators with exponential memory. Now we assume that I = Zn . For i = (i1 , . . . , in ) ∈ Zn we set |i| = |i1 |+· · ·+|in |. We say that a matrix {Tij } is a matrix with exponentially decreasing memory or, briefly, a matrix with exponential memory if there exist N < ∞ and ν > 0 (of course, N and ν depend on the matrix) such that αh = sup kTij k ≤ N e−ν|h| for all h ∈ Zn . (2) i−j=h

¡ ¢ ¯ i the set of all matrices {Tij } with exponential We denote by e = e Zn , Ej , E memory. ¡ ¢ ¯ i ) the set of all operators induced by We denote by e = e lq (Zn , Ej ), lq (Zn , E matrices of the class e. Let G be a locally compact compactly generated abelian group, i.e., (see 1.6.1) the group of the¢ form Ra × Zb × K. Let us consider the space L¡pq (G, E) '¢ ¡ a+b lq Z , Lp (Q, E) , where Q ¡= [0, 1]c ×¢{0} × K. We denote by © e = e ¡Lpq (G, E)¢ the set of all operators T ∈ B Lpq (G, E) induced by matrices Tij ∈ B Lp (Q, E) : ª a+b of the class e. i, j ∈ I = Z We call an operator T ∈ e an operator with exponential memory.

56

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Proposition. (a) A matrix {Tij } belongs to the class e if and only if there exists γ > 0 such that X {Tij } γ = eγ|h| αh < ∞. (3) h∈Zn

(b) The product of matrices of the class e is a matrix of the class e. In ¯ i for all i then e is an algebra. (This algebra is not particular, if Ei = E complete in the norm k · k or · .) (c) One has df ⊆ e ⊆ s; and df is dense in the norm · ¢. Each matrix ¡ s in n ¯ i) , 1 ≤ q ≤ ∞ {Tij } ∈ e induces an operator T ∈ B lq (Z , Ej ), lq (Zn , E or q = 0; the correspondence {Tij } 7→ T preserves linear operations and ¯ i for all i it is a morphism from the algebra e into multiplication; if Ei = E the algebra B. ¡ ¢ ¯ i ) is a linear (d) The set of all operators of the class e = e lq (Zn , Ej ), lq (Zn , E space; the product of operators of the class e is also an operator of the ¯ i for all i then e is an algebra. class e. In particular, if Ei = E Proof. (a) Clearly, (3) implies (2) with ν = γ. Conversely, assume that (2) is satisfied. We set γ = ν/2. Then we have X X eγ|h| N e−ν|h| eγ|h| αh = h∈Zn

h∈Zn

=N

X

e−γ|h| < ∞.

h∈Zn

(b) With the notation of the proof of 1.6.8(b) we have X X X X eγ|h| sup kTij k · kBjk k ≤ eγ|h| αm βh−m h∈Zn

i−k=h

h∈Zn

j∈I

≤

X X

m∈Zn

eγ|m| eγ|h−m| αm βh−m

h∈Zn m∈Zn

=

µX

m∈Zn

e

γ|m|

αm

¶µ X

¶ e

γ|h−m|

βh−m

h∈Zn

which is less than infinity for sufficiently small γ > 0 by assumption. (c) is evident. (d) follows from (a), (b), and (c). ¤ Since df ⊆ e ⊆ s the closure of e in norm coincides with d. In 3.4.2 we shall prove that e(lq ) is a full subalgebra of the algebra B(lq ), see also 5.2.12. 1.6.10. The representation of Cq as a space of sequences. To define the classes df , d, s, and e on Cq we describe a representation of Cq as lq (I, C♦i ), where, roughly speaking, C♦i is the space of continuous functions on Qi . Unfortunately, it is impossible to choose C♦i in the natural way. Nevertheless, the classes df , d, s, and e do not depend on this choice.

1.6. INFINITE MATRICES

57

Let G = Rc × V and I = Zc × D (see 1.6.3). We define the operators Πi , i = (i1 , . . . , ic , d) ∈ I, acting on the spaces Cq = Cq (G, E) as follows. If G = R, for i ∈ Z we set  0 for t ≤ i,     ¡ ¢ x(t) − (i + 1 − t)x(i) for t ∈ [i, i + 1], Πi x (t) =  (i + 2 − t)x(i + 1) for t ∈ [i + 1, i + 2],    0 for t ≥ i + 2. Now we assume that G = Rc × V. We define Π(i1 ,0,...,0;0) as acting on the first argument of x ∈ Cq (G) by the same rule as Πi acts on x ∈ Cq (R). Namely, for g = (t1 , t2 , . . . , tc ; v) ∈ Rc × V = G and i = (i1 , . . . , ic , d) ∈ I we set ¡ ¢ Π(i1 ,0,...,0;0) x (g)  0 for g ≤ i1 ,     x(g) − (i1 + 1 − t)x(i1 , t2 , . . . , tc ; v) for t1 ∈ [i1 , i1 + 1], =  (i1 + 2 − t1 )x(i1 + 1, t2 , . . . , tc ; v) for t1 ∈ [i1 + 1, i1 + 2],    0 for t1 ≥ i1 + 2. We define Π(0,i2 ,...,0;0) , . . . , Π(0,...,ic ;0) analogously. Further, we set ½ ¡ ¢ x(g) for v ∈ d, Π(0,0,...,0;d) x (g) = 0 for v ∈ / d, where g = (t1 , t2 , . . . , tc ; v) ∈ G. We observe that all these operators pairwise commute. Finally, for i = (i1 , . . . , ic ; d) ∈ I we set Πi = Π(i1 ,0,...,0;0) · . . . · Π(0,...,ic ;0) · Π(0,0,...,0;d) . We denote by C♦i the image of Πi and endow C♦i with the supremum norm. Proposition. The family { Πi : i ∈ I } possesses the following properties: (a) Each Πi , i ∈ I, is a projector, i.e., (Πi )2 = Πi . Moreover, Πi Πj = 0 for i 6= j. The family { Πi : i ∈ I } is bounded. (b) The function Πi x is equal to zero outside [i1 , i1 + 2] × · · · × [ic , ic + 2] × d; here i = (i ¢ P1 , . . . , ic , d). P ¡ (c) One has i∈I Πi = 1. ¡(For¢any x ∈ Cq we define i∈I Πi x (g) pointwise; P note that by (b) i∈I Πi x (g) is finite for any g ∈ G.) (d) The mapping Φ : x 7→ { Πi x } is a topological isomorphism from Cq onto P −1 lq (I, C♦i ). We note that Φ {xi } = i∈I xi (we define this sum pointwise). Clearly, kΦk, kΦ−1 k ≤ 2c . (e) If G = R the projectors Πi are causal, i.e., ¡ ¢ x(s) = 0, s < t, ⇒ Πi x (s) = 0, s < t. Consequently in this case the operators Φ and Φ−1 are also causal, i.e., x(s) = 0, xi = 0,

s < n, i < n,

⇒ ⇒

¡

(Φx)i = 0, i < n, ¢ Φ x (s) = 0, s < n. −1

58

I. FUNCTIONAL ANALYSIS PRELIMINARIES

¡ ¢ We define the classes df , d, s, and e (or more detailed df Cq (G, E) and so on), to be induced by matrices { Tij ∈ B(C♦j , C♦i ) } of the corresponding class. We claim that these definitions do not depend on the choice of the family {Πi } satisfying (a)–(c). Proof. It is straightforward to verify that (a)–(e) holds. Moreover, (d) follows from (a)–(c). Let us prove the independence of the definitions of the classes df , d, s, and e e i } satisfying from the choice of {Πi }. Assume we have two families {Πi } and {Π e♦i be the images of Πi and Π e i ; and let Φ : Cq → lq (I, C♦i ) (a)–(d); let C♦i and C e : Cq → lq (I, C e♦i ) be the isomorphisms in (d). For an arbitrary T ∈ B(Cq ) we and Φ e♦i ) → lq (I, C e♦i ) consider the operators T1 : lq (I, C♦i ) → lq (I, C♦i ) and T2 : lq (I, C defined by the formulae T1 = ΦT Φ−1

and

e Φ e −1 . T2 = ΦT

Clearly, they are connected by the identity e −1 T1 ΦΦ e −1 . T2 = ΦΦ

(4)

e −1 : lq (I, C♦i ) → lq (I, C e♦i ). By Let us consider the matrix of the operator ΦΦ the definition of a matrix one has e −1 )ij = Q e i ΦΦ e −1 Jj , (ΦΦ e i : lq → C e♦i is the natural where Jj : C♦i → lq is the natural embedding, and Q −1 ei Φ e=Π e i : Cq → C e♦i . Thus projection. Clearly, Φ Jj = 1 : C♦j → Cq and Q e −1 )ij = Π e i : C♦j → C e♦i . (ΦΦ We assume for the sake of definiteness that i = (i1 , . . . , ic ; d) and j = (j1 , . . . , jc ; b). e i x is determined by the values of x on [i1 , i1 + 2] × · · · × From (b) it follows that Π e i is zero on C♦j if |im − jm | ≥ 2 for at least one m, or [ic , ic + 2] × d. Therefore Π e −1 )ij } has only a finite number of non-zero diagonals, d 6= b. Thus the matrix {(ΦΦ e −1 )ij }. i.e., it belongs to the class df . Clearly, the same is true for the matrix {(ΦΦ e −1 and ΦΦ e −1 are restored by their matrices. We note that the operators ΦΦ Let us return to (4). Clearly, if T1 belongs to df , d, e, or e then by virtue of 1.6.6–9 T2 also does. By symmetry we can exchange T1 with T2 , and vice versa. ¤ ei = Remark. (a) We recall that by definition Qi = (k + [0, 1]c ) × d and Q c (k + (0, 1] ) × d, i = (k, d) ∈ I. We denote by Cx (Qi ) the subspace of the space e i. C(Qi ) consisting of all functions x ∈ C(Qi ) which are equal to zero on Qi \ Q Clearly, the space C♦i is isomorphic to the space Cx (Qi ). (b) We set C♦ = C♦0 . Let us fix d¯ ∈ d for each d ∈ D. Then we can associate ¯ which is an element of C♦0 , with a function xi = Πi x the function t 7→ xi (t−(k, d)) cf. 1.6.3. Thus we represent Cq as lq (I, C♦ ) with the space C♦ independent of i. Note that the isomorphism lq (I, C♦i ) ' lq (I, C♦ ) depends on the choice of d¯ ∈ d.

59

1.6. INFINITE MATRICES

1.6.11. The cut in Cq (R). In the previous proof we have seen that the isomorphism Φ and its inverse Φ−1 can be considered as operators of the class df . Here we describe the isomorphism Υ between Cq and lq of the class e. Sometimes Υ is more convenient than Φ because it possesses some additional algebraic properties (see 3.1.2 below). We shall use Υ only for the case G = R. So let G = R. For any a ∈ R we define the projector Pa : Cq → Cq as follows ½ ¡ ¢ 0 for t < a, Pa x (t) = x(t) − x(a)e−(t−a) for t ≥ a. Clearly, Pa is causal, i.e., x(s) = 0,

s < t,

⇒

(Pa x)(s) = 0,

s < t.

Furthermore, the family {Pa } possesses the special property Pa Pb = Pb Pa = Pb

for a < b.

The family {Pa } is related closely to the family {Πi } considered in the previous ˇ i = Pi − Pi+1 , i ∈ Z, possess some properties of Πi . And subsection. Namely, Π P conversely, Pˆi = k≥i Πk , i ∈ Z, possess some properties of Pa . Proposition. Let C♥i denote the image of the projector Pi − Pi+1 . We endow C♥i with the supremum norm. Then the mapping Υ : x 7→ { (Pi − Pi+1 )x : i ∈ Z } is a topological isomorphism from Cq onto lq (Z, C♥i ). Proof. The explicit formula   0 ¡ ¢ x(t) − x(i)e−(t−i) (Pi − Pi+1 )x (t) =   x(i + 1)e−(t−i−1) − x(i)e−(t−i)

for t ≤ i, for i ≤ t ≤ i + 1, for t ≥ i + 1

shows that Υ acts continuously from Cq into lq . Let us consider {xi } ∈ lq . We note that xi (t) = xi (i + 1)e−(t−i−1) for t ≥ i + 1. Assume that t ∈ [k, k + 1]. Then ¯ ¯¡ −1 ¢ ¯ ¯X ¯ ¯ Υ {xi } (t)¯ = ¯¯ xi (t)¯ i≤k

≤ |xk (t)| + ≤

X

X

|xi (i + 1)e−(t−i−1) |

i≤k−1

kxi ke−(k−i−1) .

i≤k

In particular, this series converges uniformly on [k, k + 1]. Consequently the function Υ−1 {xi } is continuous. We recall from 1.6.2 ¡ that the norm¢ on Cq (R, E) is defined by virtue of the natural mapping into lq Z, C([i, i + 1], E) . The last estimate shows that this mapping belongs to the class e. By 1.6.9(c) it follows that Υ−1 {xi } is bounded in the norm of Cq . ¤

60

I. FUNCTIONAL ANALYSIS PRELIMINARIES

1.6.12. The shift operators. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let G be a locally compact abelian group of the form Rc × V. Proposition. We consider the shift operator ¡

¢ Sh x (t) = x(t − h),

h ∈ G.

(a) The shift operator acts continuously on the spaces Lp (G, E), Lpq (G, E), and Cq (G, E). Moreover, Sh : Lpq → Lpq ≤ 2c and Sh : Cq → Cq ≤ 3c , and Sh ∈ df . Nevertheless, kSh : Cq → Cq k ≤ 2c . (b) If p, q 6= ∞ the operator Sh depends on h strongly continuously (see 1.1.5 for the definition). We define the shift operator Sh on the conjugate spaces (Lp )0 , (Lpq )0 , and (Cq )0 as the conjugate of S−h . Proof. (a) Clearly, Sh acts on the spaces Lp , C, and C0 with kSh k = 1. To define it correctly on Lpq we make use of its matrix representation. Assume G = R and h ∈ m + (0, 1], where m ∈ Z. It is straightforward to verify that the values of Sh x on i + (0, 1], i ∈ Z, are determined by the values of x on [i − h, i − h + 1) ⊆ (i − m − 1, i − m + 1). Thus the matrix of Sh contains only two non-zero diagonals, namely, (Sh )ij can be non-zero only if m ≤ i − j ≤ m + 1. In a similar way, if G = Rc and h ∈ m + (0, 1]c , where m ∈ Zc , then (Sh )ij can be non-zero only if m ≤ i − j ≤ m + 1 (the relation ‘≤’ is considered in the componentwise sense, i.e., m1 ≤ i1 − j1 ≤ m1 + 1, . . . , mc ≤ ic − jc ≤ mc + 1). Thus the matrix of Sh contains at most 2c non-zero diagonals. If G = V (see 1.6.2) then (Sh )ij can be non-zero only if i − j = h + K ∈ D. Thus the matrix contains only one non-zero diagonal. In the case of an arbitrary G of the form Rc × V an entry (Sh )ij can be non-zero only if m ≤ k − l ≤ m + 1 and d − b = e + K, where h = (m, e), i = (k, d), and j = (l, b). Thus the matrix of Sh contains at most 2c non-zero diagonals. Since kSh : Lp (G, E) → Lp (G, E)k = 1 we have k(Sh )ij k ≤ 1. Nevertheless, if one uses the isomorphism Cq (R) ' lq (Z, C♦i ), in general the matrix of Sh contains three non-zero diagonals. But if one uses the isomorphism Cq (R) ' lq (Z, C♥i ) it contains at most two non-zero diagonals. The verification is reduced to direct calculations. Therefore the matrix of Sh in Cq (G, E) may contain at most 3c non-zero diagonals. The estimate kSh : Cq → Cq k ≤ 2c holds since Cq is embedded isometrically in L∞q . Evidently, in all cases Sh is restored by its matrix. (b) If p, q = 6 ∞ the subspace C00 is dense in Lp , Lpq , and Cq . Clearly, for x ∈ C00 , Sh x depends on h continuously in the norm of C and consequently in the norms of Lp , Lpq , and Cq . Since the family {Sh } is uniformly bounded it depends on h strongly continuously over the whole of Lp , Lpq , and Cq . ¤ Remark. The norm on Lpq is not shift invariant, i.e., kSh xk 6= kxk. But, clearly, a shift invariant norm exists. This is, e.g., kxk∗ = sup{ kSh xk : h ∈ G }.

1.7. TENSOR PRODUCTS

61

1.7. Tensor products Tensor products are an abstract model for spaces of vector functions and functions of several variables, see 1.7.4. In this book we apply them mainly to vectorvalued situations. This causes asymmetry in our notation. The principal results of this section are theorems 1.7.8 and 1.7.10 which generalize the spectral mapping theorem and the Gel0 fand transform to the operator-valued case. 1.7.1. Dual families. We recall that the Kronecker symbol is the family of numbers δkl , k, l = 1, . . . , n, such that δkl = 1 if k = l, and δkl = 0 if k 6= l. Let X be a Banach space and X 0 be the conjugate of it. Families x1 , . . . , xn ∈ X and x01 , . . . , x0n ∈ X 0 are called dual if hxk , x0l i = δkl . Proposition. (a) A family x1 , . . . , xn ∈ X is linearly independent if and only if it has a dual family x01 , . . . , x0n ∈ X 0 . (b) A family x01 , . . . , x0n ∈ X 0 is linearly independent if and only if it has a dual family x1 , . . . , xn ∈ X. Proof. Clearly, the existence of a dual family implies linear independence. We prove the converse. (a) follows from the Hanh–Banach theorem. (b) is reduced to (a) as follows. We set N = ∩nl=1 Ker x0l and denote by e0n are x e0l : X/N → C the functionals induced by x0l (see 1.2.2). Obviously, x e01 , . . . , x linearly independent. We observe that X/N is finite-dimensional. Consequently e0n has a dual family, say x e1 , . . . , x en ∈ X/N . X/N is reflexive. Thus by (a) x e01 , . . . , x We pick arbitrary xk ∈ x ek . Then the family x1 , . . . , xn is dual of x01 , . . . , x0n . ¤ 1.7.2. The definition of an algebraic tensor product. Let E and X be linear spaces. We denote by V the linear space of all formal sums X v= λ(e,x) (e ⊗ x), e∈E, x∈X

where λ(e,x) ∈ C, and a finite number of summands only is non-zero. Clearly, one can imagine PnV as the space l00 (E × X, C). For simplicity of notation we shall write v as v = k=1 λk (ek ⊗ xk ). We denote by V0 the span in V of all expressions of the form (e1 + e2 ) ⊗ x − e1 ⊗ x − e2 ⊗ x,

λ(e ⊗ x) − (λe) ⊗ x,

e ⊗ (x1 + x2 ) − e ⊗ x1 − e ⊗ x2 ,

λ(e ⊗ x) − e ⊗ (λx).

(We omit the factors 1.) The quotient space V /V0 is called the (algebraic) tensor product of E and X and denoted by E ⊗ X. The natural projection of e ⊗ x into V /V0 is denoted by the same symbol e⊗x and called the tensor product of e and x. With this convention any element v ∈ E ⊗ X can be represented as a finite sum n X v= ek ⊗ xk . k=1

Notice that by definition such a representation is not unique.

62

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Lemma. Let E P and X be Banach spaces. Then any v ∈ E ⊗ X possesses the n representation v = k=1 ek ⊗ xk with e1 , . . . , en being linearly independent. For such a representation v = 0 implies xk = 0. Pn Proof. We consider v = k=1 ek ⊗ xk . We denote by F the span of e1 , . . . , en and choose Pm a basis f1 , . . . , fm in F . Representing each ek as the linear combination ek = l=1 λkl fl we arrive at the required equality v=

n µX m X k=1

¶ λkl fl

⊗ xk =

l=1

m X l=1

fl ⊗

µX n

¶ λkl xk .

k=1

Pn Now we consider v = k=1 ek ⊗ xk with linearly independent e1 , . . . , en . We show that v = 0 implies xk = 0. We pick e01 , . . . , e0n ∈ E0 such that hek , e0l i = δkl . Let V and V0 be the same as in the definition of the tensor product. We consider the mapping e0l ⊗ 1 : V → X defined on e ⊗ x ∈ V as he, e0l ix and continued by linearity to the whole of V . Obviously, V0 ⊆ Ker e0l ⊗ 1. Thus e0l ⊗ 1 induces the mapping from E ⊗ X into X. Clearly, the last mapping takes v to xl . Therefore if v = 0 then xl = 0, too. ¤ Example. (a) Let E and X be finite-dimensional spaces with bases e1 , e2 , . . . , en and x1 , x2 , . . . , xm , respectively. Then, clearly, the collection {ek ⊗ xl } forms a basis in E ⊗ X. Thus the dimension of E ⊗PX is equal to nm. (b) Let E be a n finite-dimensional any element k=1 ak ⊗ b0k ∈ E ⊗ E0 induces the Pn space. Then operator x 7→ k=1 ak hx, b0k i. Thus we arrive at the natural isomorphism of the space E ⊗ E0 and the algebra of all linear operators acting on E. (c) Let E be a finite-dimensional space with a basis e1 ,P e2 , . . . , en . Then any element v ∈ E ⊗ X n can be represented uniquely in the form k=1 ek ⊗ xk , where xk ∈ X. Thus E ⊗ X is isomorphic to the direct sum X ⊕ · · · ⊕ X of n copies of X. 1.7.3. The definition of a topological tensor product. Now let E and X be Banach spaces. A non-degenerate norm α(·) = k · kα on E ⊗ X is called a cross-norm if ke ⊗ xkα = kek · kxk

for all e ∈ E and x ∈ X.

(1)

¡ ¢1/2 Example. We consider R2 with the Euclidean norm kxk2 = |x1 |2 + |x2 |2 . 2 2 2 Let {e1 , e2 } be an orthonomal basis in R . By the example in 1.7.2 each v ∈ R ⊗R P2 can be represented uniquely in the form v = i,j=1 cij (ei ⊗ ej ), cij ∈ R. It is ¡ ¢1/2 straightforward to verify that the semi-norm kvk = |c11 + c22 |2 + |c12 − c21 |2 satisfies (1). We stress that this semi-norm is degenerate. We denote the space E⊗X equipped with a cross-norm α by the symbol E⊗α X. If both E and X are infinite-dimensional the space E ⊗α X is non-complete. We denote the completion of E⊗α X by the symbol E ⊗α X and call it the (topological ) tensor product of E and X. A cross-norm on E ⊗ X is essentially not unique. So there can exist different completions of E ⊗ X.

63

1.7. TENSOR PRODUCTS

For v ∈ E ⊗ X we define the quantities ε(v) = kvkε and π(v) = kvkπ as follows: ¾ ½ ¯X n ¯ ¯ 0 0 ¯ 0 0 0 0 0 0 kvkε = sup ¯ hek , e i · hxk , x i¯ : e ∈ E , ke k ≤ 1, x ∈ X , kx k ≤ 1 ,

(2)

k=1

Pn where v = k=1 ek ⊗ xk and the supremumP is taken over all e0 ∈ E0 , ke0 k = 1, and n 0 0 x0 ∈ X 0 , kx0 k = 1. We stress Pnthat the sum k=1 hek , e i · hxk , x i does not depend on the representation v = k=1 ek ⊗ xk , cf. 1.7.5 below. Further, we set kvkπ = inf

n nX

kek k · kxk k : v =

k=1

n X

o ek ⊗ x k ,

k=1

where the infimum is taken over all representations v =

Pn

k=1 ek

⊗ xk .

Proposition. (a) For any semi-norm α(·) = k · kα on E ⊗ X satisfying (1) one has kvkα ≤ kvkπ

for all v ∈ E ⊗ X.

(b) k · kε and k · kπ are cross-norms. (c) Any element v ∈ E ⊗π X can be represented in the form v=

∞ X

ek ⊗ xk ,

(3)

k=1

P∞

And k=1 kek k · kxk k < ∞, cf. 1.1.2. P∞ kvk is the infimum of ke k · kx k over all representations v = k k k=1 k=1 ek ⊗ xk .

where P∞

Of course, representation (3) is not unique. Proof. (a) follows immediately from the definition of π. (b) Clearly, k · kε and k · kπ are semi-norms on E ⊗ X. Evidently, k · kε satisfies (1). We show that Pn k·kπ also satisfies (1). Assume that v = e⊗x has Pnanother representation v = k=1 ek ⊗ xk . It suffices to show that kek · kxk ≤ k=1 kek k · kxk k. We take e0 ∈ E 0 , ke0 k = 1, and x0 ∈ X 0 , kx0 k = 1, such that he, e0 i = kek and hx, x0 i = kxk. Clearly, kek · kxk = he, e0 i · hx, x0 i = ≤

n X k=1 n X

hek , e0 i · hxk , x0 i kek k · kxk k.

k=1

Let us verify that ε is non-degenerate. We consider an arbitrary v ∈ E ⊗ X. We show that kvkε = 0 implies v = 0. By 1.7.2 we choose a representation v =

64

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Pn

⊗ xk with linearly independent e1 , . . . , en . We pickPe0l , l = 1, . . . , n, such n that hek , e0l i = δkl . For any x0 ∈ X 0 , from kvkε = 0 we have k=1 hek , e0l ihxk , x0 i = 0, which implies hxl , x0 i = 0. Hence xl = 0 for all l. Consequently v = 0. Since kvkε ≤ kvkπ for all v, the cross-norm k · kπ is non-degenerate, too. P (c) We denote by V the linear space of all finite formal sums v = λe⊗x (e ⊗ x) P (see 1.7.2) and endow V with the l1 -norm kvk = |λe⊗x | · kek · kxk. Clearly, k · kπ is the norm of the quotient space on V /V0 . We denote by V the completion of V and by V0 the closure of V0 in V . Obviously, the space V /V0 is isometrically isomorphic to the dense subspace V /V0 of V /V0 . Hence the completion of V /V0 to observe that one P may imagine V as the space coincides with V /V0 . It remains P of all countable sums v = λe⊗x (e ⊗ x) with kvk = |λe⊗x | · kek · kxk < ∞. ¤ k=1 ek

Example. If E and X are Hilbert spaces then the inner product n DX

ek ⊗ xk ,

k=1

m X

n X m E X cl ⊗ yl = hek , cl i · hxk , yl i

l=1

k=1 l=1

on E ⊗ X, where ek , cl ∈ E and xk , yl ∈ X, induces a cross-norm. 1.7.4. Examples of topological tensor products Theorem ([Gro, ch. 1], [Scha, ch. 4, §6]). Let E be a Banach space. (a) Let K and N be compact topological spaces. Then E ⊗ε C(K, C) ' C(K, E)

and

C(K, C) ⊗ε C(N, C) ' C(K × N, C).

(b) Let T and S be locally compact topological spaces. Then E ⊗ε C0 (T, C) ' C0 (T, E)

and

C0 (T, C) ⊗ε C0 (S, C) ' C0 (T × S, C).

(c) Let T and S be locally compact topological spaces with positive measures. Then E ⊗π L1 (T, C) ' L1 (T, E)

and

L1 (T, C) ⊗π L1 (S, C) ' L1 (T × S, C).

These isomorphisms are natural and isometric. Proof. The second isomorphisms follow from the first ones and 1.5.8 and 1.5.9. Pn (a) Let V denote the space of all finite formal sums k=1 Pn λk (ek ⊗ xk ), see 1.7.2. Pn Clearly, the natural assignment k=1 λk (ek ⊗ xk ) 7→ k=1 λk ek xk from V to C(K, E) maps the subspace V0 ⊆ V to zero. Thus we obtain the quotient mapping Φ : E ⊗ C(K, C) → C(K, E). Pn We take v = k=1 ek ⊗ xk ∈ E ⊗ C(K, C) and consider its image ¡ ¢an arbitrary Pn z(t) = Φv (t) = k=1 ek xk (t).

1.7. TENSOR PRODUCTS

65

¡ ¢0 Let e0 ∈ E0 and x0 ∈ C(K, C) . We set ze0 (t) = hz(t), e0 i. Clearly, |ze0 (t)| ≤ |z(t)| · ke0 k. With this notation we have n n ¯X ¯ ¯D³X ´ E¯ ¯ ¯ ¯ 0 0 ¯ hek , e0 ixk , x0 ¯ ¯ hek , e i · hxk , x i¯ = ¯ k=1

¯ k=10 ¯ = ¯hze0 , x i¯ ≤ kzk · ke0 k · kx0 k.

Taking in this inequality the supremum over all e0 , ke0 k ≤ 1, and x0 , kx0 k ≤ 1, we obtain the estimate kvkε ≤ kzk. Next, we assume that the maximum of the function t 7→ |z(t)| is attained at a ¡ ¢0 point t0 . Let δt0 ∈ C(K, C) be the functional defined by the rule hx, δt0 i = x(t0 ). We choose e0 ∈ E0 such that ke0 k = 1 and hz(t0 ), e0 i = |z(t0 )| = kzk. Then n ¯X ¯ ¯ ¯ ¯ hek , e0 i · hxk , δt0 i¯ = kzk. k=1

Taking the supremum we arrive at the opposite inequality kvkε ≥ kzk. Thus Φ is an isometry. Evidently, the image of Φ contains all finite-dimensional functions x ∈ C(K, E). By 1.5.3 such functions form a dense subset of C(K, E). Consequently the extension of Φ by continuity is an isomorphism from E ⊗ε C(K, C) onto C(K, E). (b) Let Te be the one point compactification of T , see 1.1.1. We recall that C0 (T, E) can be interpreted as the subspace of C(Te, E) consisting of all functions vanishing at ∞. Clearly, the restriction of the isomorphism from (a) to this subspace is the one needed. Pn (c) Let us take an arbitrary v = k=1 ek ⊗ xk ∈ E ⊗ C00 (T, C) and consider its Pn image z = k=1 ek xk P in C00 (T, E). We endow C00 (T, C) and C00 (T, E) with the n L1 -norms. Evidently, k=1 kek k · kxk kL1 ≥ kzkL1 . Hence by the definition of π kvkπ ≥ kzkL1

for all v ∈ E ⊗ C00 (T, C).

Consequently the mapping Φ : v 7→ z is continuous, and therefore it can be extended to Φ : E ⊗π L1 (T, C) → L1 (T, E). Next, we assume that xk are characteristic functions of pairwise disjoint summable subsets Ek ⊆ T . Then, clearly, kvkπ ≤ kzkL1 . Since the family of all such z is dense in L1 (T, E), the image of Φ coincides with L1 (T, E) and Φ is an isomorphism. ¤

66

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Remark. Assume T is a locally compact space with a positive measure, and 1 ≤ p ≤ ∞ or p = 0. Clearly, the natural embedding j : E ⊗ Lp (T, C) → Lp (T, E) induces a cross-norm on E⊗Lp (T, C). If 1 ≤ p < ∞, the completion of E⊗Lp (T, C) by this cross-norm is isometrically isomorphic to Lp (T, E) because the image of E ⊗ Lp (T, C) is dense in Lp (T, E). If p = 0, ∞ this is not the case. Namely, one can show that the space E ⊗ε L∞ (T, C) is isometrically isomorphic to the subspace of L∞ (T, E) consisting of all functions with compact locally essential image, and analogously for p = 0. Cf. the example in 1.4.12. Similarly, if T is not compact the space E ⊗ε C(T, C) is isometrically isomorphic to the proper subspace of C(T, E) consisting of all functions with compact image. 1.7.5. conjugate cross-norm. Let us consider an arbitrary element Pm The 0 0 v = l=1 el ⊗ xl ∈ E0 ⊗ X 0 . It is easy to verify that the rule 0

n X

ek ⊗ xk 7→

k=1

n DX

ek ⊗ xk ,

m X

k=1

l=1

e0l

⊗

x0l

E =

m X n X

hek , e0l i · hxk , x0l i

l=1 k=1

defines a functional on E ⊗ X. (Strictly speaking, one should first define the action of v 0 on V and then observe that v 0 P is equal to zero on V0 . Moreover, one should m check that equivalent formal sums l=1 λl (e0l ⊗ x0l ) induce the same functional. These verifications are plain and omitted.) We recall that we consider E ⊗ X without norm. Thus we have the natural mapping j : E0 ⊗ X 0 → (E ⊗ X)∗ , where (E ⊗ X)∗ is the space of all linear (not obligatorily bounded) functionals on E ⊗ X. Proposition. The natural mapping j : E0 ⊗ X 0 → (E ⊗ X)∗ is injective. Proof. We consider an arbitrary v 0 ∈ E0 ⊗ X 0 . We show that jv 0 = 0 implies P n v 0 = 0. By virtue of 1.7.2 we choose a representation v 0 = k=1 e0k ⊗ x0k with e1 , . . . , en such that hel , e0k i = δkl . For all linearly independent e01 , . . . , e0n­ and pick ® P n x ∈ X, from jv 0 = 0 we have el ⊗ x, k=1 e0k ⊗ x0k = hx, x0l i = 0, which in turn implies x0l = 0. ¤ According to this proposition we identify E0 ⊗X 0 with the subspace of (E⊗X)∗ , and we do not distinguish elements of E0 ⊗ X 0 and functionals by them. Pn induced 0 0 0 Assume we have a cross-norm α on E ⊗ X. For any v = k=1 ek ⊗ xk ∈ E0 ⊗ X 0 we define α0 (v 0 ) = kv 0 kα0 to be the norm of the functional jv 0 on E ⊗α X, i.e., kv 0 kα0 = sup{ |hv, v 0 i| : kvkα ≤ 1, v ∈ E ⊗α X } or more detailed m °X ° ° ° e0l ⊗ x0l ° ° l=1

α0

½¯X ¾ m X n n ¯ °X ° ¯ ° 0 0 ¯ ° = sup ¯ hek , el i · hxk , xl i¯ : ° ek ⊗ xk ° ≤ 1 . l=1 k=1

k=1

α

(We do not exclude the case kv 0 kα0 = ∞.) We say that a cross-norm α on E ⊗ X is ∗-uniform if for all e0 ∈ E0 and x0 ∈ X ke0 ⊗ x0 kα0 = ke0 k · kx0 k.

67

1.7. TENSOR PRODUCTS

0 0 (Note that the inequality ke0 k · kx0 k is always true.) Clearly, in this Pm 0 ke0 ⊗ x 0kα0 ≥ case all elements l=1 el ⊗xl ∈ E ⊗X 0 induce bounded functionals, i.e., the image of j : E0 ⊗ X 0 → (E ⊗ X)∗ is situated in (E ⊗α X)0 . We stress that the definition of a ∗-uniform cross-norm α means that the norm α0 on E0 ⊗ X 0 induced by α is a cross-norm, too. We denote the completion of E0 ⊗α0 X 0 by E0 ⊗α0 X 0 and identify E0 ⊗α0 X 0 with the closed subspace of (E ⊗α X)0 .

1.7.6. The Schatten theorem Theorem ([Schn]). Let E and X be Banach spaces. (a) The norm π 0 on E0 ⊗ X 0 coincides with ε. In particular, π is ∗-uniform. (b) The cross-norm ε is ∗-uniform. Consequently by 1.7.3(a) kv 0 kε0 ≤ kv 0 kπ for all v 0 ∈ E0 ⊗ X 0 . (c) A cross-norm α on E ⊗ X is ∗-uniform if and only if kvkε ≤ kvkα ≤ kvkπ

for all v ∈ E ⊗ X.

(4)

(d) If the space E is finite-dimensional, all ∗-uniform cross-norms α on E ⊗ X are equivalent and the space E ⊗α X is complete. Remark. Assertion (a) implies evidently that E0 ⊗ε X 0 can be identified naturally with a closed subspace of (E ⊗π X)0 , but in general it is not true that E0 ⊗ε X 0 coincides with the whole of (E ⊗π X)0 . We give a counter-example. If it were true then for any Banach space E the conjugate of L1 (T, E) ' E ⊗π L1 (T, C) would be isomorphic to E0 ⊗ε L∞ (T, C). But E0 ⊗ε L∞ (T, C) only is the closed subspace of L∞ (T, E0 ) consisting of all functions with compact locally essential image, see remarks in 1.4.12 and 1.7.4. Moreover, L∞ (T, E0 ) itself only is a closed subspace of L1 (T, E)0 , see [Bou3 , ch. 6, §2, 6] and [Edw, §8.18]. Proof. (a) We consider the cross-norm π on E ⊗ X. We show that the conjugate norm 0

kv kπ0

½¯ X ¾ m X n n ¯ °X ° ¯ ° ° 0 0 ¯ = sup ¯ hek , el i · hxk , xl i¯ : ek ∈ E, xk ∈ X, ° ek ⊗ xk ° ≤ 1 l=1 k=1

k=1

π

Pm 0 0 0 0 0 of v 0 = l=1 el ⊗ xl ∈ E ⊗ X coincides with kv kε . We observe that by the definition of π we can rewrite kv 0 kπ0 equivalently as 0

kv kπ0

½¯X ¾ m X n n ¯ X ¯ 0 0 ¯ = sup ¯ hek , el i · hxk , xl i¯ : ek ∈ E, xk ∈ X, kek k · kxk k ≤ 1 . l=1 k=1

k=1

Next, we show that kv 0 kπ0 coincides with 0

kv k∗ε

½ ¯X ¾ m ¯ ¯ 0 0 ¯ = sup ¯ he, el i · hx, xl i¯ : e ∈ E, kek ≤ 1, x ∈ X, kxk ≤ 1 . l=1

68

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Indeed, kv 0 kπ0 ≥ kv 0 k∗ε because in the latter formula for k · kπ0 the supremum is taken over the wider set. The opposite inequality follows from the estimate m ¯X ¯ ¯ 0 0 ¯ ¯ hek , el i · hxk , xl i¯ ≤ kv 0 k∗ε · kek k · kxk k. l=1

Finally, we show that kv 0 k∗ε = kv 0 kε . We consider kv 0 kε . By definition (2) ½ ¯X ¾ m ¯ ¯ 0 00 0 00 0 ¯ 00 00 00 00 00 00 kv kε = sup ¯ he , el i · hx , xl i¯ : e ∈ E , ke k ≤ 1, x ∈ X , kx k ≤ 1 . l=1 0

Therefore kv kε ≥ kv 0 k∗ε . The opposite inequality follows from 1.1.9. (b) Let us consider v 0 ∈ E0 ⊗ X 0 of the form v 0 = e0 ⊗ x0 . We must show that ke0 ⊗ x0 kε0 ≤ ke0 k · kx0 k. Without loss of generality we may assume that ke0 k = 1 and kx0 k = 1. By the definition of ε it follows that kvkε ≤ 1, v ∈ E ⊗ X, implies |hv, e0 ⊗ x0 i| ≤ 1. Thus ke0 ⊗ x0 kε0 ≤ 1. (c) Let assumption (4) be fulfilled. From (4) we have kv 0 kε0 ≥ kv 0 kα0 ≥ kv 0 kπ0

for all v 0 ∈ E0 ⊗ X 0 .

By (a) and (b), π 0 = ε and ε0 ≤ π. Since π and ε are cross-norms, so is α0 . Conversely, assume α0 is a cross-norm. We prove (4). By 1.7.3(a) the inequality kvkα ≤ kvkπ holds for any cross-norm α. So it remains to prove that kvkα ≥ kvkε . We assume the contrary: let kvkα < kvkε for some v ∈ E ⊗ X. Then by (2) there exist e0 ∈ E0 , ke0 k ≤ 1, and x0 ∈ X 0 , kx0 k ≤ 1, such that kvkα < hv, e0 ⊗ x0 i. From this inequality, by the definition of k · kα0 we obtain hv, e0 ⊗ x0 i ke0 ⊗ x0 kα0 ≥ > 1. kvkα This contradicts the inequality ke0 ⊗ x0 kα0 ≤ ke0 k · kx0 k ≤ 1. (d) Let e1 , . . . , en be a basis in E. We represent E ⊗ X as the direct sum n X = X ⊕ · · · ⊕ X, see the example in 1.7.2. We consider on X n ' E ⊗ X the norms n X kvk1 = kek k · kxk k and kvk∞ = max kek k · kxk k, k

k=1

Pn where v = k=1 ek ⊗ xk . Clearly, k · k1 and k · k∞ are equivalent. It is easy to see that kvkπ ≤ kvk1

for all v ∈ E ⊗ X.

Next, we fix v ∈ E ⊗ X and assume that maxk kek k · kxk k is attained at k = k0 . We set e0 = e0k0 , where e01 , . . . , e0n is the dual basis in E0 , and take x0 ∈ X 0 , kx0 k = 1, Pn 0 0 0 0 such that hxk0 , x0 i = kxk0 k. Then k=1 hek , e /ke ki · hxk , x i = kxk0 k/kek0 k. Hence from (2) it follows that kvkε ≥ kxk0 k/ke0k0 k = (kxk0 k · kek0 k)/(ke0k0 k · kek0 k) = kvk∞ /(ke0k0 k · kek0 k). Consequently kvk∞ ≤ (max ke0k k · kek k) · kvkε k

for all v ∈ E ⊗ X.

Finally, we recall from (4) that kvkε ≤ kvkπ . Thus k·kπ and k·kε are equivalent to the Banach norms k · k1 and k · k∞ . ¤

1.7. TENSOR PRODUCTS

69

Example. Let E be finite-dimensional. We claim that the conjugate of E⊗π X is isometrically isomorphic to E0 ⊗ε X 0 and the conjugate of E⊗ε X is isometrically isomorphic to E0 ⊗π X 0 . The first assertion follows immediately from (a). We prove the second one. By (d) the space E ⊗ε X is complete and the norm ε on it is equivalent to the norm π. Hence the cross-norms ε0 and π 0 = ε (see (a)) on E0 ⊗ X 0 are equivalent, too. Applying (d) once more we obtain that ε0 on E0 ⊗ X 0 is equivalent to π. We prove that they are equal. We consider the space E0 ⊗ X 0 with the cross-norm π. By (a) it induces the cross-norm ε on E ⊗ X 00 . We begin with the proving of the auxiliary assertion: the 00 natural Pn embedding of E ⊗ε X in E ⊗ε X is isometric. By the definition (2), for v = k=1 ek ⊗ xk ∈ E ⊗ X we have ½ ¯X ¾ n ¯ ¯ 0 0 ¯ 0 0 kvkE ⊗ε X = sup ¯ hek , e i · hxk , x i¯ : ke k ≤ 1, kx k ≤ 1 , k=1

kvkE ⊗ε X 00 0

0

0

½ ¯X ¾ n ¯ ¯ 0 000 ¯ 0 000 = sup ¯ hek , e i · hxk , x i¯ : ke k ≤ 1, kx k ≤ 1 ; 0

k=1 000

here e ∈ E , x ∈ X , and x ∈ X 000 . Since any x0 ∈ X 0 can be extended to x000 ∈ X 000 with kx000 k = kx0 k we have kvkE ⊗ε X ≤ kvkE ⊗ε X 00 . On the other hand, since the restriction of x000 ∈ X 000 to X is an element x0 ∈ X 0 with kx0 k ≤ kx000 k, we have the opposite inequality kvkE ⊗ε X 00 ≤ kvkE ⊗ε X . We assume the contrary: let ε0 be not equal to π on E0 ⊗ X 0 . First we show that ε00 is then not equal to π 0 on E00 ⊗ X 00 . Indeed (see (b)), there exists v 0 ∈ E0 ⊗ X 0 such that kv 0 kε0 < 1 and kv 0 kπ = 1. Then by the Hanh–Banach theorem there exists v 00 ∈ E00 ⊗ X 00 such that hv 00 , v 0 i = 1 and kv 00 kπ0 = 1. Clearly, kv 00 kε00 > 1. Thus the unit balls Bε00 and Bπ0 in E00 ⊗ X 00 = E ⊗ X 00 induced by ε00 and π 0 , respectively, do not coincide. On the other hand, by our auxiliary assertion the intersections of Bε00 and Bπ0 with E ⊗ X is the same set — the unit ball in E ⊗ X with respect to ε. By 1.1.9 its ∗-weak closure is the unit ball in E00 ⊗ X 00 . We observe that the ∗-weak topology is defined by the set E0 ⊗X 0 and does not depend on the norm implicitly. Thus the closure must be the same. This is a contradiction. 1.7.7. The tensor product of operators. Let E and X be P Banach spaces, m and let Al ∈ B(E) and Ml ∈ B(X), l = 1, . . . , m. Then the element l=1 Al ⊗Ml ∈ B(E) ⊗ B(X) defines the linear operator µX ¶µX ¶ X m n m X n Al ⊗ M l ek ⊗ xk = (Al ek ) ⊗ (Ml xk ) l=1

k=1

l=1 k=1

Pm acting on E⊗X. (Strictly speaking, one must first define operator l=1 Al ⊗Ml Pthe m as acting on V — see 1.7.2 — and then observe that l=1 Al ⊗PMl maps V0 into m itself. Moreover, one should check that Pmthe operator induced by l=1 Al ⊗Ml does not depend on the representation l=1 Al ⊗ Ml . These verifications are plain.) We recall that we consider the space E ⊗ X without norm. Therefore we obtain the natural mapping j : B(E) ⊗ B(X) → L(E ⊗ X), where L(E ⊗ X) is the space of all linear operators acting on the linear space E ⊗ X.

70

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Proposition. The natural mapping j : B(E) ⊗ B(X) → L(E ⊗ X) is injective. Proof. Let us consider an arbitrary T ∈ B(E) ⊗ B(X), and show jT = 0 Pthat m implies T = 0. By virtue of 1.7.2 we choose a representation T = l=1 Al ⊗ Ml with linearly independent A1 , . . . , Am . First of all, by induction on m we show that for any linearly independent family A1 , . . . , Am there exists a finite-dimensional subspace Em ⊆ E such that the restrictions A1 , . . . , Am : Em → E are linearly independent. Indeed, for m = 1 we have A1 e1 6= 0 for some e1 ∈ E since A1 6= 0. We define E1 to be the span of e1 . We assume that our assertion is true for m and show that it is also true for m + 1. We consider the restriction of Am+1 to Em . If the restrictions of A1 , . . . , Am+1 to Em are linearly independent we can set Em+1 = Em . So it remains to consider the case where the restrictions of A1 , . . . , Am+1 to Em are linearly dependent. In the latter case Am+1 = c1 A1 + · · · + cm Am on Em for some c1 , . . . , cm ∈ C. We stress that this representation is unique since A1 , . . . , Am are linearly independent on Em . We consider the kernel of the operator D = Am+1 − c1 A1 − · · · − cm Am over the whole of E. It can not coincide with E. Therefore Dem+1 6= 0 for some em+1 . We define Em+1 to be the span of Em and em+1 . Clearly, the restrictions of A1 , . . . , Am+1 to Em+1 are linearly independent. We denote by Fm ⊆ E the span of A1 Em , . . . , Am Em . By what has been proved the operators A1 , . . . , Am : Em → Fm are linearly Pn independent. 0 0 Next, P for any A ∈ L(Em , Fm ) and g = k=1 ek ⊗ fk ∈ Em ⊗ Fm we set n 0 τg (A) = k=1 hAek , fk i. Thus we obtain the mapping τ : g 7→ τg from Em ⊗ F0m to the conjugate of L(Em , Fm ). For the reason of dimension, τ is an isomorphism. Let us choose Em ⊗ F0m such that gk (Al ) = δkl . We take a reprePgn1k, . . . , gm ∈ 0 sentation gk = i=1 eki ⊗ fki . Let us fix k. From the assumption jT = 0 it follows that for all i = 1, . . . , nk and 0 an arbitrary x ∈ X we have T eki ⊗ xP= 0. Therefore hT eki ⊗ x, fki ⊗ x0 i = 0 for all n k 0 hT eki ⊗ x, fki ⊗ x0 i = 0. Straightforward x0 ∈ X 0 . Summing up we arrive at i=1 calculations show that, on the other hand, the left hand side of the last expression equals hMk x, x0 i. Consequently Mk = 0, which implies T = 0. ¤ According to this proposition we identify B(E) ⊗ B(X) with the subspace of L(E ⊗ X), and we do not distinguish elements of B(E) ⊗ B(X) and operators induced by them. Pm Assume we have a cross-norm α on E ⊗ X. For any T = l=1 Al ⊗ Ml ∈ B(E) ⊗ B(X) we define α e(T ) = kT kα˜ to be the norm of the operator jT on E ⊗α X. In other words, kT kα˜ = sup{ kT vkα : kvkα ≤ 1 }. (We do not exclude the case kT kα˜ = ∞.) We say that a cross-norm α on E ⊗ X is uniform if for all A ∈ B(E) and M ∈ B(X) kA ⊗ M : E ⊗α X → E ⊗α Xk = kAk · kM k. (Note that the inequality kA⊗M : P E⊗α X → E⊗α Xk ≥ kAk·kM k is always true.) m Clearly, in this case all elements l=1 Al ⊗ Ml ∈ B(E) ⊗ B(X) induce bounded

1.7. TENSOR PRODUCTS

71

operators on E ⊗α X. Thus one may think that the image of j lies in B(E⊗α X). We stress that the definition of a uniform cross-norm α means that the norm α e on B(E) ⊗ B(X) induced by the embedding j : B(E) ⊗ B(X) → B(E⊗α X) is a cross-norm, too. We denote the completion of B(E) ⊗α˜ B(X) by B(E) ⊗α˜ B(X). We identify B(E) ⊗α˜ B(X) with the closed subspace of B(E ⊗α X). It is easy to see that B(E)⊗B(X) is an algebra with respect to the multiplication µX ¶ µX ¶ X m n X n m (Ak Bl ) ⊗ (Mk Nl ). Ak ⊗ Mk · Bl ⊗ Nl = k=1

l=1

k=1 l=1

Clearly, j is a morphism of algebras. Hence B(E) ⊗α˜ B(X) is a Banach algebra. Example. Clearly, π and ε are uniform. An example of a non-uniform crossnorm will be given in 1.7.12. Remark. We note that if a cross-norm α is uniform then it is ∗-uniform, too. Indeed, let us fix a one-dimensional subspace E0 ⊆ E. Clearly, the subspace B0 (E) = B(E, E0 ) consisting of all operators whose images are situated in E0 is isometrically isomorphic to E0 . A similar assertion holds for B0 (X). It remains to observe that the equality kA ⊗ M kα˜ = kAk · kM k for A ∈ B0 (E) and M ∈ B0 (X) is equivalent to ke0 ⊗ x0 kα0 = ke0 k · kx0 k for e0 ∈ E0 and x0 ∈ X 0 . 1.7.8. The spectral mapping theorem (operator-valued variant). In this subsection we generalize theorem 1.4.10 to the operator-valued case. Let E and X be¡ Banach spaces, and let T ∈ B(X) be a fixed operator. We ¢ denote by O = O σ(T ), B(E) the set of all equivalence classes of holomorphic (see 1.4.9) functions f : U → B(E), where U = U (f ) is a neighbourhood of σ(T ). We say that f, g ∈ O are equivalent if they coincide on a neighbourhood of σ(T ). Clearly, O is an algebra with respect to pointwise operations. Let α be a uniform cross-norm on E ⊗ X. For any f ∈ O we define the element f (T ) ∈ B(E ⊗α X) by the formula Z 1 f (T ) = f (λ) ⊗ (λ − T )−1 dλ, (5) 2πi Γ where Γ ⊆ U (f ) is a contour surrounding σ(T ), cf. 1.4.9. Theorem. Assume a cross-norm α on E ⊗ X is uniform. (a) The mapping ϕ : f 7→ f (T ) is a morphism from the algebra O to the algebra B(E ⊗α X). Moreover, the mapping ϕ takes the function u(λ) = A to the operator A ⊗ 1X , and the function v(λ) = Aλ to the operator A ⊗ T . Consequently if p(λ) = A0 + A1 λ + A2 λ2 + · · · + An λn then p(T ) = A0 ⊗ 1X + A1 ⊗ T + A2 ⊗ T 2 + · · · + An ⊗ T n . (b) For all f ∈ O one has ¡ ¢ S ¡ ¢ σ f (T ) = { σ f (λ) : λ ∈ σ(T ) }.

72

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Proof. (a) Clearly, the mapping ϕ preserves the linear operations. We show that ϕ also preserves the operation of multiplication, i.e., ϕ(f1 · f2 ) = ϕ(f1 )ϕ(f2 ). Let us denote briefly the resolvent (λ − T )−1 by Rλ . First, we recall from the example in 1.4.9 the Hilbert identity Rλ Rµ = −

Rλ − Rµ . λ−µ

Let f1 , f2 ∈ O. We consider µ ¶ µ ¶ Z Z 1 1 ϕ(f1 )ϕ(f2 ) = f1 (λ) ⊗ Rλ dλ · f2 (µ) ⊗ Rµ dµ . 2πi Γ1 2πi Γ2 Let us choose Γ1 and Γ2 such that Γ2 separates Γ1 (and the singularities of f1 and f2 ) from σ(T ). In such a case we have Z Z f2 (µ) 1 f1 (λ) 1 dµ = 0, λ ∈ Γ1 , and dλ = f1 (µ), µ ∈ Γ2 . 2πi Γ2 λ − µ 2πi Γ1 λ − µ Therefore we can represent ϕ(f1 )ϕ(f2 ) as Z Z ¡ ¢ ¡ ¢ 1 ϕ(f1 )ϕ(f2 ) = f (λ)f (µ) ⊗ R R dµ dλ 1 2 λ µ (2πi)2 Γ1 Γ2 Z Z ¡ ¢ ³ Rλ − Rµ ´ 1 = f (λ)f (µ) ⊗ − dµ dλ 1 2 (2πi)2 Γ1 Γ2 λ−µ Z Z ¡ ¢ ³ Rµ ´ 1 = f (λ)f (µ) ⊗ dµ dλ 1 2 (2πi)2 Γ1 Γ2 λ−µ Z Z ¡ ¢ ³ Rλ ´ 1 − f (λ)f (µ) ⊗ dµ dλ 1 2 (2πi)2 Γ1 Γ2 λ−µ ¶ Z µZ ¡ ¢ 1 f1 (λ) = ⊗ 1 dλ f2 (µ) ⊗ Rµ dµ 2 (2πi) Γ2 Γ1 λ − µ µZ ¶ Z ¡ ¢ 1 f2 (µ) − f1 (λ) ⊗ Rλ ⊗ 1 dµ dλ (2πi)2 Γ1 Γ2 λ − µ Z ¡ ¢¡ ¢ 1 f1 (µ) ⊗ 1 f2 (µ) ⊗ Rµ dµ − 0 = 2πi Γ2 Z ¡ ¢ 1 = f1 (µ)f2 (µ) ⊗ Rµ dµ 2πi Γ2 = ϕ(f1 f2 ). Now we calculate f (T ) where f is either u(λ) = A or v(λ) = Aλ. We make use of the explicit formula (5). We recall from the example in 1.4.9 that for large λ the resolvent can be represented in the form (λ − T )

−1

∞ X Tk = . λk+1 k=0

1.7. TENSOR PRODUCTS

73

Thus we obtain Z X ∞ A ⊗ Tk dλ k+1 Γ k=0 λ Z ∞ X 1 k 1 = A⊗T dλ k+1 2πi Γ λ

1 u(T ) = 2πi

k=0

and Z X ∞ A ⊗ Tk dλ λk Γ k=0 Z ∞ X 1 k 1 = A⊗T dλ. 2πi Γ λk

1 v(T ) = 2πi

k=0

It remains to recall that Z

½

for k = 1, for k = 6 1. Γ ¡ ¢ (b) Clearly, it suffices to show that 0 belongs to σ f (T ) if and only if it belongs ¢ S ¡ to { σ f (λ) : λ ∈ σ(T ) }. ¢ ¡ ¢−1 S ¡ Assume 0 ∈ / { σ f (λ) : λ ∈ σ(T ) }. Then the function g(λ) = f (λ) is holomorphic on a neighbourhood of σ(T ) (see the example in 1.4.9). By (a) the operator g(T ) is the inverseSof f¡(T ). ¢ Conversely, assume 0 ∈ { σ f (λ) : λ ∈ σ(T ) }, i.e., for some λ0 ∈ σ(T ) the operator f (λ0 ) is not invertible. Let N denote the set of all λ lying in the domain of f such that f (λ) is not invertible. By 1.3.6(c) N is closed (in the domain of f ). We have λ0 ∈ σ(T ) ∩ N . Thus the set σ(T ) ∩ N is not empty. We fix z belonging to the boundary of σ(T ) ∩ N . Clearly, z belongs either to the boundary of σ(T ) or to the boundary of N . Both cases are handled similarly. We consider, for example, the case ¡ when z ¢belongs to the boundary ∂N of N0. We set A = f (z) and g(λ) = f (λ) − A /(z − λ). (We define g(z) as −f (z). Clearly, g is differentiable at z.) Then f (λ) = A + (z − λ)g(λ) and consequently ¡ ¢ f (T ) = A ⊗ 1 + 1 ⊗ (z − T ) g(T ). 1 2πi

1 dλ = λk

1 0

We recall that z ∈ ∂N . Hence the operator A is not invertible, but in any neighbourhood of A there exists an invertible operator B. We show that this implies that both |A|+ and |A|− are equal to zero (see 1.3.1 for the definitions of | · |+ and | · |− ). We assume the contrary: let, for example, |A|+ > 0. Then by 1.3.5 there exist ε > 0 and a neighbourhood U of A such that |B|+ > ε for all B ∈ U . We pick a sequence of invertible operators Bi ∈ U that converges to A. By 1.3.2 |Bi |− = |Bi |+ > ε, since Bi are invertible. Therefore, also by 1.3.5 we have |A|− > 0. But both |A|+ > 0 and |A|− > 0 imply the invertibility of A. This

74

I. FUNCTIONAL ANALYSIS PRELIMINARIES

is a contradiction. In a similar way |A|− > 0 implies |A|+ > 0 and consequently the invertibility of A. We recall that the operator z − T is not invertible. Therefore by 1.3.2(c) either |z − T |+ = 0 or |z − T |− = 0. First we suppose that |z − T |+ = 0. As we have shown, |A|+ = 0, too. By the definition of | · |+ we can choose xi ∈ X, kxi k = 1, and ei ∈ E, kei k = 1, such that k(z − T )xi k → 0 and kAei k → 0. Then it follows ¡ ¢ kf (T )(ei ⊗ xi )k ≤ k(A ⊗ 1)(ei ⊗ xi )k + kg(T ) 1 ⊗ (z − T ) (ei ⊗ xi )k ≤ kAei k + kg(T )k · k(z − T )xi k → 0, ¡ ¢ i.e., |f (T )|+ = 0 and consequently 0 ∈ σ f (T ) . Now we suppose that |z − T |− = 0. As we have seen, |A|− = 0, too. By 1.3.4 we can choose x0i ∈ X 0 , kx0i k = 1, and e0i ∈ E0 , ke0i k = 1, such that kx0i (z − T )k → 0 and ke0i Ak → 0. We recall from the remark in 1.7.7 that since α is uniform, it is also ∗-uniform. But since α is ∗-uniform, e0i ⊗ x0i is a bounded functional on E ⊗ X and ke0i ⊗ x0i k = 1. Therefore we have ¡ ¢ k(e0i ⊗ x0i )f (T )k ≤ k(e0i ⊗ x0i )(A ⊗ 1)k + k(e0i ⊗ x0i ) 1 ⊗ (z − T ) g(T )k ≤ ke0i Ak + kx0i (z − T )k · kg(T )k → 0, ¡ ¢ i.e., |f (T )|− = 0 and consequently 0 ∈ σ f (T ) .

¤

1.7.9. Irreducible representations of a Banach algebra. Let A be a Banach algebra with a unit. A linear subspace J ⊆ A is called a left (right) ideal in A if AJ ∈ J for all A ∈ A and J ∈ J (JA ∈ J for all A ∈ A and J ∈ J, respectively). This condition is often rewritten briefly as AJ ⊆ J (JA ⊆ J). We discuss only left ideals since the theory of right ideals is a simple repetition of the theory of left ones. Clearly, the closure of a left ideal is also a left ideal. A left ideal J is called proper if J 6= A. A proper left ideal cannot contain left invertible elements. Moreover, the closure of a proper ideal is also proper since, by virtue of 1.4.2, the set of all invertible elements is open. Unless the contrary is explicitly stated we always assume that all ideals under consideration are proper. A proper left ideal is called maximal if it is not contained in a wider proper left ideal. Clearly, a maximal ideal is closed. By means of Zorn’s lemma it can be shown easily that any proper left ideal is contained in some maximal left ideal (see, for example, [Nai, ch. 2, §7.4] for the proof). Any A ∈ A induces a linear operator TA : A → A (here we consider A as a Banach space) acting by the rule TA B = AB. Clearly, kTA k ≤ kAk. It is easy to see that the mapping T : A 7→ TA is a morphism from the algebra A into the algebra B(A). The morphism T is called the left regular representation of A. More generally, let J be a closed left ideal in A. Let us consider the Banach space F = A/J. Any A ∈ A induces the linear operator τA : F → F defined by the

1.7. TENSOR PRODUCTS

75

rule τA (B + J) = AB + J. We observe that τA is the quotient operator induced by TA ; consequently by 1.2.2 kτA k ≤ kTA k ≤ kAk. Clearly, the mapping τ : A 7→ τA is a morphism from the algebra A into the algebra B(F ). The morphism τ is called the left representation of A induced by J. (If J is a right ideal the mapping τ : A → τA defined by the rule τA (B +J) = BA+J is an anti-morphism of algebras, i.e., τAB = τB τA .) Let τ : A → B(F ) be a left representation. It is easy to see that the set Fx = { τA x : A ∈ A } is a subspace of F for any x ∈ F . The left representation τ is called irreducible if the subspace Fx coincides with F for all x ∈ F , x 6= 0. Lemma. Let A be a Banach algebra with a unit and J be a maximal left ideal in A. (a) The left representation τ induced by J is irreducible. (b) If an element R ∈ A commutes with all A ∈ A, i.e., RA = AR for all A ∈ A then the operator τR is a scalar one, i.e., τR = λ1F , where λ ∈ C. Similar facts hold for right ideals. Proof. (a) We take an arbitrary x ∈ F = A/J, x 6= 0, and consider the subspace Fx . Next, we consider the pre-image I = q −1 (Fx ) of Fx under the natural projection q : A → F . Clearly, Fx is invariant under all operators τA , A ∈ A. Hence I is a left ideal in A (perhaps, non-proper). Evidently, J ⊆ I. Moreover, I is strictly wider than J; indeed, since x 6= 0 and A contains a unit we have x ∈ Fx and thus Fx 6= {0}. Since J is maximal this implies I = A. But then Fx = F . (b) Assume that an element R ∈ A commutes with all A ∈ A. We note that then τR τA = τA τR for all A ∈ A. We show that if τR is not invertible then τR = 0. Let Ker τR 6= {0}, i.e., τR x = 0 for some x ∈ F , x 6= 0. Then τR (τA x) = τA τR x = 0 for all A ∈ A. In other words, all vectors of the form τA x, A ∈ A, belong to Ker τR . By (a) the set of such vectors coincides with F . Thus τR = 0. Suppose τR 6= 0. We choose x ∈ F such that τR x 6= 0 and consider the set of all vectors of the form τR (τA x), A ∈ A. Clearly, it is contained in Im τR . Since τR τA x = τA (τR x) this set coincides with Fy , where y = τR x. But since τ is irreducible and y 6= 0, by (a), Fy coincides with F . Thus in this case Im τR = F . We note that dim F 6= 0 because J is proper. Therefore (see the example in 1.4.9) σ(τR ) is not empty. Let λ ∈ σ(τR ), i.e., the operator λ−τR be not invertible. Clearly, λ − R commutes with all A ∈ A. As we have proved above, this implies that λ − τR = 0, i.e., τR = λ = λ1F . ¤ 1.7.10. The Bochner–Phillips theorem. In this subsection we present a generalization of the theorem 1.4.11 to the operator-valued case. Let B and M be Banach algebras. Clearly (cf. 1.7.7), B ⊗ M is also an algebra with respect to the natural multiplication µX ¶ µX ¶ X n X n m m Ak ⊗ Mk · Bl ⊗ Nl = (Ak Bl ) ⊗ (Mk Nl ). k=1

l=1

k=1 l=1

76

I. FUNCTIONAL ANALYSIS PRELIMINARIES

We say that a cross-norm α on B ⊗ M is compatible with multiplication if kuvkα ≤ kukα · kvkα

for all u, v ∈ B ⊗ M.

Clearly, in this case B ⊗α M is a Banach algebra. Let ϕ : B → B1 and ψ : M → M1 be morphisms of algebras. Then, obviously, ϕ ⊗ ψ : B ⊗ M → B1 ⊗ M1 is also a morphism of algebras. Furthermore, if ϕ ⊗ ψ is continuous with respect to cross-norms α and α1 on B ⊗ M and B1 ⊗ M1 , respectively, then ϕ ⊗ ψ possesses the extension by continuity to a morphism of algebras ϕ ⊗ ψ : B ⊗α M → B1 ⊗α1 M1 . Examples. (a) Let a cross-norm α on E ⊗ X be uniform. Then the induced cross-norm α e on B(E) ⊗ B(X) is compatible with multiplication. (b) Clearly, the cross-norm π is compatible with multiplication for arbitrary B and M. (c) Let T be a compact Hausdorff space, and let B be an arbitrary Banach algebra. Then the cross-norm ε on B ⊗ C(T, C) is compatible with multiplication by virtue of 1.7.4. Theorem. Let B be an arbitrary Banach algebra with a unit. And let M be a commutative Banach algebra with a unit, and X = X(M) be its space of characters (see 1.4.11). And let a cross-norm α on B ⊗ M satisfy the two following assumptions: (a) α is compatible with multiplication. (b) For any character ζ ∈ X(M) the mapping 1 ⊗ ζ : B ⊗α M → B ⊗ C ' B is continuous. Here 1 is the identical morphism from B into itself. Then an element T ∈ B ⊗α M is left (right) invertible in B ⊗α M if and only if the element (1 ⊗ ζ)T ∈ B ⊗ε C ' B is left (right) invertible for all ζ ∈ X(M). Consequently ¢ ª S© ¡ σ (1 ⊗ ζ)T : ζ ∈ X . σ(T ) = Remark. Usually assumption (b) is fulfilled in the following stronger form: (b0 ) The morphism of algebras Θ defined by the rule Θ = 1 ⊗ ϑ : B ⊗α M → B ⊗ε C(X, C) ' C(X, B) is continuous. Here ϑ : M → C(X, C) is the Gel0 fand transform. In such a case Θ can be extended by continuity to the morphism Θ = 1 ⊗ ϑ : B ⊗α M → C(X, B). We call this extension the operator-valued Gel0 fand transform. Let ΘT denote the morphism Θ applied to T ∈ B ⊗α M. We note that ΘT ∈ C(X, B). The conclusion of the theorem can now be reformulated similarly to that of theorem 1.4.11: σ(T ) = σ(ΘT ) ¢ ª S© ¡ = σ ΘT (ζ) : ζ ∈ X .

1.7. TENSOR PRODUCTS

77

Proof. We denote briefly the algebra B ⊗α M by A. Assume the element (1 ⊗ ζ)T is left invertible for all ζ ∈ X(M). We show that T is then left invertible. Assume the contrary: let T be not left invertible in A. Then { AT : A ∈ A } is a proper left ideal in A. We extend it to a maximal left ideal J. Since A contains a unit T ∈ J. We denote by τ : A → B(A/J) the left representation induced by J (see 1.7.9). We break the following argument into four steps. (i) We consider the subalgebra 1 ⊗ M = { 1 ⊗ M : M ∈ M } of A; here 1 ∈ B. Clearly, elements R ∈ 1 ⊗ M commute with all A ∈ A, i.e., RA = AR for all A ∈ A. By 1.7.9(b) this implies that the representation τ maps all elements R ∈ 1 ⊗ M to scalar operators. Thus the restriction of τ to 1 ⊗ M is similar to a morphism from the algebra M into the algebra C. We recall that such a morphism of algebras is called a character of M (see 1.4.11). We denote this character by ζ. (ii) We denote by I the least closed left ideal in the algebra A generated by the set 1 ⊗ Ker ζ = { 1 ⊗ M : M ∈ M, ζ(M ) = 0 }, here 1 ∈ B. We show that I ⊆ J. Since J is a closed left ideal it suffices to show that R ∈ 1 ⊗ Ker ζ implies R ∈ J. By virtue of (i), R = 1 ⊗ M ∈ 1 ⊗ Ker ζ implies τR ' ζ(M ) = 0. In other word, τR (A + J) = RA + J ⊆ J

for all A ∈ A.

Substituting in this inclusion A = 1A we obtain that R belongs to J. (iii) Next, we show that Ker(1⊗ζ) ⊆ I; where 1⊗ζ : B ⊗α M → B⊗C ' B; note that here we use assumption (b). We recall from the formulation of the theorem that 1 is the identical morphism from B into itself. Assume R ∈ A belongs to Ker(1 ⊗ ζ). We pick a sequence Ri =

ni X

Bki ⊗ Mki ,

Bki ∈ B, Mki ∈ M,

k=1

that converges to R (we recall that R ∈ A = B⊗α M). From the assumption R ∈ Ker(1 ⊗ ζ) and the continuity of 1 ⊗ ζ we have (1 ⊗ ζ)Ri → 0. We note that (1 ⊗ ζ)Ri =

ni X

Bki ⊗ ζ(Mki ) =

k=1

µX ni

¶ Bki · ζ(Mki ) ⊗ 1C .

k=1

Pni

Thus k=1 Bki · ζ(Mki ) → 0. We consider the sequence Ai ∈ B ⊗ M defined by the formula µX ¶ ni ni X ¡ ¢ Ai = Bki ⊗ ζ(Mki )1M = Bki · ζ(Mki ) ⊗ 1M . k=1

From

Pni k=1

k=1

Bki · ζ(Mki ) → 0 it follows that Ai → 0. We observe that Ri − Ai =

ni X k=1

¡ ¢ Bki ⊗ Mki − ζ(Mki )1M .

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I. FUNCTIONAL ANALYSIS PRELIMINARIES

¡ ¢ Clearly, 1 ⊗ Mki − ζ(Mki )1M ∈ 1 ⊗ Ker ζ. Since I is a left ideal this implies Ri − Ai ∈ I. Passing to the limit in this inclusion, we obtain R − 0 ∈ I, i.e., R ∈ I. (iv) We recall that the element (1 ⊗ ζ)T is left invertible. This means that there exists Q ∈ A such that (1 ⊗ ζ)(QT − 1A ) = 0. By (iii) this implies QT − 1A ∈ I. But by (ii) I ⊆ J. Therefore QT − 1A ∈ J. Since T ∈ J and J is a left ideal we have 1A ∈ J. This is a contradiction. The converse statement follows from 1.4.6. Indeed, since 1 ⊗ ζ is a morphism of algebras, the invertibility of T implies the invertibility of (1 ⊗ ζ)T . ¤ 1.7.11. The tensor product of a direct sum Proposition. Let X be a Banach space, and let Y and Z be closed subspaces of it. Assume that X = Y ⊕ Z with kxk = kyk + kzk,

x ∈ X,

for the corresponding representation x = y +z. Then the space E ⊗π X is naturally isometrically isomorphic to (E ⊗π Y ) ⊕ (E ⊗π Z) with kvkE ⊗π X = kukE ⊗π Y + kwkE ⊗π Z ,

v ∈ E ⊗π X,

for the corresponding representation v = u + w. Proof. We define the complementary projectors PY , PZ : X → X onto Y and Z, respectively, induced by the direct sum X = Y ⊕ Z. It is straightforward to verify that the equality X = Im PY ⊕ Im PZ can be rewritten equivalently as the collection of the identities PY2 = PY ,

PZ2 = PZ ,

PY PZ = PZ PY = 0,

and

PY + PZ = 1.

Clearly, they imply (1 ⊗ PY )2 = 1 ⊗ PY , (1 ⊗ PZ )2 = 1 ⊗ PZ , (1 ⊗ PY )(1 ⊗ PZ ) = (1 ⊗ PZ )(1 ⊗ PY ) = 0, and 1 ⊗ PY + 1 ⊗ PZ = 1 ⊗ 1, which mean that E ⊗π X = Im(1 ⊗ PY ) ⊕ Im(1 ⊗ PZ ). (We recall from 1.7.7 that the cross-norm π is uniform.) We show, e.g., that Im(1 ⊗ PY ) is isometrically isomorphic to E ⊗π Y . Let J : Y → X be the natural embedding. We consider also the projection Q : X → Y parallel to Z. Clearly, kJk ≤ 1, kQk ≤ 1, QJ = 1Y , and JQ = PY . Therefore for the operators 1 ⊗ J : E ⊗π Y → E ⊗π X and 1 ⊗ Q : E ⊗π X → E ⊗π Y we have k1 ⊗ Jk ≤ 1, k1 ⊗ Qk ≤ 1, (1 ⊗ Q)(1 ⊗ J) = 1 ⊗ 1Y , and (1 ⊗ J)(1 ⊗ Q) = 1 ⊗ PY . From the identity (1 ⊗ Q)(1 ⊗ J) = 1 ⊗ 1Y it follows that 1 ⊗ Q and 1 ⊗ J establish isomorphism between Im(1 ⊗ J) and E ⊗π Y . The estimates k1 ⊗ Jk ≤ 1 and k1 ⊗ Qk ≤ 1 show that this isomorphism is isometric. Finally, the identity (1 ⊗ J)(1 ⊗ Q) = 1 ⊗ PY implies that Im(1 ⊗ PY ) = Im(1 ⊗ J). Thus Im(1 ⊗ PY ) is isometrically isomorphic to E ⊗π Y . According to 1.7.3(c) we represent an element v ∈ E ⊗π X in the form v = P∞ k=1 ek ⊗ xk , and then represent xk as xk = yk + zk with yk ∈ Y and zk ∈ Z. By

79

1.7. TENSOR PRODUCTS

assumption, ∞ X

kek k · kxk k =

k=1

=

∞ X k=1 ∞ X

¡ ¢ kek k · kyk k + kzk k kek k · kyk k +

k=1

∞ X

kek k · kzk k

k=1

≥ kukE ⊗π Y . + kwkE ⊗π Z . P∞ Taking then the infimum over all representations v = k=1 ek ⊗ xk of v, we obtain the inequality kvk ≥ kuk + kwk. The opposite inequality is trivial. ¤ 1.7.12. A remark on the complexification. In 1.1.2 we have announced that the field of scalars is always C. The present subsection is an exception. The suggestion that all spaces under consideration are complex is often imposed in operator theory. This imply that the real theory can be reduced easily to the complex one by means of complexification. And such is usually the case. But we would like to call attention to the problem: the norm of the complexification of an operator depends on the choice of a norm on the complexification of the space. Let X be a real linear space. The complexification of X is the complex linear space X C of all formal expressions z = x1 + ix2 , x1 , x2 ∈ X, with the natural operations of addition and multiplication by complex scalars. Thus as a real space X C is the direct sum X ⊕ X. One can also imagine X C as the tensor product C ⊗ X of real linear spaces C and X, and interpret the multiplication by a scalar α ∈ C as the multiplication by the operator α ⊗ 1. We adhere to the last point of view. Let X and Y be real linear spaces, and let T : X → Y be a linear operator. The complexification of T is the operator T C : X C → Y C defined as the tensor product 1⊗T: C⊗X →C⊗Y. Let X be a real Banach space with the norm k · k = k · kX . We say that a complex norm α(·) = k · kα on X C is a complex continuation of the norm k · kX on X if k1 ⊗ xkα = kxkX for all x ∈ X. Since α is a complex norm we have kλ ⊗ xk = |λ| · kxk for all λ ∈ C and x ∈ X C (not only x ∈ X); in particular, α is a cross-norm. The simplest example of a complex continuation of a norm is kx1 + ix2 k2 = (kx1 k2 + kx2 k2 )1/2 . It is easy to see that X C is Banach with respect to k · k2 . Clearly, any uniform cross-norm (for instance, π and ε) induces a complex continuation of a norm. Remark. Note that the norms kx1 + ix2 k1 = kx1 k + kx2 k

and

kx1 + ix2 k∞ = max(kx1 k, kx2 k)

are not complex continuations. Indeed, for x ∈ √X we have k(1+i)xk1 = 2kxk √ 1 and k(1 + i)xk∞ = kxk∞ instead of k(1 + i)xk1 = 2kxk1 and k(1 + i)xk∞ = 2kxk∞ .

80

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Example. For some concrete X there exist natural complex continuations of the norm on X. (a) If X is Euclidean, the natural complex norm on X C is k · k2 . In such a case C X is a Hilbert space. (b) If X is Lp (T, R), p < ∞, where T is locally compact topological space with a positive measure (see 1.5.5), then X C can be identified with Lp (T, C), and the natural complex norm on Lp (T, C) is ³Z kx + iykLp =

b

´1/p |x(t) + iy(t)|p dt ,

a

i.e., the Lp -norm of the function t 7→ |x(t) + iy(t)|. (c) If X is L∞ (T, R) or C(T, R) then one can identify X C with L∞ (T, C) or C(T, C), respectively. The natural complex norm of x + iy ∈ X C is the supremum norm of the function t 7→ |x(t) + iy(t)|. If X is L∞0 or C0 , the natural norms are analogous. Below we assume that α(·) = k·kα is a complex continuations of the norm on X equivalent to k · k2 . If α is uniform we have kT C kα = kT k for all T ∈ B(X). But in general this is not the case as the following example shows. Example. (a) Let X be R2 with l∞ -norm kxk = max(|x1 |, |x2 |) and Y be R2 with l1 -norm kyk = |y1 | + |y2 |. We represent X C and Y C as C2 and endow them with the natural complex continuations kxk = max(|x1 |, |x2 |) and kyk = |y1 |+|y2 |, respectively. We consider the operator A : X → Y defined by the matrix µ ¶ 1 1 A= . 1 −1 √ It is easy to verify that kAk = 2, but kAC k = 2 2. (b) The preceding example possesses a modification for the case more closed to our discussion. Let X be R4 with the norm kxk = max{|x1 | + |x2 |, |x3 | + |x4 |}. This is the norm of the space l1∞ = l∞ (l1 ), cf. 1.6.3. We represent X C as C4 and endow it with the natural complex continuation kxk = max{|x1 |+|x2 |, |x3 |+|x4 |}. We consider the operator B : X → X defined by the matrix 

1 1 B= 0 0

0 0 0 0

 1 0 −1 0  . 0 0 0 0

1.8. CONJUGATE SPACES

81

√ For reasons similar to that in (a), one has kBk = 2, but kB C k = 2 2. Since we define B C as 1 ⊗ B : C ⊗ X → C ⊗ X, this example is also an example of a non-uniform cross-norm. Let X and Y be real Banach spaces, and let A : X → Y be a bounded linear operator, and AC : X C → Y C be its complexification. As we have seen, the norms of A and AC may be distinct. Nevertheless, a pair of simple facts holds. First, the operators A and AC are invertible simultaneously. Furthermore, there exists a constant K such that kAk ≤ kAC kα ≤ KkAk

for all A ∈ B(X, Y ).

Second, we stress that the constant in the left inequality is 1. Therefore an estimate of the norm of the complexification of an operator is always an estimate of the norm of the initial (real) operator, too.

1.8. Conjugate spaces In this section we discuss the conjugates of functional spaces Lpq and Cq . We recall that X 0 denotes the conjugate of a Banach space X, see 1.1.7. 1.8.1. The conjugate of Lpq . For 1 < q < ∞ we determine the conjugate index q 0 from the relation 1/q + 1/q 0 = 1; we set q 0 = ∞ if q = 1; and we set q 0 = 1 if q = ∞ or q = 0. For 1 ≤ p ≤ ∞ we define p0 analogously. Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) Let I be a set, and let { Ei : i ∈ I } be a collection of Banach spaces. Then for every x ∈ lq (I, Ei ) and y ∈ lq0 (I, E0i ) the collection { hxi , yi i : i ∈ I } belongs to l1 (I, C). Furthermore, the formula hx, yi =

X hxi , yi i i∈I

defines an isometric embedding of lq0 in (lq )0 ; in particular, © ª kyklq0 = sup |hx, yi| : x ∈ lq , kxk = 1 .

(1)

If q 6= ∞ (for example, q = 0) this embedding is onto. If q 6= 0 then for any family {xi ∈ Ei } the estimate o nX 0 0 sup |hxi , yi i| : y ∈ lq (I, Ei ), kyk = 1 < ∞ i∈I

implies x ∈ lq . (b) Let T be a locally compact topological space with a positive measure λ, and let E be a Banach space. Then for every x ∈ Lp (T, E) and y ∈ Lp0 (T, E0 ) the function t 7→ hx(t), y(t)i belongs to L1 (T, C). (Here we set hx(t), y(t)i = 0 if one of the factors x(t) or y(t) is undefined, but the other is

82

I. FUNCTIONAL ANALYSIS PRELIMINARIES

zero; this agreement is important if p = ∞ or p0 = ∞ because the product may be undefined on a locally null, but not σ-finite set.) Furthermore, the formula Z hx, yi = hx(t), y(t)i dλ(t) T

defines an isometric embedding of Lp0 in (Lp )0 ; in particular, © ª kykLp0 = sup |hx, yi| : x ∈ Lp , kxk = 1 .

(2)

If E is finite-dimensional and p 6= ∞ this embedding is onto. For any measurable function x : T → E the estimate nZ

∗

sup

|hx(t), y(t)i| dλ(t) : y ∈ Lp0 (T, E0 ), kyk = 1

o ky0 k + ky ⊥ k − 3ε, which completes ¡ the proof.¢ ¡(b) It suffices¢ to recall that Lp0 and Lp∞ are defined to be l0 I, Lp (Qi , E) and l∞ I, Lp (Qi , E) and refer to (a). ¤ 1.8.4. The modulus of a measure. Let T be a locally compact topological space. A (complex ) measure or a (complex ) integral on T is an arbitrary linear functional µ : C00 (T, C) → C whose restriction to CK (T, C) is continuous in the norm of C for any compact set K ⊆ T , cf. 1.5.5. We denote the space of all complex measures on T by the symbol Mloc = Mloc (T, C). Clearly, theorem 1.5.8 remains valid for complex measures. We denote by C00 (T, R) the real subspace of C00 (T, C) consisting of all com+ pactly supported continuous functions z : T → R, and by C00 the subset of C00 (T, R) consisting of all functions taking their values in [0, +∞). A measure R µ ∈ Mloc (T, C) is called real if x dµ ∈ R for all x ∈ C00 (T, R). We denote the set of all real measures by Mloc = Mloc (T, R). Clearly, it is a real linear space, and it can be treated as the space of all continuous R linear functionals on+C00 (T, R). A complex measure µ on T is called positive if x dµ ≥ 0 for all x ∈ C00 , cf. 1.5.5. A measure µ on T is called bounded or finite if it is continuous in the norm of C on the whole of C00 (T, C). Since the closure of C00 in C is C0 , this definition is ¡ ¢0 equivalent to the following one: a bounded measure is an element of C0 (T, C) . We denote by M = M(T, C) the space of all bounded complex measures on T ¡ ¢0 endowed with the norm of C (T, C) . We denote the value of µ ∈ M on x ∈ C0 0 R both by hx, µi and x dµ. Proposition. Let µ ∈ Mloc (T, C). + (a) ([Bou3 , ch. 3, §1, 6]) For z ∈ C00 we set Z ¯ n¯ o ¯ ¯ hz, |µ|i = sup ¯ x dµ¯ : |x(t)| ≤ z(t), x ∈ C00 (T, C) . The functional z 7→ hz, |µ|i has a unique continuation to a linear bounded functional on C00 (T, C), i.e., to a measure. The measure |µ| is the smallest

86

I. FUNCTIONAL ANALYSIS PRELIMINARIES

positive measure which dominates µ, i.e., ¯Z ¯ Z ¯ ¯ for all x ∈ C00 (T, C), ¯ x dµ¯ ≤ |x| d|µ|

(4)

where |x|(t) = |x(t)|. (b) For all µ ∈ M(T, C) kµk = k |µ| k = |µ|(T ). R R (c) We define the complex conjugate measure µ ¯ by the rule x d¯ µ= x ¯ dµ. The measures µr = (µ + µ ¯)/2 and µi = (µ − µ ¯)/(2i) are real; they are called the real and imaginary parts of µ, respectively. Then we set µ± r = ± ± ± (|µr |±µr )/2 and µi = (|µi |±µi )/2. The measures µr and µi are positive, and + − − µ = µ+ r − µr + i(µi − µi ). Thus any complex measure is a linear combination of positive measures. ± Finally, kµ± r k ≤ kµk and kµi k ≤ kµk provided µ ∈ M(T, C). Proof. (b) Inequality (4) implies that kµk ≤ k |µ| k. Conversely, we assume that z ∈ C00 (T, R). Then representing z as z = z1 − z2 with z1 = max{z, 0} and z2 = max{−z, 0}, it is easy to see that ¯Z ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ z d|µ|¯ ≤ ¯ z1 d|µ|¯ + ¯ z2 d|µ|¯ Z = (z1 + z2 ) d|µ| ≤ kz1 + z2 k · kµk = kzk · kµk. Now we consider an arbitrary (complex) z ∈ C00 (T, C). We take u ∈ C, |u| = 1, R such that uz d|µ| ≥ 0. Then we have ¯Z ¯ Z ¯ ¯ ¯ z d|µ|¯ = uz d|µ| Z = Re(uz) d|µ| ≤ k Re(uz)k · kµk ≤ kzk · kµk, which implies k |µ| k ≤ kµk. Finally, since 1T ∈ M + we have Z |µ|(T ) = 1T d|µ| nZ o = sup y d|µ| : y ∈ C00 , 0 ≤ y(t) ≤ 1 = k |µ| k.

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(c) is evident. ¤ The measure |µ| is called the modulus of µ. Let T be a locally compact topological space, and let E be a Banach space. Let µ ∈ Mloc (T, C). We define the space L1 (T, E, µ) to be L1 (T, E, |µ|). Pn If x ∈ C00 (T, E) is a finite-dimensional function (1.5.3)R we represent Pn x asRx = k=1 ek xk , where ek ∈ E and xk ∈ C00 (T, C). Then we set x dµ = k=1 ek xk dµ, cf. 1.7.7. Clearly, for any e0 ∈ E0 , ke0 k ≤ 1, we have Z Z ¯ ¯ ¯ ¯ ¯h x dµ, e0 i¯ ≤ ¯ hx(t), e0 i dµ(t)¯ Z ≤ |x| d|µ| = kxkL1 (T,E,µ) . ¯R ¯ Consequently ¯ x dµ¯ ≤ kxkL1 (T,E,µ) . In particular, this estimate shows that our Pn definition does not depend on the representation x = k=1 ek xk . By 1.5.3 the set of finite-dimensional R functions is dense in C00 . And C00 is dense in L1 . Therefore the mapping x 7→ x dµ possesses the extension to L1 (T, E, µ) by continuity. We R call x dµ the integral of x with respect to µ. Clearly, ¯ Z ¯Z ¯ ¯ for all x ∈ L1 (T, E, µ). ¯ x dµ¯ ≤ |x| d|µ| Cf. the definition of integral on L1 (T, E, λ) in 1.5.5. By continuity, for any x ∈ L1 (T, E, µ) and a ∈ B(E) we also have Z Z a x dµ = ax(t) dµ(t). For µ ∈ Mloc (T, C) we call a set E ⊆ T µ-summable if it is |µ|-summable, i.e., R its characteristic function 1E is integrable. The number µ(E) = 1E dµ is called the (complex) measure of E. From the above estimate we have |µ(E)| ≤ |µ|(E)

for all µ-summable sets E.

1.8.5. Measures with a compact support. Let T be a locally compact topological space. Assume µ ∈ M and |µ| is the modulus of µ, see 1.8.4. We say that µ is concentrated on a µ-measurable set M ⊆ T if |µ|(T \ M ) = 0. (We say that µ is supported in M if, in addition, M is closed.) We say that µ has a compact support if it is concentrated on a compact set. Clearly, the set of all compactly supported measures is a subspace of M(T ). Proposition. Let T be a locally compact topological space. (a) The set of all compactly supported measures is dense in M(T, C). In particular, a positive (real) bounded measure can be approximated by positive (real) measures with a compact support. (b) Any measure µ ∈ M(T, C) is concentrated on a σ-compact set.

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Remark. From the following proof it is also evident that an (absolutely) continuous bounded measure can be approximated by (absolutely) continuous measures with a compact support, see 1.8.7 and 4.1.7 below for the definitions. Proof. (a)n Let of µ. By definition |µ|(T ) = ¯R µ ∈¯ M, and let |µ| be the modulus o ¯ ¯ 1 d|µ| = sup ¯ y dµ¯ : y ∈ C00 , 0 ≤ y(t) ≤ 1 . We fix ε > 0. Since |µ|(T ) < ∞ R there exists y ∈ C00 ,R0 ≤ y(t) ≤ 1, such that | y dµ| > |µ|(T )−ε. Let K = supp y. ≥ | y dµ| > |µ|(T ) − ε. We define the measure µK by the rule RClearly, |µ|(K) R x dµK = K x dµ. Clearly, µK is concentrated on K and R

¯ o n ¯Z ¯ ¯ x dµ¯ : x ∈ C0 , kxk ≤ 1 kµ − µK k = sup ¯ T \K nZ o ≤ sup |x| d|µ| : x ∈ C0 , kxk ≤ 1 T \K

≤ |µ|(T \ K) = |µ|(T ) − |µ|(K) < ε. (b) Assume µS n are supported in compact sets Kn and µn → µ. Then µ is ∞ concentrated on n=1 Kn . ¤ 1.8.6. The decomposition C 0 = C00 ⊕ C0⊥ . Let T be a locally compact topological space, and let E be a Banach space. 0 = C∞ (T, E)0 is naturally isomorphic to the Theorem. The space C 0 = C∞ direct sum of C00 = C0 (T, E)0 and C0⊥ = C0 (T, E)⊥ . For the corresponding representation of y ∈ C 0 in the form y = y0 + y ⊥ with y0 ∈ C00 and y ⊥ ∈ C0⊥ , one has kyk = ky0 k + ky ⊥ k.

Proof. The proof is similar to that of 1.8.3. First it is required to show that the natural projection q : C 0 → C 0 /C0⊥ ' C00 has a right inverse J : C00 → C 0 . We show that J can be defined as follows. For µ ∈ C00 and x ∈ C we set hx, Jµi = lim hxK , µi. K

Here K ⊆ T runs over compact subsets of T ; xK ∈ C00 is an arbitrary function such that xK is equal to x on K and kxK k ≤ kxk, the existence of such an xK follows from the Urysohn theorem (e.g., one can set xK = α ex for a relevant α e); and, finally, limK means the following: limK hxK , µi = A if for any ε > 0 there exists a compact set N ⊆ T such that for any compact set K ⊇ N one has |hxK , µi − A| ≤ ε. To prove the existence of the limit, it suffices to show that for any ε > 0 there exists a compact set N ⊆ T such that |hx, µi| ≤ ε for all x ∈ C00 , kxk ≤ 1, and x(t) = 0 for t ∈ N .

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We assume the contrary: let there exist ε > 0 such that for any compact set N ⊆ T we have |hx, µi| > ε for some x ∈ C00 , kxk ≤ 1, x(t) = 0 for t ∈ N . Without loss of generality we may assume that hx, µi is real and hx, µi > ε. Evidently, we can pick a sequence xm ∈ C00 such that kxm k ≤ 1, P xm (t) = 0 n for t ∈ supp x1 ∪ · · · ∪ supp xm−1 , and hxm , µi > ε. We consider vn = m=1 xm . Clearly, vn ∈ C00 , kvn k ≤ 1, and hvn , µi > nε. But the arbitrariness of n contradicts the boundedness of µ. It remains to observe that qJ = 1. The proof of the equality kyk = ky0 k + ky ⊥ k is similar to that of 1.8.3. Namely, let y ∈ C 0 and y = y0 + y ⊥ be the decomposition according to the direct sum. Evidently, kyk ≤ ky0 k + ky ⊥ k. We prove the opposite inequality. Given ε > 0 we choose xε ∈ C0 , kxε k = 1, such that hxε , y0 i > ky0 k − ε. Since x ∈ C0 , without loss of generality we may assume that xε is compactly supported, namely, xε (t) = 0 for all t ∈ / K, where K ⊆ T is a compact set. Next, we choose ε ε x ∈ C, kx k = 1, such that hxε , y ⊥ i > ky ⊥ k − ε. Now, since y ⊥ vanishes on C0 , without loss of generality we may assume that xε (t) = 0 for all t ∈ K. Moreover, by 1.8.5, without loss of generality we may assume that |hxε , y0 i| < ε. Clearly, we have kxε + xε k = 1. On the other hand, hxε + xε , y0 + y ⊥ i = hxε , y0 i + hxε , y0 i + hxε , y ⊥ i + hxε , y ⊥ i. By what has been proved above, |hxε + xε , y0 + y ⊥ i| > ky0 k + ky ⊥ k − 3ε, which completes the proof. ¤ Remark. In the course of the proof we have established that any µ ∈ C00 possesses a natural continuation to the functionalR Jµ : C(T, E) → C. It is easy to verify that if E = C then hx, Jµi coincides with x dµ. A similar interpretation takes place for the proof of 1.8.3. 1.8.7. Absolutely continuous measures. Let T be a locally compact topological space, and let λ be a fixed positive measure on T . We call a function g : T → C locally integrable if for any compact set K ⊆ G the restriction of g to K is integrable. Clearly, g is measurable. We allow g to be defined locally a.e.. We denote by L1 loc = L1 loc (T, C, λ) the set of all locally integrable functions. And we denote by L1 loc = L1 loc (T, C, λ) the linear space of equivalence classes of all locally integrable functions with identification l.a.e.. A measure µ ∈ Mloc (T, C) is called absolutely continuous with respect to λ if there exists g ∈ L1 loc (T, C, λ) such that for all x ∈ C00 (T, C) Z Z x dµ = xg dλ. (5) (Here and below we set x(t)g(t) = 0 provided one of the factors is undefined, but the other is equal to zero.) Clearly, if g and f coincide l.a.e. they induce the same measure µ. We call g a density of µ and briefly write (5) as dµ(t) = g(t) dλ(t)

or

µ = g λ.

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Proposition. Let λ be a positive measure on T , and let E be a Banach space. Let a measure µ ∈ Mloc (T, C) be absolutely continuous with respect to the measure λ with the density g ∈ L1 loc (T, C, λ). Then (a) ([Bou3 , ch. 5, §5, 2, proposition 2]) |µ| = |g|λ. (b) ([Bou3 , ch. 5, §5, 3]) For any x : T → E one has x ∈ L1 (T, E, µ) if and only R R if xg ∈ L1 (T, E, λ); and for such an x, the equality x dµ = xg dλ holds. (c) If g ∈ L1 (T, C, λ) then µ ∈ M(T, C). In such a case, for any x : T → E one has x ∈ L1 (T, E, µ) if and only if xg ∈ L1 (T, E, λ); and (5) holds. Moreover, kµkM = kgkL1 . ¯R ¯ Proof. (c) Let g ∈ L1 (T, C, λ). Then ¯ xg dλ¯ ≤ kxk · kgkL1 for all x ∈ C00 . Thus the measure µ = gλ is bounded; namely, kgλkM ≤ kgkL1 . Since µ is bounded we have L1 (T, E, µ) = L1 (T, E, µ). On the other hand, since g ∈ L1 (T, C, λ) the function g is equal to zero outside a σ-finite set. Therefore so is xg. Hence xg ∈ L1 (T, E, λ) implies xg ∈ L1 (T, E, λ). Thus by (b) x ∈ L1 (T, E, µ) if and only if xg ∈ L1 (T, E, λ). Finally, we prove that kµkM = kgkL1 . Indeed, by 1.8.1 and 1.8.2 we have kgkL1

¯ o n ¯Z ¯ ¯ = sup ¯ xg dλ¯ : x ∈ L∞ (T, C, λ), kxk ≤ 1 ¯ n ¯Z o ¯ ¯ = sup ¯ xg dλ¯ : x ∈ C00 (T, C), kxk ≤ 1 = kgλkM .

¤

1.8.8. The Lebesgue–Radon–Nikodym theorem Theorem ([Bou3 , ch. 5, §5, 5], [Hew, theorem 12.17]). Let T be a locally compact topological space, and let λ and µ be positive measures on T . The following conditions are equivalent. (a) The measure µ is absolutely continuous with respect to λ. (b) For any h ∈ L1 (T, C, µ), h(t) ≥ 0, and Rany ε > 0 there exists R ∗ δ > 0 such ∗ that for all x : T → C, 0 ≤ x(t) ≤ h(t), x dλ ≤ δ implies x dµ ≤ ε. (c) For any h ∈ C00 (T, C), h(t) ≥ 0, and any Rε > 0 there exists Rδ > 0 such that for all x ∈ C00 (T, C), 0 ≤ x(t) ≤ h(t), x dλ ≤ δ implies x dµ ≤ ε. (d) Any locally λ-null set is locally µ-null. (e) Any λ-null set is µ-null. 1.8.9. Mutually singular measures. Let T be a locally compact topological space. We say that a measure µ ∈ Mloc (T, C) is concentrated on a µ-measurable set M if the set T \ M is µ-locally null, cf. 1.8.5. Measures µ, ν ∈ Mloc (T, C) are called mutually singular if they are concentrated on disjoint sets M and N . Proposition. Let µ, ν ∈ Mloc (T, C) be mutually singular measures. Then (a) ([Bou3 , ch. 5, §5, 7]) The sets M and N (on which µ and ν are concentrated) can be chosen universally measurable.

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(b) One has |µ + ν| = |µ| + |ν|. If µ, ν ∈ M this identity implies kµ + νk = kµk + kνk. Proof. (b) Clearly, to prove the first identity it suffices to show that hz, |µ + ν|i = hz, |µ| + |ν|i + for all z ∈ C00 (see 1.8.4 for the definition of hz, |µ|i). By the definition of the modulus of a measure we have |µ + ν| ≤ |µ| + |ν|, i.e., hz, |µ + ν|i ≤ hz, |µ| + |ν|i + for all z ∈ C00 . We prove the opposite inequality. + We fix z ∈ C00 and ε > 0. By the definition of the modulus of a measure we choose xµ and xν in C00 such that |xµ (t)|, |xν (t)| ≤ z(t) for t ∈ T , and

¯ ¯Z ¯ ¯ ¯ xµ dµ¯ > hz, |µ|i − ε

and

¯Z ¯ ¯ ¯ ¯ xν dν ¯ > hz, |ν|i − ε.

R Clearly, without loss of generality we may assume that the numbers xµ dµ and R xν dν are real and positive. Next, since C00 is dense in L1 we can find a continuous function αM which approximates the characteristic function 1M both in L1 (T, C, µ) and L1 (T, C, ν) to within ε, i.e., kαM − 1M kL1 < ε. It is easy to see that for the function αN (t) = 1 − αM (t) we have the similar inequality kαN − 1N kL1 < ε. Without loss of generality we may assume that αM and αN take their values in [0, 1]. Let K = supt∈T |z(t)|. Then |xµ (t)|, |xν (t)| ≤ K, too. We observe that ¯Z ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ αN xν dµ¯ ≤ ¯ (αN − 1N )xν dµ¯ + ¯ 1N xν dµ¯ Z ≤ |αN (t) − 1N (t)| · |xν (t)| d|µ| + 0 < εKkµk and similarly Z ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ xµ dµ − αM xµ dµ¯ = ¯ αN xµ dµ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ≤ ¯ (αN − 1N )xµ dµ¯ + ¯ 1N xµ dµ¯ Z ≤ |αN (t) − 1N (t)| · |xµ (t)| d|µ| + 0 < εKkµk.

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Consequently Z ¯Z ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ αM xµ dµ − hz, |µ|i¯ ≤ ¯ αM xµ dµ − xµ dµ¯ + ¯ xµ dµ − hz, |µ|i¯ ≤ εKkµk + ε. Similar estimates hold for ν. Multiplying functions xµ and xν by relevant complex factors, without loss of generality we may assume that the numbers hαM xµ , µi and hαN xν , νi are real and positive. We set x = αM xµ + αN xν . From the preceding estimates we have ¯Z ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ ¯ x dµ − hz, |µ|i ≤ α x dµ − hz, |µ|i + α x dµ ¯ ¯ ¯ ¯ ¯ ¯ M µ N ν ≤ ε + 2εKkµk and similar estimates hold for ν. Clearly, |x(t)| = |αM (t)xµ (t) + αN (t)xν (t)| ≤ αM (t)|xµ (t)| + αN (t)|xν (t)| ≤ z(t). Therefore Z ¯Z ¯ ¯ ¯ hz, |µ + ν|i ≥ ¯ x dµ + x dν ¯ ¯ ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ≥ ¯hz, |µ|i + hz, |ν|i¯ − ¯ x dµ − hz, |µ|i¯ − ¯ x dν − hz, |ν|i¯ ≥ hz, |µ|i + hz, |ν|i − 2ε − 4εKkµk, which implies the required inequality hz, |µ + ν|i ≥ hz, |µ| + |ν|i. R To prove the identity kµ+νk = kµk+kνk it suffices to observe that 1 dλ = kλk for all positive measures λ. ¤ 1.8.10. The Lebesgue decomposition M = Ms ⊕ Mac . Let T be a locally compact topological space with a fixed positive measure λ. We denote by Mac = Mac (T, C) = Mac (T, C, λ) the subspace of M(T, C, λ) consisting of all measures absolutely continuous with respect to λ. We say that a measure µ ∈ M(T, C) is singular with respect to λ if µ and λ are mutually singular (see 1.8.9) or, equivalently, if µ is concentrated on a λ-locally null set M ⊆ T . Since µ is bounded the set M is µ-summable and, by 1.8.5(b), it may be taken λ-null. We denote by Ms = Ms (T, C) = Ms (T, C, λ) the subset of M consisting of all singular measures.

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Theorem. (a) The space Mac (T, C, λ) is naturally isometrically isomorphic to the space L1 (T, C, λ). Namely, µ ∈ Mac (T, C, λ) if and only if µ = gλ for some g ∈ L1 (T, C, λ). (b) The set Ms (T, C, λ) is a closed subspace of M(T, C, λ). (c) ([Bou3 , ch. 5, §5, 7]) The space M(T, C) is isomorphic to the direct sum of its subspaces Ms (T, C, λ) and Mac (T, C, λ). For the corresponding representation of µ ∈ M in the form µ = µs + µac , one has |µ| = |µs | + |µac |

and

kµk = kµs k + kµac k.

(d) The space C 0 = C(T, C)0 has a complemented subspace which is naturally isometrically isomorphic to L1 = L1 (T, C, λ); we denote this subspace by the same symbol L1 . For the natural projector P : C 0 → C 0 onto L1 and any y ∈ C 0 one has kyk = kP yk + k(1 − P )yk. Proof. (a) By 1.8.7(c) the belonging g ∈ L1 (T, C, λ) implies gλ ∈ M(T, C) with kgλkM = kgkL1 . Conversely, assume µ ∈ Mac (T, C, λ). By definition this implies that µ ∈ M and µ = gλ with g ∈ L1 loc (T, C, λ). First we show that g ∈ L1 . Let α : T → [0, 1] be a continuous function with a compact support. Then, clearly, αg ∈ L1 (T, C, λ). By the definition of the norm kαgλkM ≤ kgλkM . Consequently by 1.8.7(c) we have kαgkL1 = kαgλkM ≤ kgλkM . Since α is arbitrary this implies that g ∈ L1 with kgkL1 ≤ kgλkM . Clearly, in the representation µ = gλ one can change g ∈ L1 to g˜ ∈ L1 . (b) follows easily from 1.1.2. (d) follows from 1.8.6 and (c). ¤ 1.8.11. The Yosida–Hewitt theorem. Let T be a locally compact topological space with a positive measure λ, and let E be a Banach space. We say that a functional ξ ∈ L0∞ = L∞ (T, E, λ)0 is absolutely continuous with respect to λ if it can be represented in the form Z hx, ξi =

hx(t), g(t)i dλ(t)

for all x ∈ L∞ ,

(6)

where g ∈ L1 = L1 (T, E0 , λ). To some extent an absolutely continuous functional can be interpreted as an extension of a bounded absolutely continuous vectorvalued measure from C00 to L∞ , cf. 1.8.2. By 1.8.1 the subspace of all absolutely continuous functionals ξ ∈ L0∞ is isometrically isomorphic to L1 ; we denote it by the same symbol L1 .

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Theorem. Let E be a finite-dimensional Banach space. (a) ([YoH, theorem 1.24]). Let T be a compact (not locally compact) topological space with a positive measure λ. The subspace L1 of all absolutely continuous functionals is complemented in L0∞ = L∞ (T, E, λ)0 . Moreover, there exists a projector P : L0∞ → L0∞ onto L1 which possesses the property kξk = kP ξk + k(1 − P )ξk for all ξ ∈ L0∞ . (b) Let G be a locally compact abelian group of the form Rc × V. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and p0 and q 0 be the conjugate indices. Then the subspace Lp0 q0 = Lp0 q0 (G, E0 ) (see 1.8.1) is complemented in L0pq = Lpq (G, E)0 . Moreover, there exists a projector P : L0pq → L0pq onto Lp0 q0 which possesses the property kP k ≤ 1. Proof. (a) First, we consider the case E = C. By 1.4.12 L∞ = L∞ (T, C) is isometrically isomorphic to C = C(Tb, C), where Tb is a compact topological space. Consequently the conjugate spaces L0∞ and C 0 are isometrically isomorphic, too. In particular, since T is compact the measure λ on T induces the bounded R functional λ : x 7→ x dλ on L∞ , and by virtue of the isomorphism L∞ (T ) ' C(Tb) ˆ on C(Tb), i.e., a measure on Tb. the functional λ By 1.8.10 there exists a projector P from C(Tb)0 to itself onto the subspace ˆ of measures on Tb absolutely continuous with respect to λ, ˆ with the Mac (Tb, C, λ) ˆ = kP ξk ˆ + k(1 − P )ξk ˆ for all ξˆ ∈ C(Tb)0 . Thus it suffices to show that property kξk 0 a functional ξ ∈ L∞ is absolutely continuous with respect to λ if and only if the ˆ corresponding functional ξˆ ∈ C 0 is absolutely continuous with respect to λ. ˆ First, we suppose Let ξˆ be a measure absolutely continuous with respect to λ. ˆ additionally that ξ is positive. We make use of the Lebesgue–Radon–Nikodym theorem (see 1.8.8). We observe that since Tb is compact, the function h(t) = 1 can be considered as the general case in 1.8.8(c). Thus, by virtue of the isomorphism ˆ and ξˆ can be rewritten equivalently between L∞ and C, condition 1.8.8(c) for λ as follows. (f) For any ε > 0 there existsR δ > 0 such that for all x ∈ L∞ (T, C, λ), 0 ≤ x(t) ≤ 1, the inequality x dλ ≤ δ implies hx, ξi ≤ ε. Clearly, (f) implies assumption 1.8.8(c) for λ and the restriction of ξ to C(T ). Hence the restriction of ξ to C(T ) ⊆ L∞ (T ) is an absolutely continuous measure. Therefore ξ on C(T ) has the form (6). By 1.8.2 ξ has a unique continuous extension from C(T ) to L∞ (T ). But formula (6) defines such an extension. Thus ξ is an absolutely continuous functional. The case of a non-positive ξ follows from 1.8.4. Conversely, let ξ be a functional absolutely continuous with respect to λ, i.e., ξ can be identified with the measure gλ, g ∈ L1 . We suppose, in addition, that ξ is positive. Then from assertion 1.8.8(b) applied to λ and gλ it follows assertion (f), which in turn means that ξˆ is absolutely continuous. The case of a non-positive ξ follows from 1.8.4. We turn to the case of an arbitrary E. We observe that if E is finite-dimensional the space L∞ (T, E, λ) is isometrically isomorphic to the space E ⊗ε L∞ (T, C, λ)

1.8. CONJUGATE SPACES

95

(cf. 1.7.4). Therefore L∞ (T, E, λ)0 can be represented as E0 ⊗π L∞ (T, C, λ)0 (see the example in 1.7.6). Now the assertion follows from 1.7.11. (b) By 1.8.1, for p 6= ∞ and q 6= ∞ we have trivially that Lp0 q0 = L0pq . The case q = ∞ is reduced to the case q = 0 by 1.8.3. Thus it remains to consider the case ¢ p = ∞ and q 6= ∞. By virtue of 1.8.1 ¡ we represent L0pq as lq0 I, L∞ (Qi , E)0 and then according to (a) we represent L∞ (Qi , E)0 as Fs i ⊕ Fac i with ¡ kξi k = kξs0 ¢i k + kξac ¡ i k. Clearly, ¢the norm of the 0 0 natural projector ¢ P from lq 0 I, L∞ (Qi , E) ' lq I, Fs i ⊕ Fac i into itself onto ¡ 0 0 lq I, {0} ⊕ Fac i ' L1q (G, E ) is equal to unity. ¤ 1.8.12. The space Mq0 . Let G be a locally compact abelian group of the form Rc × V, and let E be a finite-dimensional Banach space. Assume 1 ≤ q < ∞ or q = 0, but q 6= ∞. We denote by Mq0 = Mq0 (G, E0 ) the conjugate of the space Cq (G, E). Since C00 is dense in Cq , one may interpret elements of Mq0 as vector-valued measures. In particular, M1 (G, C) is the space M(G, C) of all bounded measures. By 1.7.6(d) Mq0 (G, E0 ) is topologically isomorphic to E0 ⊗ Mq0 (G, C); moreover, by the example in 1.7.6 the isomorphism M1 (G, E0 ) ' E0 ⊗π M1 (G, C) is isometric. By virtue of this isomorphism the notion of measure concentrated on a set (see 1.8.5) carries over to vector-valued measures. We denote by Mi = MQe i (G, E0 ) the space of all vector-valued measures e i. µ ∈ M1 (G, E0 ) concentrated on Q Let 1 ≤ q ≤ ∞ or q = 0, and let q 0 be the conjugate index. Since the space Cq = Cq (G, E) is embedded isometrically in L∞q (G, E), by 1.8.1 any function y ∈ L1q0 = L1q0 (G, E0 ) induces a functional on Cq . Moreover, by 1.8.2 thus defined embedding of L1q0 (G, E0 ) in Cq (G, E)0 is isometric. Theorem. Let E be a finite-dimensional Banach space. (a) For q 6= ∞ the space Mq0 (G, E0 ) is topologically isomorphic to the space lq0 (I, Mi ). (b) For all 1 ≤ q ≤ ∞ or q = 0 the space L1q0 = L1q0 (G, E0 ) is complemented as a subspace of Cq0 = Cq (G, E)0 . Moreover, there exists a projector P : Cq0 → Cq0 onto L1q0 which possesses the property kP k ≤ 1. Proof. (a) Since Cq ¢is topologically isomorphic to lq (I, C♦i ) (see 1.6.10) we ¡ have Mq0 ' lq0 I, (C♦i )0 . So it suffices to prove that (C♦i )0 ' Mi . Next we observe that the vector-valued space C♦i can be represented as the tensor product of E and the scalar space C♦i . Therefore (see the example in 1.7.6) it suffices to consider the case E = C. e i induces a functional f on C♦i with Clearly, any measure µ concentrated on Q kf k ≤ kµk. Conversely, assume f is a functional C♦i . We observe that C♦i is isometrically isomorphic to the subspace Cx (Qi ) of the space C(Qi ) consisting of all functions e i . By virtue of this isomorphism we consider which are equal to zero on Qi \ Q the corresponding functional on Cx (Qi ) and its extension by the Hanh–Banach theorem to the whole of C(Qi ). It is induced by a measure µ on Qi . Replacing µ to e i ) = 0. 1Qi \Qe i µ (see 1.8.7), without loss of generality we may assume that |µ|(Qi \ Q

96

I. FUNCTIONAL ANALYSIS PRELIMINARIES

Let CG\Qi q = CG\Qi q (G, C) denote the subspace of Cq = Cq (G, C) consisting of all functions being equal to zero on Qi . We observe that C(Qi ) is isomorphic to Cq /CG\Qi q because any continuous function x : Qi → C has an extension to a function x ˜ ∈ Cq (G, C) by the Urysohn theorem. Thus µ can be interpreted as an element of (Cq )0 = Mq0 vanishing on CG\Qi q . By the definition of |µ| (see 1.8.4) |µ| is equal to zero on CG\Qi q , too. Consequently since G \ Qi is open (and therefore its characteristic function is lower semi-continuous, by the definition of e i) = 0 the Lebesgue integral) we have |µ|(G \ Qi ) = 0. Since, in addition, |µ|(Qi \ Q e i. we conclude that µ is concentrated on Q (b) The case of q = ∞ is reduced to the case q = 0 by 1.8.6. We recall from 1.6.3 that the norm ¡ on Cq =¢ Cq (G, E) is defined by cmeans of the natural embedding in lq = lq I, C(Qi , E) , where Qi = (k + [0, 1] ) × d, (k, d) ∈ I = Zc × D. We identify Cq with its image in lq . Then by 1.2.9 and 1.3.9 Cq0 can be identified with lq0 /Cq⊥ . ¡ ¢ Since q 6= ∞, by 1.8.1(a) lq0 can be identified with lq0 I, C(Qi , E)0 . In turn (see the example in 1.7.6) C(Qi , E)0 ' E0 ⊗π M(Qi , C). Next, by 1.8.10 M(Qi , C) ' Ms (Qi , C) ⊕ Mac (Qi , C). Hence M(Qi , E0 ) ' Ms (Qi , E0 ) ⊕ Mac (Qi , E0 ) (see 1.7.11), where Ms and Mac has the natural meaning. Moreover, the projector π on M(Qi , E0 ) onto Mac (Qi , E0 ) possesses the property kµk = kπµk + k(1 − π)µk. (We also note that Mac (Qi , E0 ) is isometrically isomorphic to L1 (Qi , E0 ).) Thus ¡ ¢ ¡ ¢ lq0 I, M(Qi , E0 ) ' lq0 I, Ms (Qi , E0 ) ⊕ Mac (Qi , E0 ) . ¡ ¢ 0 0 I, {0} ⊕ Mac (Qi , E ) From this representation it follows that the subspace l ' q ¡ ¢ ¡ ¢ 0 0 0 0 0 0 lq I, L1 (Qi , E ) ' L1q (G, E ) is complemented in lq I, M(Qi , E ) and the norm of the natural projector Π : lq0 → lq0 onto L1q0 is less than or equal to 1. We prove that ¡ ¢ Cq⊥ ⊆ lq0 I, Ms (Qi , E0 ) ⊕ {0} , i.e., Cq⊥ ⊆ Ker Π. This will imply that the projector P : Cq0 → Cq0 onto L1q0 can be e : lq0 /Cq⊥ → lq0 /Cq⊥ , see 1.2.2. And the proof defined to be the quotient operator Π will be complete. For any i ∈ I we denote by Ci¡ the subspace of all¢ functions z ∈ C(Qi , E) vanishing on the boundary ∂Qi = k¡ + [0, 1]c \ ¢(0, 1)c × d. Clearly, lq (I, Ci ) is contained in the image of Cq in lq I, C(Qi , E) . Therefore Cq⊥ ⊆ lq (I, Ci )⊥ ¡ ¢ and it suffices to show that lq (I, Ci )⊥ ⊆ lq0 I, Ms (Qi , E0 ) ⊕ {0} . The subspace lq (I, Ci )⊥ is more convenient than Cq⊥ because it is easy to see that lq (I, Ci )⊥ = ⊥ ⊥ ⊆ Ms (Qi , E0 ). ). Thus it remains to show that Ci lq0 (I, Ci Clearly, it suffices to consider the case E = C. Assume a measure µ ∈ M(Qi , C) is equal to zero on Ci . Then by the¡ definition¢of |µ| we have that |µ| is equal to zero on Ci , too. Since the set Q0i = k +(0, 1)c ×d is open its characteristic function is lower semi-continuous. Therefore by the definition of the integral of a lower semi-continuous function (see 1.5.5) |µ|(Q0i ) = 0, i.e., |µ| and µ are concentrated on the set ∂Qi which has zero Haar measure. Thus µ ∈ Ms (Qi , C). ¤

CHAPTER II

THE INITIAL VALUE PROBLEM

We begin our investigation of functional differential equations with the initial value problem because of three reasons. First, it is an analogy of a classical topic in ordinary differential equations. Second, this is a comparatively simple problem. Third, the initial value problem is a necessary language for discussion of some other questions, e.g., stability. It should be mentioned that the initial value problem for functional differential equations is not as important for applications as the initial value problem for ordinary differential equations; so the third point seems to be the most important. There are many reasons for regarding functional equations as functional differential equations of the zero order. In particular, the theory of the initial value problem for functional equations does not differ essentially from that for functional differential equations. Moreover, as we shall see in 2.5.9 and 2.5.10, the solubility of a functional differential equation is reduced to the solubility of a functional equation. Therefore we consider both cases simultaneously. In applications an equation usually arises as an analytic expression. But in order to treat it mathematically one must specify what is meant by ‘derivative’, by ‘solution’, and so on. In this book we adhere to the methods of operator theory. So to begin the investigation of an equation, first of all we present it in the form Lx = f , i.e., put into correspondence with the equation an operator L acting from a functional space X to a functional space Y . The spaces X and Y may be chosen in different ways. And this choice may affect the mathematical properties of the equation. In this connection we shall discuss in §6.3 the difference between the approaches based on different functional spaces. Therefore we consider the initial value problem in a number of pairs (X, Y ). A non-trivial situation with the solubility of the initial value problem arises in the case of the functional differential equations of neutral type. Equations of neutral type are usually written in two forms: D

d x + Bx = f dt

and

d Dx + Bx = f. dt

We refer to them as equations with internal and external differentiation, respectively. These two forms of equation need different spaces X and Y . The initial value problem is a local one. So all facts we prove for the spaces of the type C and Lp hold for the spaces Cq and Lpq which differ from C and Lp only in the global sense.

97

98

II. THE INITIAL VALUE PROBLEM

The investigation of unique solubility of the initial value problem and stability can to a considerable extent be reduced to the investigation of causal invertibility of associated operators. Therefore we begin our discussion with the properties of causal invertibility and causal spectrum. We consider two kinds of solubility — unique solubility and evolutionary solubility. The first one is a straightforward analogue of the usual formulation of unique solubility for an ordinary differential equation. The notion of evolutionary solubility imposes the additional condition of causal dependence of the solution x upon the forcing function f . For ordinary differential equations these notions are equivalent, but for equations of neutral type this is not the case. In this book we deal mainly with the evolutionary solubility.

2.1. The causal invertibility We have already mentioned the notion of causal operator several times; in 2.1.1 we give a general definition. We say that a causal operator T is causally invertible if the operator T is invertible and the inverse operator is causal. This section is devoted to the systematic discussion of causal invertibility. 2.1.1. The definition of a causal operator. Let X and Y be linear spaces of functions defined on R (or on a part of R). The main examples of X and Y are Cq ,

Cq1 ,

Cq−1 ,

Lpq ,

1 Wpq ,

−1 Wpq .

But we shall try to write in such a way that generalizations to other functional spaces will be more or less evident. A linear operator T : X → Y is called causal if for all x ∈ X and t ¡ ¢ x(s) = 0, s < t, ⇒ T x (s) = 0, s < t. If X and Y are Banach spaces we denote by B+ = B+ (X, Y ) the set of all causal operators T ∈ B(X, Y ). If X = Y we employ the brief notation B+ = B+ (X). In this section, except whenever otherwise specified we assume that X and Y are linear spaces without norm. Example. The simplest and the most important examples of causal operators are the operators Z t ¡ ¢ N f (t) = n(t, s)f (s) ds, −∞

¡ ¢ Dx (t) =

∞ X

am (t)x(t − m),

m=0

¡ ¢ Lx (t) = x(t) ˙ + a(t)x(t), ¡

∞ ∞ X X ¢ Lx (t) = am (t)x(t ˙ − m) + bm (t)x(t − m), m=0

¡

¢ d Lx (t) = dt

µX ∞ m=0

m=0

¶

am (t)x(t − m) +

∞ X m=0

bm (t)x(t − m)

2.1. THE CAUSAL INVERTIBILITY

99

considered as acting between appropriate spaces. When one begins to investigate the initial value problem for a functional differential equation T x = f , a technical problem arises. Assume we have an equation T x = f on R and would like to consider it on a segment [a, b]. How should the operator T be applied to functions x with the domain [a, b] if T is defined initially only on functions x with the domain R? The causality of T helps to overcome this difficulty. First, it is convenient to reformulate the definition of a causal operator in a more formal way. Let Xt and Yt denote the subspaces of X and Y , respectively, consisting of all functions which are equal to zero on (−∞, t). Clearly, with this notation a linear operator T : X → Y is causal if T Xt ⊆ Yt

for all t.

(1)

This approach allows one to consider causal operators in the following abstract situation. Let X be a linear space. We call the collection { Xt ⊆ X : t ∈ R } of subspaces of X a T -direction (i.e., a time-direction) on X or simply a direction if Xa ⊇ Xb

for all a ≤ b.

If X is a Banach space we assume that the subspaces Xt are closed. For completeness we set X−∞ = X and X+∞ = {0}. Now let X and Y be a pair of spaces with directions. We say that a linear operator T : X → Y is causal if the condition (1) is satisfied. Clearly, for t = +∞ and t = −∞ the condition (1) is fulfilled for any linear operator T . Let X be a linear space with a direction, and let −∞ ≤ a < b ≤ +∞. We denote by Xa/b the quotient space Xa /Xb . We identify the space Xa/+∞ = Xa /X+∞ = Xa /{0} with Xa ; and we denote briefly the space X−∞/b = X−∞ /Xb = X/Xb by X/b . We denote by Qb = Qa/b : Xa → Xa/b the natural projection. Clearly, Xa/b is a Banach space provided so is X. Remark. Evidently, one can identify C−∞/b with C(−∞, b], and (Lp )−∞/b with Lp (−∞, b], and (Lp )a/b with Lp [a, b], and Ca/b with the subspace Ca (−∞, b] of C(−∞, b] consisting of all functions which are equal to zero on (−∞, a]. A similar representation occurs for the spaces Cq and Lpq . The cases of the spaces −1 Cq−1 and Wpq are more complicated (see 2.3.8 and 2.3.11). Let X and Y be linear spaces with directions, and let T : X → Y be a linear causal operator, and let −∞ ≤ a < b ≤ +∞. Let Ta : Xa → Ya denote the natural restriction of T (see the formula (1)), T/b : X/b → Y/b denote the quotient operator (1.2.2) induced by T , and Ta/b : Xa/b → Ya/b denote the quotient operator induced by Ta . The availability of a = −∞ and b = +∞ allows one to regard the last construction as a common one. We call the operator Ta/b the restriction of the operator T to the segment [a, b]. Clearly (see 1.2.2), kTa/b k ≤ kT k if X and Y are Banach spaces and T : X → Y is a bounded operator.

100

II. THE INITIAL VALUE PROBLEM

Example. On the space Lpq = Lpq (R, E) or on the space Cq = Cq (R, E) we consider the Volterra integral operator ¡

¢ N x (t) =

Z

t

n(t, s) x(s) ds,

t ∈ R,

−∞

with a measurable kernel n : R × R → B(E). We assume that the kernel n satisfies conditions which ensure the boundedness of N , see, e.g., 1.5.12, 4.4.11 or 5.4.3. If one identifies the space (Lp )a/b with Lp [a, b] (we stress that the corresponding isomorphism is isometric), the operator Na/b can be represented as ¡ ¢ Na/b x (t) =

Z

t

n(t, s) x(s) ds,

t ∈ [a, b].

a

In a similar way, if one identifies the space Ca/b with the subspace Ca [a, b] of C[a, b] consisting of all functions x which are equal to zero at a (the isomorphism between Ca/b and Ca [a, b] is also isometric) then the operator Na/b can be represented by the same formula. Let X be a linear space with a direction. Clearly, for any fixed a ∈ R the family { Xt ∩ Xa : − ∞ ≤ t ≤ +∞ } forms a direction on the space Xa . In a similar way, the natural projections of the subspaces Xt into X/b form a direction on X/b . Finally, the natural projections of the subspaces Xt ∩ Xa into Xa/b = Xa /Xb form a direction on Xa/b . Of course, these directions are trivial for t < a and t > b. Nevertheless, the consideration of directions with a superfluous set of indices allows one to treat all examples as special cases of the common definition. Let X and Y be linear spaces with directions, and let T : X → Y be a linear causal operator, and let −∞ ≤ a < b ≤ +∞. Evidently, Ta/b Xt/b ⊆ Yt/b , i.e., Ta/b is causal, too. Proposition. Let X, Y , and Z be linear spaces with directions, and assume that T : X → Y and R : Y → Z are linear causal operators. Then the operator RT is also causal, and (RT )a/b = Ra/b Ta/b

for all a and b.

Proof. The proof is plain. ¤ We say that spaces X and X1 with directions are causally isomorphic if there exists an invertible operator Θ : X → X1 such that Θ maps Xt onto (X1 )t and Θ−1 maps (X1 )t onto Xt for all t. If X and X1 are Banach we assume additionally that Θ is bounded. For example, the natural projections (Xa/b )t of the subspaces Xt ∩ Xa into Xa/b = Xa /Xb are causally (isometrically) isomorphic to Xt/b = Xt /Xb , −∞ ≤ a ≤ t ≤ b ≤ +∞. Another example: (Xa /Xc )/(Xb /Xc ) is causally

2.1. THE CAUSAL INVERTIBILITY

101

(isometrically) isomorphic to Xa /Xb , −∞ ≤ a < b < c ≤ +∞. For simplicity of notation, usually we do not distinguish causally isomorphic spaces. Let X, Y , X1 , and Y1 be spaces with directions. We say that causal operators T : X → Y and T1 : X1 → Y1 are causally similar if there exist causal isomorphisms ΘX : X → X1 and ΘY : Y → Y1 such that ΘY T = T1 ΘX . Usually we do not distinguish causally similar operators. In the sequel, to simplify notation we shall often designate the operator Ta/b by the symbol T . There will be no confusion because in these cases we shall apply the operator T to elements of Xa/b . Remark. (a) Let X and Y be functional spaces on R. We denote by X t and Y t the subspaces of X and Y consisting of all functions which are equal to zero on (t, +∞). Clearly, { X t } and { Y t } are inverse T -directions, i.e., X a ⊆ X b for all a ≤ b; they correspond to the inverse direction of time. Operators T : X → Y satisfying T X t ⊆ Y t for all t with respect to such directions are called anticausal. Anticausal operators arise naturally in the following situation. Assume we have a T -direction on a Banach space X. We recall from 1.2.9 that the conjugate space (X 0 )t of X/t = X/Xt can be considered as a subspace of X 0 . It is easy to see that the family { (X 0 )t } is an inverse T -direction. Thus the conjugate of a causal operator is anticausal. (b) Some functional spaces X require the set of indices for a T -direction different from R. We consider an example. Let X be the space M = M(R, C) of all bounded complex measures on R, see 1.8.4 for the definition. In this ¡ ¢ case one must distinguish¡ between¢ the subspaces Mt−0 = { µ : |µ| (−∞, t) = 0 } and Mt+0 = { µ : |µ| (−∞, t] = 0 }. Thus we have the natural set of indices { t − 0, t + 0 : t ∈ R }. We recall that M is the conjugate of C0 = C0 (R, C). Therefore the conjugate T 0 of an operator T ∈ B+ (C0 ) acts on M. We note that the T -direction on C0 induces the only of the ¡ inverse ¢ T -direction on M consisting S subspaces Mt−0 =¡ { µ : |µ| ¢[t, +∞) = 0 }. Nevertheless, s 2 is reduced to the case n = 2 by means of induction. The converse statement follows from 2.1.3. ¤ 2.1.8. Local causal invertibility. Let T : X → Y be a causal operator. We say that the operator T is left (right) causally invertible at a point t ∈ R if T is causally invertible on the segment [t − δ, t] (respectively, [t, t + δ]) for at least one δ > 0. We observe that, by 2.1.3, in this case T is causally invertible on segments [t − η, t] (respectively, [t, t + η]) for all sufficiently small η > 0. We say that the operator T is causally invertible at −∞ (+∞) if T is causally invertible on the segment [−∞, t] (respectively, [t, +∞]) for at least one t ∈ R. We note that, by 2.1.3, in this case T is causally invertible on segments [−∞, s] (respectively, [s, +∞]) for all sufficiently small (large) s. Corollary. Let T : X → Y be a causal operator, and let −∞ ≤ a < b ≤ +∞. The operator T is causally invertible on [a, b] if and only if it is left causally invertible at each point t ∈ (a, b] and is right causally invertible at each point t ∈ [a, b). Here, by right causal invertibility at −∞ we merely mean causal invertibility at −∞. And similarly so for left causal invertibility at +∞. Proof. Assume that T is causally invertible at all points, i.e., for any t ∈ (a, b] there exists a left neighbourhood [t − δ, t] (or [A, +∞] if t = b = +∞) on which T is causally invertible, and for any t ∈ [a, b) there exists a right neighbourhood [t, t + δ] (or [−∞, B] if t = a = −∞) on which T is causally invertible. By virtue of the compactness of [a, b] one can choose a finite covering of [a, b] by these neighbourhoods. From the preceding corollary it follows that T is causally invertible on [a, b]. The converse statement follows from 2.1.3. ¤

106

II. THE INITIAL VALUE PROBLEM

2.1.9. Local ordinary invertibility. Let T : X → Y be a causal operator. We say that the operator T is left (right) invertible at a point t ∈ R if T is invertible on any segment [t − δ, t] (respectively, [t, t + δ]) for all (!) sufficiently small δ > 0. We say that the operator T is invertible at −∞ (+∞) if T is invertible on segments [−∞, t] (respectively, [t, +∞]) for all sufficiently small (large) t. Proposition. Let T : X → Y be a causal operator. The operator T is left (right) invertible at t ∈ R if and only if it is left (right) causally invertible at t ∈ R. The operator T is invertible at −∞ (+∞) if and only if it is causally invertible at −∞ (+∞). Proof. By 2.1.3 local causal invertibility implies local ordinary invertibility. The converse statement follows from 2.1.5 applied to a relevant operator Ta/b (see 2.1.2). ¤

2.2. The causal spectrum In this section we consider spectral properties of causal invertibility. If X = Y are Banach spaces the set of all causal operators forms a Banach algebra. This observation allows one to apply methods of spectral theory to the investigation of causal invertibility. 2.2.1. Causal invertibility is a spectral property. Below we assume that X is a Banach space with a direction. We recall from 2.1.1 that in this case the subspaces Xt are assumed to be closed. We also recall that we denote by B+ = B+ (X) the set of all causal operators T ∈ B(X). Proposition. B+ (X) is a Banach algebra. Proof. The proof is plain (cf. 2.1.1). See 1.4.1 for the definition of a Banach algebra. ¤ We stress that the example in 2.1.2 shows that the subalgebra B+ of B is not full, see the definition of a full subalgebra in 1.4.4. 2.2.2. The definition of the causal spectrum. We call the spectrum of T ∈ B+ in the Banach algebra B+ the causal spectrum of T and denote it by σ + (T ). Let −∞ ≤ a < b ≤ +∞. We denote by σ + (Ta/b ) the spectrum of Ta/b in the Banach algebra B+ (Xa/b ), and by σ(Ta/b ) the (ordinary) spectrum of Ta/b in the Banach algebra B(Xa/b ); if Xa/b = {0} we set σ + (Ta/b ) = σ(Ta/b ) = ∅. We call σ + (Ta/b ) the causal spectrum of T on [a, b], and we call σ(Ta/b ) the spectrum of T on [a, b]. The causal spectrum possesses all properties of the spectrum in a Banach algebra, see §1.4. For example, it is a compact subset of C. We recall the following property: Proposition. For every T ∈ B+ the set σ + (Ta/b ) is the union of σ(Ta/b ) and the (possibly empty) collection of bounded components of the complement of σ(Ta/b ). In particular, the boundary of σ + (Ta/b ) lies in σ(Ta/b ). Proof. This is a special case of 1.4.5.

¤

107

2.2. THE CAUSAL SPECTRUM

2.2.3. The causal spectrum as the union of ordinary spectra Theorem. Let T ∈ B+ and −∞ ≤ a < b ≤ +∞. Then ª S© σ + (Ta/b ) = σ(Tc/d ) : [c, d] ⊆ [a, b] . Proof. The proof follows immediately from 2.1.5.

¤

2.2.4. The local causal spectrum. Let T ∈ B+ and t ∈ R. We set T + σ + (Tt−δ/t ), σ + (T, t+) = σ (Tt/t+δ ), δ>0 δ>0 ¡ ¢ T + σ (T−∞/t ), σ + (T, −∞) = σ + T, (−∞)+ = t>−∞ ¡ ¢ T + σ + (T, +∞) = σ + T, (+∞)− = σ (Tt/+∞ ).

σ + (T, t−) =

T

t0

0 0. We have that the restriction x ¯n of xn and the restriction y¯n of x˙ n to [−r, r] belong to L1 [−r, r], and k¯ xn kL1 ≤ ckxn kL and k¯ yn kL1 ≤ ckx˙ n kL for some c independent of n (but dependent on r). We observe that the operator of integration Z t ¡ ¢ Iy (t) = y(s) ds 0

acts continuously on L1 [−r, r]. Therefore, for example, we have µX ¶ X ∞ ∞ I y¯n = I y¯n n=1

n=1

114

II. THE INITIAL VALUE PROBLEM

as the equality of elements of L1 [−r, r]. On the other hand, by the definition of the Lebesgue derivative there exist en ∈ E such that ¡ ¢ x ¯n (t) − en = I y¯n (t), t ∈ [−r, r]. P∞ Let us treat e as constant functions in L [−r, r]. We recall that the series ¯n n 1 n=1 x P∞ and n=1 y¯n converge in L1 [−r, r]. Therefore we have the equality µX ¶ ∞ ∞ ∞ ∞ X X X x ¯n − en = I y¯n = I y¯n n=1

n=1

n=1

n=1

P∞

in L1 [−r, r]. In particular, the series n=1 en converges in L1 [−r, r], and consequently it converges in E to some e. Finally, we obtain the equality µX ¶ µ ³X ∞ ∞ ´¶ x ¯n (t) − e = I y¯n (t), n=1

n=1

¡ ¢ or, equivalently, x(t) − e = Iy (t) almost everywhere on [−r, r] and consequently almost everywhere on R. Thus y is the Lebesgue derivative of x. ¤ 2.3.3. The derivative of a product in Wp1 loc . We denote by Wp1 loc = Wp1 loc (R, E), 1 ≤ p ≤ ∞, the linear space of all continuous functions x : R → E whose restrictions to any segment [a, b], −∞ < a < b < +∞, belong to Wp1 [a, b]. 1 Clearly, Wpq ⊆ Wp1 loc ⊆ W11 loc for all p and q. ¡ ¢ Lemma. Let U ∈ Wp1 loc (R, C) and V ∈ Wp1 loc (R, E), or U ∈ Wp1 loc R, B(E) ¡ ¢ ¡ ¢ and V ∈ Wp1 loc (R, E), or U ∈ Wp1 loc R, B(E) and V ∈ Wp1 loc R, B(E) . Then for all a, t ∈ R Z a

t

Z t ˙ U (s)V (s) ds = U (t)V (t) − U (a)V (a) − U˙ (s)V (s) ds, a ¡ ¢. U (t)V (t) = U˙ (t)V (t) + U (t)V˙ (t).

Proof. Obviously, the formulae are equivalent. If U, V ∈ C 1 the second formula follows immediately from the classical definition of a derivative. The first formula is a consequence of it. Assume U, V ∈ Wp1 loc . We denote by u and v the functions U˙ and V˙ , respectively. By assumption we have u, v ∈ Lp loc . Since C[a, t] is dense in L1 [a, t] we can pick sequences un , vn ∈ C[a, t] which converge in L1 [a, t] to u and v, respectively. For r ∈ [a, t] we set Z r Z r Un (r) = U (a) + un (s) ds, Vn (r) = V (a) + vn (s) ds. a

a

Clearly, Un → U and Vn → V in C[a, t]. Passing to the limit in the formula of integration by parts for Un , Vn ∈ C 1 , we obtain the formula of integration by parts for U, V ∈ Wp1 loc . ¤

2.3. SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

115

1 2.3.4. The isomorphism U between Cq1 and Cq , and Wpq and Lpq

Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) The operator U x = x˙ + x 1 is a topological isomorphism from Wpq onto Lpq and from Cq1 onto Cq . In both cases the inverse operator is defined by the formula

¡ −1 ¢ U f (t) =

Z

+∞

e−s f (t − s) ds

0

Z

t

=

e−(t−s) f (s) ds.

−∞ 1 (b) The space Wpq is embedded topologically in Cq .

Proof. (a) First, we show that the operator U −1 defined by the preceding formula acts continuously from Lpq into Lpq and from Cq into Cq for all p and q. We represent U −1 f as ¡

U

−1

¢ f (t) = e−t

Z

t

es f (s) ds.

(1)

−∞

We consider the matrix

©

¡ ¢ª (U −1 )ij ∈ B Lp [0, 1] with the entries

Z 1 ´ −(i−j) −t U f (t) = e e es f (s) ds ij 0 Z t ³¡ ¢ ´ −1 −t U f (t) = e es f (s) ds ij 0 ¡ −1 ¢ U =0 ij ³¡

−1

¢

for j < i, for j = i, for j > i

(we recall that here t ∈ [0, 1]). Clearly, this is a matrix of the class e (see 1.6.9). Therefore it defines a bounded operator, say Λ, acting on Lpq . It is straightforward verification that Λ coincides with the initial operator U −1 . Hence U −1 acts continuously from Lpq to Lpq . Next, we observe that from the representation (1) it follows easily that U −1 f , f ∈ Lpq , is a continuous function. Since, in particular, U −1 acts on L∞q , the operator U −1 maps Cq into itself. 1 Second, we show that U −1 acts from Lpq into Wpq and from Cq into Cq1 , and −1 U U = 1. Applying 2.3.3 to the right hand side of (1) we obtain ¶. µ Z t ¡ −1 ¢. d s −t e f (s) ds U f (t) = e dt −∞ ¡ ¢ = − U −1 f (t) + f (t).

116

II. THE INITIAL VALUE PROBLEM

¡ ¢. 1 Consequently U −1 f belongs to Lpq (or to Cq ) and thus U −1 f belongs to Wpq (or ¡ ¢ ¡ ¢ . 1 −1 −1 to Cq ). Moreover, the preceding equality implies that U f (t) + U f = f , i.e., U U −1 = 1. Finally, we show that U −1 U = 1, i.e., Z e

−t

t

¡ ¢ es x(s) ˙ + x(s) ds = x(t).

−∞

Again from 2.3.3 we have Z

t

Z s

t

−N

e x(s) ˙ ds = e x(t) − e

t

x(−N ) −

−N

es x(s) ds.

−N

Since x˙ ∈ Lpq ⊆ L1∞ the function x increases at infinity no faster than a linear one. Therefore limN →+∞ e−N x(−N ) = 0. Letting N → +∞ we obtain Z

t

Z s

t

t

e x(s) ˙ ds = e x(t) − −∞

es x(s) ds,

−∞

which implies U −1 U = 1. Rt 1 (b) Assume x ∈ Wpq . From the representation x(t) = x(0) + 0 x(s) ˙ ds it is −1 clear that x is continuous. It is straightforward to verify that (U )ij acts from L1 [0, 1] to L∞ [0, 1] and the matrix { (U −1 )ij } belongs to e(Z, L1 [0, 1], L∞ [0, 1]), cf. 5.3.8 below. Thus U −1 maps Lpq ⊆ L1q into L∞q . Hence by (a) it follows that 1 ⊆ Cq . The continuity of the embedding is evident. ¤ Wpq 2.3.5. The distribution derivative. Let ψ : R → C be a function. The closure of the set { t : ψ(t) 6= 0 } is called the support of ψ and is denoted by the symbol supp ψ. We denote by D = D(R, C) the linear space of all infinitely many times differentiable functions ψ : R → C with a compact support. We say that a sequence ψk ∈ D converges to a function ψ ∈ D (in the sense of D) if: (a) the supports of ψk are uniformly bounded, i.e., there exists a segment [a, b] such that supp ψk ⊆ [a, b] for all k; (n) (b) the sequence ψk converges uniformly to ψ (n) for all n = 0, 1, 2, . . . ; here ψ (n) is the n-th derivative of ψ, in particular, ψ (0) is ψ itself. We note that the operator of differentiation ψ 7→ ψ˙ acts continuously on D, i.e., if ψn ∈ D converges to ψ ∈ D then ψ˙ n converges to ψ˙ (the proof is apparent). Let E be a fixed Banach space with the norm | · |. Let f : D → E be a linear (vector-valued) functional. As usual (cf. 1.1.7) we denote the value of the functional f on ψ by hψ, f i. We say that the functional f is continuous if hψk , f i converges to hψ, f i whenever ψk converges to ψ. Clearly, if f is continuous at the point ψ = 0, it is continuous at all points. Any continuous linear functional f : D → E is called a distribution on R (with values in E). By misuse of language, sometimes we shall call elements of D0 functions. We denote by D0 = D0 (R, E) the space of all distributions f .

2.3. SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

117

We say that a sequence fk ∈ D0 converges to f ∈ D0 if hψ, fk i → hψ, f i for all ψ ∈ D. Remark. It can be shown (see, e.g., [Rud, theorem 6.4]) that the convergence in D is induced by a locally convex topology. In this topology the zero of D has no a countable neighbourhood base. Nevertheless, a linear functional f on D is continuous if and only if hψk , f i converges to hψ, f i whenever ψk converges to ψ (see [Rud, theorem 6.6] for the explanation). It is interesting that usually the discussion of distributions does not need the explicit mentioning of this topology. We also note that the topology on D0 is the ∗-weak topology, see 1.1.7. Let f ∈ D0 . The functional f˙ defined by the rule ˙ fi hψ, f˙i = −hψ, is called the derivative of the functional f . It is straightforward to verify that f˙ is actually linear and continuous; thus it belongs to D0 . Let x ∈ L1 loc (in particular, x may be a continuous function, or x ∈ Cq , or x ∈ Lpq ). It is easy to see that the functional fx defined by the rule Z +∞ hψ, fx i = ψ(t) x(t) dt −∞ 0

belongs to D . The distribution fx induced by a function x ∈ L1 loc is called regular, cf. 1.8.7. It is easy to see that fx = fy implies that x = y a.e.. Thus the mapping x 7→ fx defines the embeddings of L1 loc and its subspaces Cq and Lpq in D0 . We call the derivative f˙x ∈ D0 of the functional fx ∈ D0 the distribution derivative of x ∈ L1 loc . 1 Now assume that x ∈ Cq1 or x ∈ Wpq . From definitions and 2.3.3 it follows that f˙x = fx˙ , i.e., the classical derivative and the Lebesgue derivative (see 2.3.1) are distribution derivatives, too. The following proposition, in particular, asserts that the converse is also true. We say that F ∈ D0 is a primitive of f ∈ D0 if F 0 = f. Let f, g ∈ D0 . We say that f is equal to g on an open subset M ⊆ R or, briefly, f = g on M if hψ, f i = hψ, gi for all ψ ∈ D such that supp ψ ⊆ M . (We stress that it makes sense only the equality of distributions on open sets M .) Clearly, if f and g are regular, this definition is equivalent to the equality a.e.. Let α : R → C be an infinitely many times differentiable function and f ∈ D0 . We define the product αf by the rule hψ, αf i = hαψ, f i. It is straightforward to verify that αf is really a linear continuous functional, i.e., an element of D0 . Clearly, if f is regular, this definition agrees with the usual one. −1 −1 Remark. If f ∈ Cq−1 or f ∈ Wpq (for the definitions of Cq−1 and Wpq see 2.3.6 below), the conditions on α in the definition of αf can be relaxed. Indeed, by the following proposition, for f = u˙ + v we have αf = α(u˙ + v) = (αu). − αu ˙ + αv. Note that the right hand side of the last formula can be taken as the definition of αf not only for infinitely many times differentiable α.

118

II. THE INITIAL VALUE PROBLEM

Proposition. Let −∞ ≤ a < b ≤ +∞. (a) Any distribution f ∈ D0 has a primitive. (b) Let F ∈ D0 , and let F˙ be equal to zero on (a, b). Then F coincides on (a, b) with a constant function, i.e., there exists c ∈ E such that F = fx on (a, b), where x(t) = c, t ∈ R. Consequently two primitives F1 and F2 of the same f ∈ D0 differ by a constant, i.e., F1 − F2 is a regular distribution induced by a constant function. (c) Assume that F ∈ D0 has a regular derivative on (a, b), i.e., there exists y ∈ L1 loc such that F˙ = fy on (a, b). Then F is regular on (a, b) itself, namely, F = fx on (a, b) for a continuous x. Moreover, the Lebesgue derivative of x exists and coincides on (a, b) with y in the sense of L1 loc . 1 In particular, if x ∈ Wpq then the distribution derivative of x is regular and coincides with fx˙ , where x˙ is the Lebesgue derivative of x. (d) Let α be an infinitely many times differentiable function and f ∈ D0 . Then . (αf ) = αf ˙ + αf˙. ˙ = { ψ˙ : ψ ∈ D }. We consider the functional m : D → C Proof. (a) We set D R +∞ ˙ coincides with defined as follows: m(ψ) = −∞ ψ(s) ds. It is easy to see that D ˙ → D by the rule the kernel of m. We define the operator I : D Z t ¡ ¢ Iψ (t) = ψ(s) ds. −∞

It is easy to verify that I is continuous, i.e., Iψn → Iψ provided ψn → ψ. Clearly, I is the two-sided inverse of the operator of differentiation ψ 7→ ψ˙ considered as ˙ acting from D to D. By the definition of a primitive the functional F is a primitive of f if hϕ, ˙ F i = −hϕ, f i

for all ϕ ∈ D.

Performing the change ϕ = Iψ or ψ = ϕ˙ we obtain the equivalent definition: hψ, F i = −hIψ, f i

˙ for all ψ ∈ D.

Thus our aim is reduced to the continuation of the functional F defined by the ˙ to D. latter formula from D We fix a function ψ0 ∈ ¢D(a, b) such that m(ψ0 ) = 1. We observe that the ¡ ˙ We set operator ψ 7→ ψ − m(ψ)ψ0 on D is a continuous projector onto D. ¡ ¢ hψ, Fei = −hI ψ − m(ψ)ψ0 , f i. ˙ It is easy to Clearly, Fe is defined on the whole of D and coincides with F on D. 0 see that Fe is continuous. Thus Fe ∈ D is the required continuation.

2.3. SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

119

(b) We denote by D(a, b) the subspace of D whose elements ψ are supported ˙ ˙ in (a, b); we set D(a, b) = { ψ˙ : ψ ∈ D(a, b) }. Clearly, D(a, b) consists of all ψ ∈ D(a, b) with the zero mean on (a, b), i.e., coincides with the kernel of the Rb functional m(ψ) = a ψ(s) ds. We fix a function ψ0 ∈ D(a, b) such that m(ψ0 ) = 1. For any ψ ∈ D(a, b) we have the representation ¢ ¡ ψ = ψ − m(ψ)ψ0 + m(ψ)ψ0 . ¢ ¡ ˙ We stress that in this formula ψ − m(ψ)ψ0 ∈ D(a, b). ˙ F i = 0 for all ψ ∈ D(a, b), The equality of F˙ to zero on (a, b) means that −hψ, ˙ or, in other words, that F is equal to zero on D(a, b). We consider the constant function x(t) = c, where c = hψ0 , F i, and we consider the corresponding distribution fx . For any ψ ∈ D(a, b) we have ¡ ¢ hψ, fx i = c · m ψ − m(ψ)ψ0 + m(ψ)ψ0 ¡ ¢ = c · 0 + c · m m(ψ)ψ0 = c · m(ψ)m(ψ0 ) = c · m(ψ) = m(ψ)hψ0 , F i, ­¡ ¢ ® hψ, F i = ψ − m(ψ)ψ0 + m(ψ)ψ0 , F = hm(ψ)ψ0 , F i = m(ψ)hψ0 , F i. Thus fx is equal to F on (a, b). (c) We set

Z

t

X(t) =

y(s) ds. a

Then F and fX have the same derivative on (a, b). Hence by (b) F − fX is equal to a constant function z on (a, b). Thus F = fX+z on (a, b). Clearly, the Lebesgue derivative of x = X + z on (a, b) is equal to X˙ = y. (d) By definition . ˙ αf i = −hαψ, ˙ fi hψ, (αf ) i = −hψ, . = −h(αψ) − αψ, ˙ fi = hαψ, f˙i + hψ, αf ˙ i. ¤ −1 2.3.6. The spaces Cq−1 and Wpq . Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let L stand for Cq (R, E) or Lpq (R, E). We denote by W −1 = W −1 (R, E) the space of all f ∈ D0 representable in the form f = u˙ + v, where u, v ∈ L and u˙ is the distribution derivative of u. Note that the representation f = u˙ + v is not unique. We define the norm on W −1 by the formula © ª kf kW −1 = inf kukL + kvkL : f = u˙ + v, u, v ∈ L . −1 −1 If L = Lpq we denote the space W −1 by Wpq = Wpq (R, E). If L = Cq we denote −1 −1 −1 −1 −1 the space W by Cq = Cq (R, E); in particular, Wp−1 is Wpp and C −1 is C∞ .

120

II. THE INITIAL VALUE PROBLEM

−1 Proposition. The spaces Cq−1 and Wpq , 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0, are complete, i.e., Banach. 1 Proof. Let L stand for Cq or Lpq , let W 1 stand for Cq1 or Wpq , and let W −1 −1 stand for Cq−1 or Wpq , respectively. We consider the Banach space L⊕L consisting of all pairs (u, v) with the norm

k(u, v)k = kukL + kvkL . We consider the mapping Q : (u, v) 7→ u˙ + v from L ⊕ L into W −1 (we do not verify that Q is bounded). Clearly, Q acts onto. We show that (u, v) ∈ Ker Q if and only if u ∈ W 1 and v = −u. ˙ Actually, assume (u, v) ∈ Ker Q. Then u˙ + v = 0, and hence the distribution derivative u˙ of u is equal to −v ∈ L. Thus u˙ is regular. By 2.3.5(c) u is also regular, and one can mean by u˙ the Lebesgue derivative. Thus u ∈ W 1 . The converse is apparent. We note that the mapping u 7→ (u, −u) ˙ from W 1 onto Ker Q is an isometric isomorphism. Consequently Ker Q is a closed subspace, and the quotient space (L ⊕ L)/ Ker Q is a Banach space (see 1.1.3). To complete the proof it remains to remark that by the definitions of the norms e : (L ⊕ L)/ Ker Q → W −1 on W −1 and on a quotient space the induced operator Q is an isometric isomorphism. ¤ −1 equivalently as quotient Remark. One may define the spaces Cq−1 and Wpq spaces (L ⊕ L)/ Ker Q (see the proof), i.e., without mentioning the space D0 . This approach is simpler from the technical point of view, but not as visual as the chosen one. Such an idea of constructing of distributions is employed in [Boc1 ]. −1 . We recall that the 2.3.7. The equality to zero on (a, b) in Cq−1 and Wpq equality of distributions on (a, b) has been defined in 2.3.5.

Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let −∞ ≤ a < b ≤ −1 +∞. Let L stand for Cq or Lpq , and let W −1 stand for Cq−1 or Wpq , respectively. −1 A function f ∈ W is equal to zero on (a, b) if and only if it can be represented in the form f = u˙ + v, where both u, v ∈ L equal zero on (a, b) in the usual sense. Proof. Let f be equal to zero on (a, b), and let f = u˙ + v, u, v ∈ L, be an arbitrary representation of f . Then by 2.3.5(c) −v is the Lebesgue derivative of u 1 on (a, b). Clearly, there exists w ∈ W 1 (as usual W 1 stands for Cq1 or Wpq ) such that: (i) w(t) = u(t) for t ∈ (a, b); (ii) w(t) = 0 for t 6= (a − 1, b + 1) (we mean −∞ − 1 = −∞, +∞ + 1 = +∞). We consider the functions u1 = u − w and v1 = v + w. ˙ Clearly, u˙ 1 + v1 = u˙ + v = f , and both u1 and v1 are equal to zero on (a, b). The converse is evident. ¤ 2.3.8. The cut in Cq−1 . In this subsection we show that the space Cq−1 possesses a property similar to that of Lp . Namely, any f ∈ Cq−1 can be represented uniquely in the form f = f− + f+ , where f− is equal to zero on (−∞, a), and f+ is equal to zero on (a, +∞).

2.3. SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

121

We consider arbitrary a ∈ R and f ∈ Cq−1 . First we show that f has a representation f = u˙ + v, u, v ∈ Cq , such that u(a) = 0 and v(a) = 0. Indeed, let f = u˙ + v be an arbitrary representation. We pick a continuously differentiable and compactly supported function w : R → E such that w(a) = −u(a) and w(a) ˙ = v(a). For example, w(t) =

(¡ ¢¡ ¢2 (t − a)v(a) − u(a) 1 − (t − a)2 0

for |t − a| ≤ 1, for |t − a| ≥ 1.

Clearly, the representation f = (u + w). + (v − w) ˙ possesses the desired property. For any a ∈ R we define the operator Pa : Cq−1 → Cq−1 as follows. For f ∈ Cq−1 we take a representation f = u˙ + v, u, v ∈ Cq , such that u(a) = 0 and v(a) = 0. Then we set Pa f = u˙ 1 + v1 , where

½ u1 (t) =

0 u(t)

for t ≤ a, for t ≥ a;

½ v1 (t) =

0 for t ≤ a, v(t) for t ≥ a.

We show that this definition is correct, i.e., it does not depend on the choice of the representation f = u+v ˙ with u(a) = 0 and v(a) = 0. Indeed, assume we have a pair of representations f = u+v ˙ and f = x+y ˙ with u(a) = v(a) = x(a) = y(a) = 0. We must show that u˙ 1 + v1 = x˙ 1 + y1 . We have the equality u˙ + v = x˙ + y in the sense of distributions. Consequently (u − x). + v − y = 0. By 2.3.5(c) and 2.3.1 u − x is continuously differentiable. In particular, (u − x). (a) = 0 since v(a) = y(a) = 0. These imply that the function u1 − x1 is continuously differentiable and (u1 − x1 ). (t) + (v1 − y1 )(t) = 0 for all t ∈ R, which means the independence of the definition of Pa f from the choice of the representation f = u˙ + v. Proposition. Let 1 ≤ q ≤ ∞ or q = 0. The operator Pa : Cq−1 → Cq−1 is a projector possessing the following properties. (a) The image of Pa is the subspace (Cq−1 )a of all distributions which are equal to zero on (−∞, a). (b) The kernel of Pa is the subspace (Cq−1 )a of all distributions which are equal to zero on (a, +∞). In particular, it follows that Pa f coincides with f on (a, +∞). (c) Pa Pb = Pb Pa = Pb for a < b. (d) The projector Pa is causal. The projector Pa is determined uniquely by (a) and (b). In particular, we obtain that Cq−1 = (Cq−1 )a ⊕ (Cq−1 )a . Proof. Clearly, Pa2 = Pa . Thus actually Pa is a projector. (a) Clearly, Pa f is equal to zero on (−∞, a) for all f , i.e., Im Pa ⊆ (Cq−1 )a . Conversely, assume f = u˙ + v is equal to zero on (−∞, a). Then by 2.3.7 we may additionally assume that u and v are equal to zero on (−∞, a). By continuity we

122

II. THE INITIAL VALUE PROBLEM

have u(a) = 0 and v(a) = 0. Therefore u1 = u and v1 = v. Hence Pa f = f . Thus f ∈ Im Pa . (b) is proved similarly to (a). (c) follows from (a) and (b). (d) is evident. It remains to recall that a projector is determined uniquely by its image and kernel. ¤ Remark. Clearly, Pa is also anticausal, i.e., f (s) = 0,

s > t,

⇒

(Pa f )(s) = 0,

s > t.

−1 and Lpq 2.3.9. The isomorphism U between Cq−1 and Cq , and Wpq

Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) The operator U x = x˙ + x −1 is a topological isomorphism from Lpq onto Wpq and from Cq onto Cq−1 . In both cases the inverse operator is defined by the formula Z +∞ ¡ −1 ¢ ¡ ¢ U f (t) = u(t) + e−s v(t − s) − u(t − s) ds 0 Z t ¡ ¢ = u(t) + e−(t−s) v(s) − u(s) ds, −∞ −1

where f = u˙ + v. Clearly, U is causal. (b) The space Lpq is embedded topologically in Cq−1 . −1 Proof. (a) Let L stand for Cq or Lpq , and let W −1 stand for Cq−1 or Wpq , −1 respectively. Clearly, U acts continuously from L into W . Furthermore, by the definition of the norm on W −1 we have kx˙ + xkW −1 ≤ kxkL + kxkL = 2kxkL for all x ∈ W −1 . Thus kU : L → W −1 k ≤ 2. ¯ (in order We denote temporary the operator U : W 1 → L (see 2.3.4) by U −1 to distinguish it from the operator U : L → W ). We consider the operator Λ : W −1 → L defined by the rule

¯ −1 v − U ¯ −1 u, Λf = u + U where f = u˙ + v. We show that this definition is correct, i.e., Λf = 0 if u˙ + v = 0. ¯ −1 v − U ¯ −1 u = Indeed, in this case, by 2.3.5 u ∈ W 1 and we have Λf = u + U −1 −1 −1 ¯ u˙ − U ¯ u=u−U ¯ (u˙ + u) = u − u = 0. u−U We show that U Λ = 1. For f = u˙ + v we have ¯ −1 v − U ¯ −1 u) U (Λf ) = U (u + U = Uu + v − u = (u˙ + u) + v − u = u˙ + v = f.

2.3. SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

123

Next, we show that ΛU = 1. For x ∈ L we have Λ(U x) = Λ(x˙ + x) ¯ −1 x − U ¯ −1 x =x+U = x. Thus Λ is the inverse of U . It remains to observe that the formula for U −1 coincides with the definition of Λ. (b) follows from (a) and 2.3.4(b). ¤ −1 2.3.10. Atoms in Wpq . By the atom at a point a ∈ R with a value α ∈ E we mean the distribution δa,α = α ⊗ δa defined by the rule

hψ, δa,α i = ψ(a)α. Simple calculations show that δa,α can be represented in the form δa,α = u˙ + u, where ½ 0 for t < a, u(t) = ua,α (t) = (2) αe−(t−a) for t > a. −1 Thus δa,α ∈ Wpq .

Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. A distribution −1 g ∈ Wpq is equal to zero both on (−∞, a) and (a, +∞) if and only if g = δa,α . −1 Proof. We assume that g ∈ Wpq is equal to zero both on (−∞, a) and −1 −1 (a, +∞). Let x = U g, where U is defined as in 2.3.9. Since U −1 is causal, x(t) = 0 for t < a. On the other hand, the assumption g(t) = 0 for t > a implies −1 x(t) ˙ + x(t) = 0 on (a, +∞). We recall that g ∈ Wpq . Thus x ∈ Lpq . From the equation x(t) ˙ + x(t) = 0 we conclude that x is infinitely many times differentiable on (a, +∞) and hence coincides on (a, +∞) with the function (2) for some α. Thus g = U u = δa,α . The converse statement is evident. ¤ −1 2.3.11. The cut in Wpq . In this subsection we discuss problems similar −1 to that of 2.3.8 for the space Wpq . Unfortunately, the complete analogue of −1 proposition 2.3.8 is not true: the projector Pa onto (Wpq )a is not unique. This fact is related closely to the non-uniqueness of the solution of the initial value −1 ), see 2.4.5 for details. problem in the pair (Lpq , Wpq −1 −1 For any a ∈ R we define the operator Pa = Pa,W 1 : Wpq → Wpq by the formula

Pa = U Pa,L U −1 , −1 where U : Lpq → Wpq is defined as in 2.3.9 and Pa,L : L → L is the projector

¡ ¢ Pa,L x (t) =

½

0 for t < a, x(t) for t ≥ a.

124

II. THE INITIAL VALUE PROBLEM

Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. The operator −1 −1 Pa : Wpq → Wpq is a projector with the following properties. −1 (a) The image of Pa is the subspace (Wpq )a of all distributions which are equal to zero on (−∞, a). −1 a (b) The kernel of Pa is contained in the subspace (Wpq ) of all distributions which are equal to zero on (a, +∞). Consequently Pa f coincides with f on (a, +∞). (c) Pa Pb = Pb Pa = Pb for a < b. (d) The projector Pa is causal.

Proof. Clearly, Pa is a projector. ¡ ¢ ¡ ¢ (b) We observe that on (a, +∞) we have U Pa,L x (t) = U x (t) for all x ∈ Lpq . Therefore ¡ ¢ ¡ ¢ Pa f (t) = U Pa,L U −1 f (t) ¡ ¢ = U U −1 f (t) = f (t) −1 on (a, +∞) for all f ∈ Wpq . We show that this implies −1 a Ker Pa ⊆ (Wpq ) .

¡ ¢ Indeed, let f ∈ Ker Pa , i.e., Pa f = 0; then on (a, +∞) we have f (t) = Pa f (t) = −1 a 0, i.e., f ∈ (Wpq ) . (a)¡Since and both U −1 and U are causal we ¢ Pa,L x is equal to zero on (−∞, a) −1 have Pa f (t) = 0 on (−∞, a) for all f ∈ Wpq . Thus −1 Im Pa ⊆ (Wpq )a . −1 Let us prove the reverse inclusion Im Pa ⊇ (Wpq )a . The previous proposition −1 a −1 states that (Wpq ) ∩ (Wpq )a consists of atoms δa,α , α ∈ E. Straightforward calculations show that Pa δa,α = δa,α . Thus δa,α ∈ Im Pa or, in other words, −1 a −1 (Wpq ) ∩ (Wpq )a ⊆ Im Pa .

We show that this inclusion implies the identity −1 )a = {0}. Ker Pa ∩ (Wpq −1 a Indeed, from Ker Pa ⊆ (Wpq ) we have

¡ ¢ −1 −1 a −1 Ker Pa ∩ (Wpq )a = Ker Pa ∩ (Wpq ) ∩ (Wpq )a ¡ −1 a ¢ −1 = Ker Pa ∩ (Wpq ) ∩ (Wpq )a ⊆ Ker Pa ∩ Im Pa = {0}.

2.3. SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS

125

−1 Assume f ∈ (Wpq )a . We represent f as f = fk + fi , where fk ∈ Ker Pa and −1 −1 fi ∈ Im Pa . Since Im Pa ⊆ (Wpq )a we have fi ∈ (Wpq )a . Hence fk = f − fi must −1 −1 also lie in (Wpq )a . But by the equality Ker Pa ∩ (Wpq )a = {0} it follows fk = 0. −1 And thus we have f = fi ∈ Im Pa . So we have proved that Im Pa ⊇ (Wpq )a . (c) and (d) follow from the definition of Pa . ¤

Remark. (i) Unfortunately, a projector Pa with the properties (a)–(d) is not unique. For example, if we define Pa as U2 Pa,L U2−1 , where U2 x = x+2x, ˙ we obtain another projector Pa . Moreover, there is no any preferable way of choosing Pa . (ii) The freedom in the choice of Pa is not very wide. If we have a pair of projectors Pa and Pea satisfying conditions (a) and (b) of the proposition then, evidently, Pea f coincides with Pa f both on (−∞, a) and (a, +∞) for all f . Hence by 2.3.10 (Pa − Pea )f is an atom δa,α . (iii) Note that the projector Pa depends on a in the shift invariant way, i.e., Sb Pa = Sa Pb

for all a and b,

¡ ¢ where Sh x (t) = x(t − h). (iv) As it has been remarked, the projector Pa is causal. The main application of Pa will be the regularization of an initial value problem in 2.4.5. In that situation we shall apply Pa to a function f which may be defined only on (a, +∞). In such a case the causality of Pa may be inconvenient. We note that there exists an anticausal projector Pa satisfying assumptions (a) and (b) of the proposition. (We recall that an operator T is anticausal if f (s) = 0, s > t, ⇒ (T f )(s) = 0, s > t.) Namely, one can define Pa to be U 0 Pa,L (U 0 )−1 , where U 0 x = −x˙ + x. The proof is similar to the preceding one. (v) In chapter 3 we shall use the following important additional property of Pa . Namely, we shall use that Pa is an operator of the class e. This fact follows from the formula Pa = U Pa,L U −1 . We do not include this property in the proposition −1 −1 in order to not discuss the class e(Wpq , Wpq ) now. 1 2.3.12. The duality between Wpq and Wp−1 0 q 0 . We consider the operators d d 0 U = dt + 1 and U = − dt + 1. It is easy to verify that U 0 as well as U establishes 1 −1 isomorphisms between Wpq and Lpq , and Lpq and Wpq , but (U 0 )−1 is anticausal:

¡

0 −1

(U )

¢

Z

Z

0

s

f (t) =

e f (t − s) ds = −∞

+∞

et−s f (s) ds

for f ∈ Lpq ,

t

(U 0 )−1 f = −u + (U 0 )−1 (v + u)

−1 for f = u˙ + v ∈ Wpq .

0 1 For x ∈ Wpq (R, E) and g = u˙ + v ∈ Wp−1 0 q 0 (R, E ) we set

hx, gi = hx, gi(W 1

−1 pq ,Wp0 q 0 )

= hU x, (U −1 )0 gi(Lpq ,Lp0 q0 ) .

(3)

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II. THE INITIAL VALUE PROBLEM

−1 Similarly, for g = u˙ + v ∈ Wpq (R, E) and x ∈ Wp10 q0 (R, E0 ) we set −1 hg, xi = hg, xi(Wpq ,W 10

p q0

)

= h(U −1 )0 g, U xi(Lpq ,Lp0 q0 ) .

(4)

Here h·, ·i(Lpq ,Lp0 q0 ) is defined as in 1.8.1. Proposition. 1 0 (a) Formula (3) defines an embedding of Wp−1 If E is finite0 q 0 in (Wpq ) . dimensional and p, q 6= ∞ (for example, q = 0), this embedding is onto. −1 0 ) . If E is finite(b) Formula (4) defines an embedding of Wp10 q0 in (Wpq dimensional and p, q 6= ∞ (for example, q = 0), this embedding is onto. Let E be finite-dimensional, and let p, q 6= ∞. 1 (c) The conjugate of the operator of the natural embedding J : Wpq → Lpq is −1 the operator of the natural embedding J : Lp0 q0 → Wp0 q0 , i.e., hx, yi(Lpq ,Lp0 q0 ) = hx, yi(W 1

−1 pq ,Wp0 q 0 )

1 for all x ∈ Wpq and y ∈ Lp0 q0 .

−1 The conjugate of the operator of the natural embedding J : Lpq → Wpq is 1 the operator of the natural embedding J : Wp0 q0 → Lp0 q0 , i.e., −1 hx, yi(Lpq ,Lp0 q0 ) = hx, yi(Wpq ,W 10

p q0

)

for all x ∈ Lpq and y ∈ Wp10 q0 .

d 1 (d) The conjugate of the operator of differentiation dt → Lpq is the : Wpq −1 d operator of differentiation − dt : Lp0 q0 → Wp0 q0 . The conjugate of the opd −1 erator of differentiation dt : Lpq → Wpq is the operator of differentiation d 1 − dt : Wp0 q0 → Lp0 q0 .

Proof. (a) and (b) follows immediately from 1.8.1, 2.3.4, and 2.3.9. (c) We prove, e.g., the first assertion. By definition hx, yi(W 1

−1 pq ,Wp0 q 0 )

= hU x, (U −1 )0 yi(Lpq ,Lp0 q0 ) .

Thus we must prove that hx, yi(Lpq ,Lp0 q0 ) = hU x, (U −1 )0 yi(Lpq ,Lp0 q0 ) .

(5)

First we consider z = (U −1 )0 y (clearly, z ∈ Wp10 q0 ) and show that hx, ˙ zi(Lpq ,Lp0 q0 ) = hx, −zi ˙ (Lpq ,Lp0 q0 ) . R +∞ 1 Indeed, by 2.3.3 we have hx, ˙ zi − hx, −zi ˙ = −∞ (xz). ds. Since Wpq ⊆ Cq and 1 Wp0 q0 ⊆ Cq0 ⊆ C (see 2.3.4), and q 6= ∞ we have that x vanish at infinity and z is bounded. Thus the integral is equal to zero.

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2.4. THE UNIQUE SOLUBILITY

Now it is evident that hU x, zi(Lpq ,Lp0 q0 ) = hx, U 0 zi(Lpq ,Lp0 q0 ) . Performing in this identity the inverse change y = U 0 z we arrive at (5). The second assertion is proved similarly. 1 (d) We prove, e.g., the first assertion. It means that for all x ∈ Wpq and y ∈ Lp0 q0 one has hx, ˙ yi(Lpq ,Lp0 q0 ) = −hx, yi ˙ (W 1 ,W −1 . Indeed, 0 0) pq

−hx, yi ˙ (W 1

−1 pq ,Wp0 q 0 )

p q

= hU x, −(U −1 )0 yi ˙ (Lpq ,Lp0 q0 ) = hU x, (U −1 )0 (−y˙ + y)i(Lpq ,Lp0 q0 ) − hU x, (U −1 )0 yi(Lpq ,Lp0 q0 ) = hU x, (U −1 )0 U 0 yi(Lpq ,Lp0 q0 ) − hU x, (U −1 )0 yi(Lpq ,Lp0 q0 ) = hU x, yi(Lpq ,Lp0 q0 ) − hU x, (U −1 )0 yi(Lpq ,Lp0 q0 ) = hx, ˙ yi(Lpq ,Lp0 q0 ) + hx, yi(Lpq ,Lp0 q0 ) − hU x, (U −1 )0 yi(Lpq ,Lp0 q0 ) .

To complete the proof it remains to refer to (5). The second assertion is proved similarly. ¤

2.4. The unique solubility In this section we discuss the features of the initial value problem in various functional spaces and the simplest type of its solubility — the unique solubility. Our main goal is to formulate the conditions of solubility in operator terms. We postpone the discussion of examples until §2.6. In this and the next section we consider the abstract equation Lx = f with a causal L. For the time being, by the main example the reader may mean the equation of the kind N X

N ¡ ¢ X ¡ ¢ am (t)x˙ τm (t) + bm (t)x σm (t) = f (t)

m=1

m=1

with τm (t) ≤ t and σm (t) ≤ t. 2.4.1. Initial value problem for an abstract Y . Let X and Y be Banach spaces of functions defined on R, and let L ∈ B+ (X, Y ). We consider the equation Lx = f.

(1)

In this book we usually assume that (X, Y ) is one of the following pairs: (Cq , Cq ),

(Lpq , Lpq ),

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ).

All these six variants are important. We use the following terminology. If (X, Y ) is (Cq , Cq ) or (Lpq , Lpq ) we say that (1) is a functional equation. And if (X, Y ) is 1 −1 one of the pairs (Cq1 , Cq ), (Cq , Cq−1 ), (Wpq , Lpq ), and (Lpq , Wpq ), we say that (1) is a functional differential equation. Since in this chapter we study local theory the value of the index q is of no importance. Therefore usually it will be omitted. To shorten the exposition we first consider the case of abstract X and Y .

128

II. THE INITIAL VALUE PROBLEM

Let −∞ < a < b < +∞. We consider the initial value problem ¡ ¢ Lx (t) = f (t), x(t) = ϕ(t),

a < t < b,

(2)

t < a.

(3)

Here the forcing function f and the initial function ϕ are given, and the solution x is unknown. The usual interpretation of these equalities is the following. Equation (2) means that the behaviour of x on the time interval (a, b) is governed by the equation. And the identity (3) means that the behaviour of x for t < a is known. We usually imagine that t = a is the present moment of time, t < a is the past, and t > a is the future. Remark. Instead of (2), (3) one can consider, for example, the problem ¡ ¢ Lx (t) = f (t), x(t) = ϕ(t),

a ≤ t ≤ b, t ≤ a.

For concrete X and Y , it can be equivalent to (2), (3) or not. For example, both initial value problems are equivalent if X = C 1 and Y = C, but¡they ¢ do not if −1 −1 X = Y = lq . Furthermore, if Y = Wp or Y = C , the condition Lx (t) = f (t), a ≤ t ≤ b, is meaningless since one can compare distributions only on open sets. Variant (2), (3) is universal for the six pairs enumerated above. Let us discuss what one can mean by the equalities (2) and (3) in the case of arbitrary spaces X and Y . First of all x and ϕ must be functions of the class X, and f must be a function of the class Y . Condition (3) means that x − ϕ ∈ Xa , i.e., x and ϕ belong to the same equivalence class in X/Xa , see 2.1.1. Thus to verify (3) we only need to know the equivalence class of ϕ, and therefore we may assume that ϕ ∈ X−∞/a . In a similar way it is sufficient to know only the equivalence class of f in Y−∞/b = Y /Yb , i.e., we may assume that f ∈ Y−∞/b . The ability of a restriction of the kind f ∈ Ya/b is also important, and it specifies the most distinctions between different concrete Y . Finally, it suffices to know the equivalence class of x in X−∞/b = X/Xb . Moreover, we always assume that we look for the solution x of (2), (3) in X−∞/b ; otherwise we cannot ensure the uniqueness of the solution of (2), (3). We recall that if x lies in C 1 , C, Wp1 , or Lp then ϕ and x can be interpreted as functions of an appropriate class on (−∞, a] and (−∞, b], respectively. It is convenient to reduce the solubility of the initial value problem (2), (3) to the homogeneous case ϕ = 0. We do it in the following way. Assume ϕ ∈ X−∞/a . Let ψ ∈ X−∞/b be an arbitrary function whose natural projection in X−∞/a coincides with ϕ. (Note that in the spaces C 1 , C, Wp1 , and Lp the function ψ can be interpreted as a continuation of ϕ from (−∞, a] to (−∞, b].) We change x in (2), (3) to x = ψ + z,

2.4. THE UNIQUE SOLUBILITY

129

where z ∈ X−∞/b is a new unknown function. Substituting x = ψ + z in (2) and (3) we obtain the new initial value problem ¡

¢ Lz (t) = g(t), z(t) = 0,

a < t < b,

(4)

t < a,

(5)

where g = f − Lψ.

(6)

We observe that condition (5) can be interpreted as z ∈ Xa/b . Thus instead of the pair of the equalities (4) and (5) we may use only (4), but look for z in Xa/b . So we arrive at the following statement. Lemma. Initial value problem (2), (3) has a unique solution x ∈ X−∞/b if and only if initial value problem (4), (5) with the forcing function (6) has a unique solution z ∈ Xa/b . It is useful to note that if f runs over Y (and ψ is fixed, e.g., ψ = 0), g also runs over Y . In conclusion we turn to g. Since f ∈ Y−∞/b and ψ ∈ X−∞/b , and L is causal we may assume that g ∈ Y−∞/b . The question whether one may assume that g ∈ Ya/b we consider for concrete Y separately. 2.4.2. Initial value problem for Y = Lp . We assume that Y = Lp , and X is Wp1 or Lp . We say that equation (1) is uniquely soluble on [a, b] if the initial value problem (2), (3) has a unique solution x ∈ Wp1 (−∞, b] (respectively, x ∈ Lp (−∞, b]) for all f ∈ Lp and ϕ ∈ Wp1 (−∞, a] (respectively, ϕ ∈ Lp (−∞, a]). We turn back to the preceding lemma. We consider the equation (4). We denote by ge the function which coincides with g on (a, b) and is equal to zero on (−∞, a]; thus ge can be considered as an element of Ya/b = (Lp )a/b . We note that equation (4) can be rewritten as the equality Lz = ge, where ge ∈ Ya/b and z ∈ Xa/b . So the preceding lemma leads to the following proposition. Proposition. Let (X, Y ) be (Wp1 , Lp ) or (Lp , Lp ). Equation (1) is uniquely soluble on [a, b] if and only if the operator La/b : Xa/b → Ya/b is invertible. Remark. We have not use any special properties of the space X (but have used that of Y ). Therefore, e.g., the proposition remains true for X = Wpn (which corresponds to equations of n-th order), X = Wp−1 (integral equations), and X = 1/2

Wp

(singular integral equations).

130

II. THE INITIAL VALUE PROBLEM

2.4.3. Initial value problem for Y = C. Next, we assume that Y = C, and X is C 1 or C. We consider ¡ ¢ the initial value problem (4), (5). From (5) and the causality of L we have Lz (t) = 0 for t < a. Since Lz is a continuous function, from here and (4) we obtain g(a) = 0. This equality is called the compatibility condition (or sewing, or matching condition). Initial value problem (4), (5) can have solutions only if g satisfies this condition. From (6) it follows that in terms of the initial value problem (2), (3) the compatibility condition can be rewritten as ¡ ¢ Lϕ (a) = f (a). Remark. The compatibility condition is usually regarded as a non-correctness of the initial value problem. Indeed, let us assume we have a device which we turn on at the moment a. Assume that the turned on device is governed by the equation (1). So it is reasonable to describe the situation by means of (2) and (3). Here ϕ is the prehistory of x which has been governed earlier by other rules. It is unlikely that the compatibility condition is obligatorily satisfied in this case, and therefore there may be no solutions of the initial value problem (2), (3). Thus initial value problem (2), (3) with Y = C is a bad mathematical model for the situation of interest. Note that the model can be improved, e.g., by the replacement of X and Y by richer functional spaces provided the transient response lie in X. On the other hand, we may consider a process which is always governed by the equation Lx = f . If we interpret a as the present moment, the compatibility condition ¡ ¢ can be looked upon as satisfied, because in the past, i.e., for t < a we had Lϕ (t) = f (t). We say that equation (1) is uniquely soluble on [a, b] if the initial value problem (2), (3) has a unique solution x ∈ C 1 (−∞, b] (respectively, x ∈ C(−∞, b]) for all f ∈ C and ϕ ∈ C 1 (−∞, a] (respectively, ϕ ∈ C(−∞, a]) satisfying the compatibility condition. In the sequel we always assume that the compatibility condition is satisfied. Let us return to the equation (4). The compatibility condition means that g(a) = 0. We denote by ge the function which coincides with g on [a, b] and is equal to zero on (−∞, a]; thus ge can be considered as element of Ya/b = Ca/b . We note that in these terms equation (4) can be rewritten as Lz = ge, where ge ∈ Ya/b and z ∈ Xa/b . Thus from 2.4.1 we again obtain the following. Proposition. Let (X, Y ) be (C 1 , C) or (C, C). Equation (1) is uniquely soluble on [a, b] if and only if the operator La/b : Xa/b → Ya/b is invertible. Remark. The proposition remains valid if one substitutes an arbitrary Banach space with direction (cf. the remark in 2.4.2) instead of X.

2.4. THE UNIQUE SOLUBILITY

131

2.4.4. Initial value problem for Y = C −1 . Now we assume that Y = C −1 and X = C, i.e., consider equation (1) with L : C → C −1 . We say that equation (1) is uniquely soluble on [a, b] if the initial value problem (2), (3) has a unique solution x ∈ C(−∞, b] for all f ∈ C −1 and ϕ ∈ C(−∞, a]. Example. (a) Let D, B ∈ B+ (C). We consider the operator Lx =

d Dx + Bx. dt

It is easy to see that L is causal and acts continuously from C into C −1 . We show that this example is general. Indeed, assume L ∈ B+ (C, C −1 ) is arbitrary. We consider the operator D = U −1 L : C → C, where U x = x˙ + x. By 2.3.9 D is d causal. It remains to note that L = U D or L = dt Dx + Dx. + (b) Let D, B ∈ B (C). We consider the initial value problem ¡

¢. ¡ ¢ Dx − u (t) + Bx (t) = v(t), x(t) = ϕ(t),

a ≤ t ≤ b, t ≤ a.

Here ϕ ∈ C(−∞, a] and u, v ∈ C[a, b] are given, and x is unknown. We say that a function x is a solution if x belongs to C(−∞, b], and the function Dx − u is continuously differentiable on [a, b] (at a and b in the one-sided sense), and the equalities are satisfied pointwise; the functions Dx and u are not assumed to be differentiable. We show that this initial value problem is equivalent to the initial value problem (2), (3). First of all we observe that by continuity the inequalities a ≤ t ≤ b and t ≤ a can be changed equivalently to a < t < b and t < a. Next, we note that if one interprets the derivative in the equation ¡ ¢. ¡ ¢ Dx − u (t) + Bx (t) = v(t), a < t < b, as the distribution derivative, one obtains the equivalent condition, because (see 2.3.5) the function is differentiable in the usual sense when the distribution derivative is continuous. Thus the equation can be rewritten as ¡ ¢. ¡ ¢ Dx (t) + Bx (t) = u(t) ˙ + v(t), a < t < b, and we arrive at the equation (2) with f = u˙ + v ∈ C −1 . We return to the end of 2.4.1. We consider equation (4). We denote by ge the function Pa g, where Pa is defined in 2.3.8. By virtue of 2.3.8 g coincides with g on (a, b) and is equal to zero on (−∞, a); thus ge can be considered as an element of Ya/b = (C −1 )a/b . We note that equation (4) can be rewritten as Lz = ge, where ge ∈ Ya/b and z ∈ Xa/b . From 2.4.1 we obtain the following proposition.

132

II. THE INITIAL VALUE PROBLEM

Proposition. Let (X, Y ) be (C, C −1 ). Equation (1) is uniquely soluble on [a, b] if and only if the operator La/b : Xa/b → Ya/b is invertible. Remark. This proposition remains valid if one substitutes an arbitrary Banach space with a direction (cf. the remarks in 2.4.2 and 2.4.3) instead of X = C. 2.4.5. Initial value problem for Y = Wp−1 . Finally, we consider the most complicated case Y = Wp−1 , i.e., we consider the equation (1) with L : Lp → Wp−1 . As above we shall try to reduce the solubility of the initial value problem on [a, b] to the invertibility of the operator La/b . Example. (a) Let D, B ∈ B+ (Lp ). We consider the operator Lx =

d Dx + Bx. dt

Clearly, L is causal and acts continuously from Lp to Wp−1 . As in the previous subsection, it is easy to show that this example is general. (b) In the theory of semi-groups induced by functional differential equations it is often considered the initial value problem ¡ ¢. ¡ ¢ µ ∗ x (t) + ν ∗ x (t) = 0, a ≤ t ≤ b, x(t) = ϕ(t),

t≤a

and its variants. Here µ and ν are bounded measures on [0, +∞) and ∗ means the convolution, see ch. 4 for the definition. For our discussion it is important that ϕ is usually assumed to be in Lp . We return to the end of 2.4.1. We consider the initial value problem (4), (5). First of all we explain why usually the solution z of (4), (5) is not unique. By (5) the function z belongs to (Lp )a/b ; therefore, since L is causal, Lz belongs to (Wp−1 )a/b , i.e., Lz is equal to zero on (−∞, a). On the other hand, condition (4) defines Lz on (a, b). And there are no other restrictions to Lz in (4), (5). But they do not determine the function Lz completely, as we have seen in 2.3.10. Therefore if the operator L is invertible (which is the most important case) then the function z is not determined uniquely. To make the solution of (4), (5) unique one should add additional conditions. It can be done in various ways. We prefer to do it in such a way that the existence of a unique solution be equivalent to the invertibility of La/b . We make use of 2.3.11. By 2.3.11(b) equality (4) is equivalent to ¡ ¢ ¡ ¢ Lz (t) = Pa g (t), a < t < b. We note that by (5) and the causality of L the function Lz is equal to zero on (−∞, a); and by 2.3.11(a) Pa g is also equal to zero on (−∞, a). We change the initial value problem (4), (5) to the equation ¡ ¢ ¡ ¢ Lz (t) = Pa g (t), t a.

We stress that since Pa is causal the index α ∈ E in δa,α is determined by the values of v(t), t < a. Next, from 2.3.11(b) we have ¡ ¢ Pa (x˙ + cx) (t) = x(t) ˙ + c(t)x(t), t > a. Therefore from the equation it follows x(t) ˙ + c(t)x(t) = v(t),

a < t < b.

135

2.4. THE UNIQUE SOLUBILITY

Thus again the solution can be represented as ½ y(t) for a < t < b, x(t) = ϕ(t) for t < a, where y is a solution of the equation x(t) ˙ + c(t)x(t) = v(t). We note that the value x(a + 0) = y(a + 0) may not coincide with the index α of δa,α . Indeed, we set ½ ϕ(t) for t < a, ψ(t) = 0 for t > a. Then as usual we perform the change of variables x = ψ + z and from the initial value problem we arrive at the equivalent equation Lz = Pa v − Pa Lψ with z ∈ (Lp )a/b and the operator Lz = z+cz ˙ acting from (Lp )a/b to (Wp−1 )a/b . Clearly, Lψ is equal to zero for t > a. Therefore by 2.3.11 and 2.3.10 Pa Lψ = δa,β for some β ∈ E. We note that since L and Pa are causal, β is determined by the values of ψ on (−∞, a). It is easy to verify that ½ 0 for t < a, z(t) = y(t) for a < t < b, where y is the solution of the equation x(t) ˙ + c(t)x(t) = v(t) satisfying the initial condition y(a + 0) = α − β. Therefore ½ ϕ(t) for t < a, x(t) = ψ(t) + z(t) = y(t) for a < t < b, with the same y. In particular, x(a + 0) = α − β. (b) Now assume that B ∈ B+ (Lp ). We consider the equation x˙ + Bx = v with v ∈ Lp . Let us interpret this equation as an equation in the pair (Lp , Wp−1 ). The situation is similar to the above. A unique solution of the initial value problem ¡ ¢ x(t) ˙ + Bx (t) = v(t), a < t < b, x(t) = ϕ(t),

t < a,

can be selected by the same additional initial condition x(a + 0) = α. A unique solution can also be selected by means of regularization. Nevertheless, we stress that if in the equation (1) f belongs to Wp−1 , the function x belongs to the class Lp on (a, b) and the condition x(a+0) = α losses its meaning. It also meaningless for the equation (1) with f ∈ Lp , but arbitrary L : Lp → Wp−1 . 2.4.6. The additivity of unique solubility. Let (X, Y ) be one of the pairs (C, C),

(Lp , Lp ),

(C 1 , C),

(Wp1 , Lp ),

(C, C −1 ),

(Lp , Wp−1 ),

and let L ∈ B+ (X, Y ). We consider equation (1). Proposition. Let −∞ < a < t < b < +∞. If equation (1) is uniquely soluble on two of the segments [a, b], [a, t], and [t, b], it is uniquely soluble on the third. Proof. The proof follows immediately from 2.1.4.

¤

136

II. THE INITIAL VALUE PROBLEM

2.5. The evolutionary solubility In this section we discuss the main definition of this chapter — the evolutionary solubility. We define evolutionary solubility to be the causal invertibility of the operator La/b : Xa/b → Ya/b . This allows us to reformulate the properties of evolutionary solubility as the properties of causal invertibility. Next (see 2.5.9 and 2.5.10), we show that the evolutionary solubility of a functional differential equation is equivalent to the causal invertibility of the operator D acting at the (highest) derivative. Since the domain and the range of D coincide the causal invertibility of D is a spectral property (see §2.2). In the next section we prove criteria for evolutionary solubility (causal invertibility) for the cases of some special operators D. 2.5.1. The definition of evolutionary solubility Example. Let

 for t ∈ / [0, 3],  t τ (t) = t/2 for t ∈ [0, 2],   2t − 3 for t ∈ [2, 3].

We stress that τ maps [0, 2] onto [0, 1]; and τ maps [2, 3] onto [1, 3]; and, finally, τ maps [0, 3] onto [0, 3]. Clearly, τ (t) ≤ t for all t; therefore the operator ¡ ¢ ¡ ¢ Dx (t) = x τ (t) is causal. It is easy to see that D acts continuously on C and Lp . We consider the functional equation ¡ ¢ x τ (t) = f (t) with f, x ∈ Lp . It is straightforward to verify that the equation is uniquely soluble on [0, 3]. We show that this is not the case on [0, 2] and [2, 3]. First, we consider the initial value problem ¡ ¢ x τ (t) = f (t), x(t) = 0,

0 < t < 2, t < 0,

on [0, 2]. The values of x(t) for t ∈ [1, 2] does not appear in the equation, since τ (t) takes its values in [0, 1] when t runs over [0, 2]. So they can be chosen in an arbitrary way. Thus the solution of the initial value problem is not unique. Now we consider the initial value problem ¡ ¢ x τ (t) = f (t), x(t) = 0,

2 < t < 3,

t < 2, ¡ ¢ on¡ [2, 3]. ¢ Since x(t) = 0 for t < 2, we have x τ (t) = 0 for t < 5/2; so the equation x τ (t) = f (t) can be satisfied only if f (t) = 0 for t ∈ (2, 5/2). Thus the solution of the initial value problem does not exist for some f .

137

2.5. THE EVOLUTIONARY SOLUBILITY

So the unique solubility may disappear away when the segment [a, b] decreases. Clearly, the replacing of Lp by another functional space cannot improve the situation. It is easy to show that the same phenomenon also holds for the more reasonable functional equation ¡ ¢ εx(t) + x τ (t) = f (t) and functional differential equations ¡ ¢ ¡ ¢ εx(t) ˙ + x˙ τ (t) + εx(t) + x τ (t) = f (t), ¡ ¢¢ ¡ ¢ d¡ εx(t) + x τ (t) + εx(t) + x τ (t) = f (t) dt provided ε is sufficiently small. We shall also see that the possibility of such examples are related closely to the fact the solution x of the initial value problem on [0, 3] depends on the forcing function f in a not causal way. (Cf. the examples in 1.4.4 and 2.1.2.) Let (X, Y ) be one of the pairs (C, C),

(Lp , Lp ),

(C 1 , C),

(Wp1 , Lp ),

(C, C −1 ),

(Lp , Wp−1 ),

and let L ∈ B+ (X, Y ). We say that the equation Lx = f.

(1)

is evolutionarily soluble on [a, b], −∞ < a < b < +∞, if it is uniquely soluble on [a, b] and the inverse of the operator La/b : Xa/b → Ya/b (which exists by the propositions of the ¡preceding section) is causal. We note that ¢ in the previous example the equation x τ (t) = f (t) is uniquely soluble on [0, 3], but not evolutionarily soluble. When we need to stress that we take f in Y and look for x in X, we say that the equation is considered, uniquely soluble, or evolutionarily soluble, respectively, in the pair (X, Y ). 2.5.2. Evolutionary solubility as complete unique solubility Theorem. The following assumptions are equivalent. (a) Equation (1) is evolutionarily soluble on [a, b]. (b) Equation (1) is uniquely soluble on any segment [c, d], [c, d] ⊆ [a, b]. (c) Equation (1) is uniquely soluble on any segment [a, t], a < t ≤ b. (d) Equation (1) is uniquely soluble on any segment [t, b], a ≤ t < b. Proof. The proof follows from the definition of evolutionary solubility, and 2.1.5, and the results of §2.4. ¤

138

II. THE INITIAL VALUE PROBLEM

2.5.3. The additivity of evolutionary solubility Theorem. Let [a, b] =

n X

[ak , bk ]

k=1

(we allow the mutual intersections of [ak , bk ] to have interior points). Equation (1) is evolutionarily soluble on [a, b] if and only if it is evolutionarily soluble on each subsegment [ak , bk ]. Proof. The proof follows immediately from 2.1.7.

¤

2.5.4. The decreasing of a segment Corollary. Let [c, d] ⊆ [a, b]. If equation (1) is evolutionarily soluble on [a, b] then it is evolutionarily soluble on [c, d]. Proof. The proof follows immediately from 2.5.3 or from 2.1.3. ¤ 2.5.5. Local evolutionary solubility. We say that equation (1) is left (right) evolutionarily soluble at a point t ∈ R if it is evolutionarily soluble on the segment [t − δ, t] ([t, t + δ]) for at least one δ > 0. We note that by 2.5.4, in this case it is evolutionarily soluble on [t − δ, t] ([t, t + δ]) for all sufficiently small δ > 0. Proposition. Equation (1) is evolutionarily soluble on [a, b] if and only if it is left evolutionarily soluble at each point t ∈ (a, b] and is right evolutionarily soluble at each point t ∈ [a, b). Proof. The proof follows immediately from 2.1.8.

¤

2.5.6. Local unique solubility. We say that equation (1) is left (right) uniquely soluble at a point t ∈ R if it is uniquely soluble on any segment [t − δ, t] (respectively, [t, t + δ]) for all sufficiently small δ > 0. Proposition. Equation (1) is left (right) uniquely soluble at t ∈ R if and only if it is left (right) evolutionarily soluble at t ∈ R. Proof. The proof follows immediately from 2.1.9.

¤

2.5.7. Small perturbations preserve the evolutionary solubility Proposition. Let L, N ∈ B+ (X, Y ). If equation (1) is evolutionarily soluble on [a, b] and kNa/b k ≤ 1/kL−1 a/b k then the equation (L − N )x = f is evolutionarily soluble on [a, b], too. Proof. The proof follows immediately from 2.2.12.

¤

2.5.8. The operator U as a causal isomorphism. We repeat once again the following fact.

2.5. THE EVOLUTIONARY SOLUBILITY

139

Proposition. The operator U x = x˙ + x is causally invertible in any of the following pairs (C 1 , C),

(Wp1 , Lp ),

(C, C −1 ),

(Lp , Wp−1 ).

Proof. The proof follows from the explicit formulae for U −1 from 2.3.4 and 2.3.9. ¤ 2.5.9. Equations with internal differentiation. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let L stand for Cq or Lpq , and let W 1 stand for Cq1 or 1 Wpq , respectively. Let D, B ∈ B+ (L). Clearly, the operator Lx = Dx˙ + Bx belongs to B+ (W 1 , L). We consider the equation with internal differentiation Dx˙ + Bx = f.

(2)

Theorem. Equation (2) is evolutionarily soluble on [a, b] if and only if the operator D is causally invertible on [a, b] or, equivalently, the equation Dy = g is evolutionarily soluble on [a, b] in the pair (L, L). In particular, evolutionary solubility does not depend on the operator B. Proof. By definition 2.5.1 the evolutionary solubility on [a, b] means the causal d invertibility of the operator L = D dt + B on [a, b]. We consider the operator T = LU −1 , where U x = x˙ + x, see 2.3.4. First, we note that L and T are causally invertible on [a, b] simultaneously. Indeed, according to 2.1.1 Ta/b = La/b (U −1 )a/b ; and according to 2.1.3 the operator U is causally invertible on [a, b]. Therefore (Ta/b )−1 = Ua/b (La/b )−1 and (La/b )−1 = (Ua/b )−1 (Ta/b )−1 . We note that the operator T can be represented as T = LU −1 = (Dd/dt + B)U −1 ¡ ¢ = D(d/dt + 1) + B − D U −1 = (DU + B − D)U −1 = D + (B − D)U −1 . By 2.2.8 and 2.2.9 (U −1 )a/b lies in the radical of the algebra B+ (La/b ). Since the radical is an ideal the operator (B − D)U −1 lies in the radical, too. But then by 1.4.8(c) the operators Ta/b and Da/b are causally invertible simultaneously. ¤ Example. Let us consider equation (2) with D = 1, i.e., x˙ + Bx = f. Such an equation is called an equation of retarded type or, simply, a retarded equation. Ordinary differential equation is a special case of retarded equation. Equation (2) with an arbitrary causal D is called an equation of neutral type or a neutral equation. Clearly, the operator D = 1 is causally invertible. Therefore retarded equation is evolutionarily soluble on any segment [a, b]. Remark. We note that the idea of the proof can be applied easily to equations of higher orders.

140

II. THE INITIAL VALUE PROBLEM

2.5.10. Equations with external differentiation. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let L stand for Cq or Lpq , and let W −1 stand for Cq−1 or −1 Wpq , respectively. Let D, B ∈ B+ (L). Clearly, the operator d Dx + Bx dt belongs to B+ (L, W −1 ). We consider the equation with external differentiation Lx =

d Dx + Bx = f. (3) dt Theorem. Equation (3) is evolutionarily soluble on [a, b] if and only if the operator D is causally invertible on [a, b] or, equivalently, the equation Dy = g is evolutionarily soluble on [a, b] in the pair (L, L). In particular, evolutionary solubility does not depend on the operator B. Proof. By definition 2.5.1 the evolutionary solubility on [a, b] means the causal d invertibility of the operator L = dt D + B on [a, b]. We consider the operator −1 T = U L, where U x = x˙ + x. As in the preceding proof, it is easy to see that L and T are causally invertible on [a, b] simultaneously. We note that the operator T can be represented as T = U −1 L = U −1 (d/dtD + B) ¡ ¢ = U −1 (d/dt + 1)D + B − D = U −1 (U D + B − D) = D + U −1 (B − D). Now, arguing as in the preceding proof one can show that the operators Ta/b and Da/b are causally invertible simultaneously. ¤ Example. Let us consider equation (3) with D = 1, i.e., x˙ + Bx = f. Here the distinction from the preceding example is that we use different functional spaces. We also call such an equation an equation of retarded type or, simply, a retarded equation, and the equation (3) with an arbitrary causal D an equation of neutral type or a neutral equation. Clearly, the operator D = 1 is causally invertible. Therefore retarded equation is evolutionarily soluble on any segment [a, b].

2.6. Criteria for evolutionary solubility At the end of the preceding section we have seen that the investigation of evolutionary solubility is reduced to the investigation of the causal invertibility of the operator D on finite segments. The operator D is a functional one, i.e., it acts from Cq or Lpq into itself. Therefore the causal invertibility of Da/b is equivalent to the condition 0 ∈ / σ + (Da/b ). Thus the problem can be reduced to the calculation of causal spectrum. We recall from 2.2.4 that σ + (Da/b ) is the union of σ + (D, t±). So the calculation of σ + (D, t±) constitutes the main part of this section.

2.6. CRITERIA FOR EVOLUTIONARY SOLUBILITY

141

2.6.1. Multiplication operators. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We call an operator of the form ¡ ¢ Ax (t) = a(t)x(t) ¡ ¢ a multiplication operator. We assume that a ∈ C R, B(E) if the ¡ operator ¡ ¢ ¢ A is considered as acting on Cq = Cq R, B(E) , and that a ∈ L R, B(E) if the ¡ ¢ ∞ operator A is considered as acting on Lpq = Lpq R, B(E) . We shall discuss multiplication operators more detailed in §5.1. Remark. If E is infinite-dimensional, the operator A acts on Cq and Lpq under weaker assumptions on a. For example, A acts on Cq if a is strongly continuous (see 1.1.5), i.e., the function t 7→ a(t)e is continuous for all e ∈ E. For simplicity we do not discuss such generalizations systematically, see also 5.1.6. Theorem. Let E be a Banach space, and let −∞ < c < d < +∞. ¡ ¢ (a) Assume a ∈ C R, B(E) . The operator A : Cq → Cq is invertible on [c, d] if and only if the¡operators ¢ a(t) are invertible for all t ∈ [c, d]. (b) Assume a ∈ L∞ R, B(E) . The operator A : Lpq → Lpq is invertible on [c, d] if and only if the operators a(t) are invertible for almost all t ∈ [c, d] and °¡ ¢−1 ° ° < ∞. ess sup° a(t) t∈[c,d]

The inverse of A on [c, d] is also a multiplication operator. In particular, A−1 is causal. Proof. (a) Let us identify (Cq )c/d with the subspace Cc [c, d] of the space C[c, d] consisting of all functions which vanish at c. Let a(t) be invertible for all ¡ ¢−1 t ∈ [c, d]. Then by the continuity of a and 1.4.2, the function b(t) = a(t) is also continuous. Thus the operator ¡ ¢ Bx (t) = b(t)x(t) acts continuously on Cc [c, d]. Clearly, B is the inverse of Ac/d . Conversely, assume that Ac/d is invertible. We show that a(t) is uniformly invertible for t ∈ (c, d]. We make use of 1.3.2. First we show that inf{ |a(t)|+ : t ∈ (c, d] } > 0. We take an arbitrary ε > 0 and suppose that there exists t∗ ∈ (c, d] such that |a(t∗ )|+ < ε. By the definition of | · |+ (see 1.3.1) we have ka(t∗ )ek < ε for some e ∈ E, kek = 1. Then from the continuity of a we obtain ka(t)ek < ε for all t in a neighbourhood U of t∗ . We consider a function x ∈ Cc [c, d] such that x(t) = h(t)e, where h : [c, d] → [0, 1] is equal to 1 at t∗ and is equal to zero outside of U . Clearly, kxk = 1 and kAxk ≤ ε. Consequently |A|+ ≤ ε. Thus if inf{ |a(t)|+ : t ∈ (c, d] } were equal to zero then |A|+ would be equal to zero, too. Suppose there exists t∗ ∈ (c, d] such that the image of a(t∗ ) does not contain a vector e ∈ E. Then Im A cannot contain functions x such that x(t∗ ) = e.

142

II. THE INITIAL VALUE PROBLEM

Thus we can claim that a(t) is invertible for all t ∈ (c, d]. Moreover, by 1.3.2(a) °¡ ¢−1 ° ° : t ∈ (c, d] } < ∞. Applying 1.3.6(c) we obtain that a(t) is invertsup{ ° a(t) ible at t = c as well. (b) We identify (Lpq )c/d with Lp [c, d]. Assume that a(t) is invertible for almost ¡ ¢−1 all t ∈ [c, d] and the function b(t) = a(t) is bounded. We note that by the ¡ ¢−1 is definition of a measurable function (see 1.5.6), the function b(t) = a(t) measurable. Therefore the operator ¡ ¢ Bx (t) = b(t)x(t) acts continuously on Lp [c, d]. Clearly, B is the inverse of Ac/d . Conversely, assume that Ac/d is invertible. We show that a(t) is uniformly invertible for almost all t ∈ [c, d]. We make use of 1.3.2. First, we show that ess inf{ |a(t)|+ : t ∈ [c, d] } > 0. We fix an arbitrary ε > 0 and suppose that the set Wε = { t ∈ (c, d) : |a(t)|+ < ε } have a positive measure. By virtue of the definition of a measurable function we choose a non-null compact subset Kε ⊆ Wε on which a is continuous. Next, arguing as in case (a) one can construct a function x ∈ Lp [c, d] vanishing outside Kε such that kxk = 1 and kAxk ≤ ε. This argument shows that ess inf{ |a(t)|+ : t ∈ [c, d] } > 0. Second, we show that ess inf{ |a(t)|− : t ∈ [c, d] } > 0. We consider the conjugate operator A0 . Let p0 be the conjugate index. We recall from 1.8.1 that Lp0 [c, d] can be identified with a subspace of (Lp [c, d])0 . By the definition of a measurable ¡ ¢0 function the function t 7→ a(t) is measurable. Hence one can consider the operator ¡ 0 ¢ ¡ ¢0 A y (t) = a(t) y(t) on Lp0 [c, d]. Clearly, A0 coincides with the restriction of A0 to (Lp [c, d])0 . Arguing ¡ ¢0 as above, one can show that |A0 |+ ≤ ess inf{ | a(t) |+ : t ∈ [c, d] }. It remains ¡ ¢0 to recall that |A0 |+ ≤ |A0 |+ by the definition of | · |+ , and |a(t)|− = | a(t) |+ by 1.3.4. Thus from 1.3.2(c) it follows that a(t) is invertible for almost all t ∈ [c, d], and °¡ ¢−1 ° ° : t ∈ [c, d] } < ∞. ¤ ess sup{ ° a(t) Remark. Obviously, assertion (b) remains valid for c = −∞ and d = +∞. Assertion (a) also remains ¡valid for ¢c = −∞ and d = +∞, but with the following correction. Suppose a ∈ C R, B(E) . The operator A : Cq → Cq is invertible (on [−∞, +∞]) if and only if the operators a(t) are invertible for all t ∈ R and °¡ ¢−1 ° ° < ∞. sup° a(t) t∈R

2.6.2. Difference operators. Let 1 ≤ p ≤ ∞, and 1 ≤ ¡q ≤ ∞ or¢ q = 0. Let L be either Cq¡ = Cq (R, ¢ E) or Lpq = Lpq (R, E). Let am ∈ C R, B(E) if L is Cq , and am ∈ L∞ R, B(E) if L is Lpq ; here m = 0, 1, . . . . Furthermore, we assume that ∞ ∞ X X sup kam (t)k < ∞ or ess sup kam (t)k < ∞, m=0

m=0

2.6. CRITERIA FOR EVOLUTIONARY SOLUBILITY

143

respectively. that hm 6= 0. We consider the operators ¡ Let¢ hm ∈ R, m = 1, 2, .¡. . . Assume ¢ Sh x (t) = x(t − hm ) and Am x (t) = am (t)x(t). We call the operator D = A0 +

∞ X

Am Shm

(1)

m=1

a difference operator. We recall from 1.6.12 that the family { Sh : h ∈ R } is uniformly bounded on Cq and Lpq . On the other hand, it is easy to see that kAm k ≤ ess sup kam (t)k. Thus series (1) converges absolutely. Clearly, ∞ X ¡ ¢ Dx (t) = a0 (t)x(t) + am (t)x(t − hm ). m=1

We denote the set of all difference operators acting on L by S = S(L). We shall discuss difference operators more detailed in §5.2. We assume that hm > 0. In this case the operator D is causal. Theorem. Let E be a Banach space, and let −∞ < c < d < +∞. (a) Let am ∈ C(R, B(E)), m = 0, 1, . . . . The operator D : Cq → Cq is causally invertible on [c, d] if and only if the operators a0 (t) are invertible for all t ∈ [c, d]. (b) Let am ∈ L∞ (R, B(E)), m = 0, 1, . . . . The operator D : Lpq → Lpq is causally invertible on [c, d] if and only if the operators a0 (t) are invertible for almost all t ∈ [c, d] and °¡ ¢−1 ° ° < ∞. ess sup° a0 (t) t∈[c,d]

Proof. We consider the shift operator ¡ ¢ Sh x (t) = x(t − h) with h > 0. We observe that (Sh )a/b = 0 provided b − a < h (see 2.1.1 for the definition of Ta/b ). Consequently by the definition of ¢G(·) (see 2.2.6) we have ¡ G(Sh , t±)¡ = 0 for all t¢∈ R, and consequently G (Sh )c/d = 0. Applying 2.2.6 we obtain G (Am Shm )c/d = 0, m = 1, 2, . . . , and, by the continuity of G, we have ∞ ³X ´ G (Am Shm )c/d = 0. m=1

¢ ¡ By virtue of 2.2.9 and 1.4.8(c) this implies that σ + (Dc/d ) = σ + (A0 )c/d . So it remains to refer to the conditions of the causal invertibility of A0 in 2.6.1. ¤

144

II. THE INITIAL VALUE PROBLEM

2.6.3. An application to an equation of neutral type Corollary. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let L be either Cq = Cq (R, E) or Lpq = Lpq (R, E). Let D ∈ B+ (L) be a difference operator of the form (1) and B ∈ B+ (L) be an arbitrary operator. Let −∞ < c < d < +∞. (a) Let am ∈ C(R, B(E)), m = 0, 1, . . . . The equation Dx˙ + Bx = f

(2)

is evolutionarily soluble on [c, d] in the pair (Cq1 , Cq ) if and only if the coefficient a0 (t) is invertible for all t ∈ [c, d]. (b) Let am ∈ C(R, B(E)), m = 0, 1, . . . . The equation d Dx + Bx = f dt

(3)

is evolutionarily soluble on [c, d] in the pair (Cq , Cq−1 ) if and only if the coefficient a0 (t) is invertible for all t ∈ [c, d]. (c) Let am ∈ L∞ (R, B(E)), m = 0, 1, . . . . Equation (2) is evolutionarily 1 , Lpq ) if and only if the coefficient a0 (t) is soluble on [c, d] in the pair (Wpq invertible for almost all t ∈ [c, d] and °¡ ¢−1 ° ° < ∞. ess sup° a0 (t) t∈[c,d]

(d) Let am ∈ L∞ (R, B(E)), m = 0, 1, . . . . Equation (3) is evolutionarily −1 ) if and only if the coefficient a0 (t) is soluble on [c, d] in the pair (Lpq , Wpq invertible for almost all t ∈ [c, d] and °¡ ¢−1 ° ° < ∞. ess sup° a0 (t) t∈[c,d]

Proof. The proof follows from 2.5.9 and 2.5.10, and 2.6.2.

¤

2.6.4. Varying retardation. Let E = C. We consider the operator ¡ ¢ ¡ ¢ T x (t) = x τ (t) .

(4)

We assume that τ : R → R is continuous and τ (t) ≤ t for all t ∈ R. Clearly, under these assumptions the operator T acts on C = C(R, C) and is bounded and causal. We consider the set K = Kτ = { t : τ (t) = t}. Clearly, K is closed. We denote the interior and the boundary of K by Int K and ∂K, respectively. We denote by ∂ + K (∂ − K) the set of all t ∈ K such that any right (left) neighbourhood of t contains points not in K. Finally, we denote by ∂ ++ K the set of all t ∈ ∂ + K such that ∀η > 0 ∃s ∈ (t, t + η)

τ (s) > t,

and we denote by ∂ +− K the complement ∂ + K \ ∂ ++ K.

2.6. CRITERIA FOR EVOLUTIONARY SOLUBILITY

145

Example. Let α ≤ 1. We consider the function ½ τ (t) =

t αt

for t ≤ 0, for t ≥ 0.

If α > 0 then 0 ∈ ∂ ++ Kτ . And if α ≤ 0 then 0 ∈ ∂ +− Kτ . Theorem. For operator (4) and t ∈ R one has (a) (b) (c) (d) (e)

σ + (T, t±) = {1} if t ∈ Int K; σ + (T, t±) = {0} if t ∈ / K; + σ (T, t+) = {0} if t ∈ ∂ +− K; σ + (T, t−) = { λ ∈ C : |λ| ≤ 1 } if t ∈ ∂ − K; σ + (T, t+) = { λ ∈ C : |λ| ≤ 1 } if t ∈ ∂ ++ K.

Proof. (a), (b), and (c) are proved similarly. For example, we prove (c). By the definition of ∂ +− K, for sufficiently small η > 0 we have Tt/t+η = 0. This implies σ + (T, t+) = {0}. Before embracing on cases (d) and (e), we prove the following auxiliary statement. Let −∞ < a < b < +∞, and let τ (t) < t for all t ∈ (a, b) (but not obligatorily for t = a and t = b), and for all n the set (a, b) ∩ τ n ((a, b)) is not empty. Then σ + (Ta/b ) = { λ ∈ C : |λ| ≤ 1 }. ¡ ¢ Here the iterations τ n are defined by the rule: τ 0 (t) = t, τ n (t) = τ τ n−1 (t) . Clearly, kT k ≤ 1. Therefore σ + (Ta/b ) ⊆ { λ ∈ C : |λ| ≤ 1 }. Conversely, from the assumption τ (t) < t and (a) we obtain σ + (T, t±) = {0} for all t ∈ (a, b). Therefore by 2.2.4 0 ∈ σ + (Ta/b ). Now let λ 6= 0 and |λ| ≤ 1. We show that λ ∈ σ + (Ta/b ). Assume the contrary: let the operator (λ − T )a/b be causally invertible. We take natural n. By assumption there exists a point c lying in ¡ an arbitrary ¢ n (a, b)∩τ (a, b) . We pick a point d ∈ (a, b) belonging to τ −n ({c}). Clearly, c < d. By 2.1.3 the causal invertibility of (λ − T )a/b implies the causal invertibility of (λ − T )c/d . Moreover, °¡ ¢ ° °¡ ¢ ° ° (λ − T )c/d −1 ° ≤ ° (λ − T )a/b −1 °. We recall that σ + (T, t±) = {0} for t ∈ (a, b). By 2.2.4 it follows σ + (Tc/d ) = {0}. Consequently (see the example in 1.4.9), the inverse of (λ−T )c/d can be represented as the absolutely convergent series ¡

(λ − T )c/d

¢−1

Tc/d (Tc/d )2 1 = + 2 + + ... . λ λ λ3

146

II. THE INITIAL VALUE PROBLEM

We rewrite this series as (for the simplicity of notation, we omit the index c/d) ¡ ¢ x(t) x(τ (t)) x(τ 2 (t)) (λ − T )−1 x (t) = + + + ... . λ λ2 λ3 By the choice of c and d, the points d, τ (d), . . . , τ n−1 (d), and τ n (d) = c lie in [c, d]. We stress that these points are distinct since τ (t) < t for t ∈ (a, b). We choose x ∈ Cc/d , kxk ≤ 1, such that x(d) = λ/|λ|, x(τ (d)) = λ2 /|λ|2 , . . . , x(τ n−1 (d)) = λn /|λ|n . We note that x(τ n (d)) = x(τ n+1 (d)) = · · · = 0 because x(t) = 0 for t ≤ c = τ n (d) and τ (t) ≤ t. Substituting this x and t = d in the preceding series we obtain ¡

¢ (λ − T )−1 x (d) = n.

°¡ ¢−1 ° °¡ ¢ ° ° ≤ ° (λ − T )a/b −1 °. Therefore k(λ − T )−1 k ≥ n We recall that ° (λ − T )c/d a/b for all n. This is a contradiction. (d) By the definition of ∂ − K, for any η > 0 there exists a ∈ (t − η, t) such that τ (a) < a. We define b to be the smallest point of K that is greater than a. Clearly, τ (b) = b and τ (s) < s for s ∈ [a, b). It is easy to see that the assumptions of our auxiliary statement are satisfied on (a, b). Therefore σ + (Ta/b ) = { λ ∈ C : |λ| ≤ 1 }. On the other hand, the inclusions σ + (Ta/b ) ⊆ σ + (Tt−η/t ) ⊆ { λ ∈ C : |λ| ≤ 1 } are plain. It remains to apply the definition of σ + (T, t−). (e) The set R\K is open. We recall that an open subset of R can be represented as a union of pairwise disjoint open intervals. We consider two cases. First: in any right neighbourhood of t there is contained a whole interval from R \ K. Then one can argue as in case (d). We turn to the second case: assume t is the left end of an interval in R \ K. By our auxiliary statement it suffices to prove that for any natural n and any η > 0, ¡ ¢ (t, t + η) ∩ τ n (t, t + η) 6= ∅. Since τ (t) ≤ t this condition is equivalent to ¡ ¢ (t, ∞) ∩ τ n (t, t + η) 6= ∅. We make use of induction on n. For n = 1, the last assertion holds by the definition of ∂ ++ K. We show that if the assertion holds for n then it holds for n + 1 as well. ¡ ¢ We take an arbitrary δ > 0 and consider the set τ n [t, t+δ) . By the assumption of induction it contains a point s > t. Furthermore, it also contains the point t. On

2.6. CRITERIA FOR EVOLUTIONARY SOLUBILITY

147

¡ ¢ the other hand, the set τ n [t, t + δ) is connected. Consequently it must contain a set of the form [t, t + η). And since τ n (t) = t we have ¡ ¢ τ n (t, t + δ) ⊇ (t, t + η). We rewrite the assumption of induction for n = 1 in the form ¡ ¢ ∀η > 0 τ −1 (t, t + ∞) ∩ (t, t + η) 6= ∅. From here and the preceding inclusion we obtain ¡ ¢ ¡ ¢ τ −1 (t, t + ∞) ∩ τ n (t, t + η) 6= ∅, which implies

¡ ¢ (t, t + ∞) ∩ τ n+1 (t, t + η) 6= ∅. ¤

2.6.5. The spectral mapping theorem for the causal spectrum. Let τ : R → R be continuous and τ (t) ≤ t for all t ∈ R. Let bm ∈ B(E), m = 0, . . . , satisfy the condition ∞ X

q m kbm k < ∞

for some q > 1.

m=1

¡ ¢ We consider the operator B ∈ B+ C(R, E) defined by the formula ¡

∞ X ¢ ¡ ¢ Bx (t) = bm x τ m (t) m=0

¡ ¢ with the constant coefficients bm ; here as usual τ 0 (t) ≡ t and τ m (t) = τ¡ τ m−1 (t) . ¢ P∞ We note ¡ m that ¢ B can be represented as the series B = m=0 Bm , where Bm x (t) = bm x τ (t) . Clearly, kBm k ≤ kbm k. Therefore the series converges absolutely, and B is well defined. We consider the auxiliary function f (λ) =

∞ X

λm bm .

m=1

It takes its values in B(E) and is defined at least for all |λ| ≤ q. Proposition. For all t ∈ R σ + (B, t±) =

¢ ª S© ¡ σ f (λ) : λ ∈ σ + (T, t±) ,

where T is defined as in the previous subsection, see formula (4). (Note that f (λ) makes sense for λ ∈ σ + (T ) since σ + (T ) ⊆ { λ ∈ C : |λ| ≤ 1 }.) Proof. We make use of 1.7.8. Let −∞ ≤ a < b ≤ +∞. We consider the space Ca/b ' Ca ([a, b], E), see the first remark in 2.1.1. It is easy to see that Ca ([a, b], E) ' E ⊗ε Ca ([a, b], C), and Ba/b can be interpreted as f (Ta/b ) in the sense of 1.7.8. Therefore we have ¢ ª S© ¡ σ f (λ) : λ ∈ σ(Ta/b ) . σ(Ba/b ) = (We stress that the spectra in here are not causal.) It remains to apply 2.2.5. ¤

148

II. THE INITIAL VALUE PROBLEM

2.6.6. The freezing of coefficients. Let E be a Banach space, and let am : R → B(E), m = 0, . . . , be continuous functions. We assume that there exists q > 1 such that for all t ∈ R, ∞ X

q m kam (t)k < ∞.

m=1

Let τ : R → R be a continuous function and τ (t) ≤ t for all t ∈ R. We consider the operator D defined by the formula ∞ X ¡ ¢ ¡ ¢ Dx (t) = am (t)x τ m (t) , m=0

¡ ¢ ¡ ¢ where τ 0 (t) ≡ t, τ m (t) = τ τ m−1 (t) . Clearly, B ∈ B+ C(R, E) . We set K = Kτ = { t : τ (t) = t}. and define Int K, ∂ − K, ∂ ++ K, and ∂ +− K as in 2.6.4. Theorem. Let −∞ < a < b < +∞. Then σ + (Da/b ) is the union of the following three sets: ¢ ª S© ¡ σ a0 (t) : t ∈ [a, b] \ K, t ∈ ∂ +− K ∩ [a, b) , ∞ ´ ª S© ³ X σ am (t) : t ∈ Int K ∩ [a, b] , (5) m=0 ∞ ´ ª S© ³ X m σ λ am (t) : t ∈ ∂ − K ∩ (a, b], t ∈ ∂ ++ K ∩ [a, b), λ ∈ C, |λ| ≤ 1 . m=0

In particular, D is causally invertible on [a, b] if and only if no one of these sets contains 0. Proof. We fix t ∈ R and set bm = am (t), m = 0, 1, . . . . We show that σ(D, t±) = σ(B, t±), where B is the operator with the constant coefficients bm considered in 2.6.5. Clearly, it suffices to show that 0 belongs to σ + (B, t±) and σ + (D, t±) simultaneously. We consider, for example, the case of t+. We observe that G(D − B, t+) = 0 (see 2.2.6 for the definition of G(·)) since the functions am (·) are continuous. We show that 0 ∈ / σ + (B, t+) implies 0 ∈ / σ + (D, t+). Indeed, we assume that the operator Bt/t+δ is causally invertible for sufficiently small δ > 0. Since k(Bt/t+δ )−1 k does not increase and k(D − B)t/t+δ k tends to zero as δ decreases, we have k(Bt/t+δ )−1 k · k(D − B)t/t+δ k < 1 for sufficiently small δ > 0. By 1.4.2 this implies that the operator Dt/t+δ is causally invertible. In a similar way one can show that 0 ∈ / σ + (D, t+) implies 0 ∈ / σ + (B, t+). Now the theorem is a straightforward consequence of 2.2.4, 2.6.4, and 2.6.5. ¤

2.6. CRITERIA FOR EVOLUTIONARY SOLUBILITY

149

Example. It is straightforward to verify that the function f (λ) = a + bλ + cλ2 with a, b, c ∈ R (!) does not vanish on the unit ball { λ : |λ| ≤ 1 } if and only if a 6= 0 and |b/a| − 1 < c/a < 1. Since σ + (B) is contained in the ball { λ : |λ| ≤ 1 }, from the theorem it follows that the operator ¡

¢ ¡ ¢ ¡ ¢ Dx (t) = a(t)x(t) + b(t)x τ (t) +c(t)x τ 2 (t)

(with continuous a, b, c, τ : R → R, τ (t) ≤ t) is causally invertible on C on all finite segments if a(t) > 0 and |b(t)/a(t)| − 1 < c(t)/a(t) < 1. Note that this criterion is rough because it does not use the behaviour of τ . Clearly, it can be strengthen if one applies the theorem more thoroughly. 2.6.7. An application to an equation of neutral type. Assume that an operator D ∈ B+ (C) is the same as in 2.6.6; and B ∈ B+ (C) is arbitrary. Corollary. Let −∞ < a < b < +∞. (a) Equation (2) is evolutionarily soluble on [a, b] in the pair (C 1 , C) if and only if no one of the sets (5) contains zero. (b) Equation (3) is evolutionarily soluble on [a, b] in the pair (C, C −1 ) if and only if no one of the sets (5) contains zero. Proof. The proof follows from 2.5.9 and 2.5.10.

¤

CHAPTER III

STABILITY

Stability is one of the most fundamental properties of equations that describe time processes. The stability of an equation means that its solutions do not vary significantly under an admissible class of perturbations. In this sense stability is a kind of continuity. The proximity of solutions and the allowable class of perturbations in the definition of stability can be interpreted in vastly different ways. Some general features of proximity and perturbations are determined by applications. But details usually can be chosen in a quite arbitrary way. Thus one can offer many variants of the definition of stability. Fortunately, the majority of these definitions are equivalent. (One of our tasks is the discussion of this equivalence.) Thus one can choose the details of the definition of stability so that the usage of mathematical methods will be the most convenient. We consider only two kinds of stability of the equation Lx = f . They are the input-output stability and exponential stability. We show that both of them are equivalent to the causal invertibility of the operator L. As we have seen in chapter 2, evolutionary solubility is equivalent to causal invertibility on finite segments. In contrast, stability is equivalent to causal invertibility at infinity. The reduction of the investigation of stability to the investigation of causal invertibility makes the application of operator theory natural and convenient. In this chapter we usually assume that equations under consideration are evolutionarily soluble on any finite interval. Under this assumption most essential features of stability turn out to be discrete, i.e., they appear in the case of equations in lq . Thus the simplest case of lq can be treated as the principal case.

3.1. Algebraic preliminaries In order to simplify further references, in this section we collect some auxiliary formulae and constructions. 3.1.1. The projectors Pa . Let X and Y be Banach spaces with directions {Xa } and {Ya }, see 2.1.1 for the definition. Assume that the subspaces Xa and Ya are complemented. Let Pa : X → X and Pa : Y → Y denote projectors onto Xa and Ya ; the same notation for Pa : X → X and Pa : Y → Y will cause no confusion since it always will be clear on which space Pa is considered. We recall that a linear operator L : X → Y is called causal if LXa ⊆ Ya

150

for all a.

(1)

151

3.1. ALGEBRAIC PRELIMINARIES

Evidently, Pa are causal. It is easy to verify that in terms of Pa definition (1) can be rewritten in any of the following equivalent ways: Pa LPa = LPa

for all a,

(1 − Pa )L(1 − Pa ) = (1 − Pa )L

for all a,

Pa L(1 − Pa ) = Pa L − LPa

for all a.

Usually the projectors Pa can be chosen so that Pa Pb = Pb Pa = Pb

for a < b

(2)

or, equivalently, (1 − Pa )(1 − Pb ) = (1 − Pb )(1 − Pa ) = 1 − Pa

for a < b.

For our main examples of functional spaces Cq = Cq (R, E),

Cq1 = Cq1 (R, E),

Cq−1 = Cq−1 (R, E),

Lpq = Lpq (R, E),

1 1 Wpq = Wpq (R, E),

−1 −1 Wpq = Wpq (R, E)

(3)

projectors Pa onto Xa satisfying (2) exist. Let us discuss them in greater detail. Assume X = Lpq . We set ¡

¢ Pa x (t) =

½

0

for t < a,

x(t)

for t ≥ a.

Clearly, Pa is a projector onto Xa , and (2) is satisfied. If X = lq (Z, Ei ) we define Pa in a similar way. 1 . We set (cf. 1.6.11) Assume X = Cq or X = Wpq ¡

¢ Pa x (t) =

½

0 x(t) − x(a)e−(t−a)

for t < a, for t ≥ a.

Clearly, Pa is a projector onto Xa , and (2) is satisfied. Assume X = Cq1 . We define Pa x to be the continuously differentiable function which is zero on (−∞, a) and which coincides with a function of the form t 7→ x(t) + (αt + β)e−t , α, β ∈ E, for t ∈ (a, +∞). We note that the coefficients α and β are determined uniquely by the values of x(a) and x(a) ˙ from the assumption of the continuous differentiability of Pa x. −1 Finally, if X = Cq−1 or X = Wpq we define Pa as in 2.3.8 and 2.3.11. Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let a ∈ R be fixed. 1 (a) Let L = Lpq and W 1 = Wpq , or L = Cq and W 1 = Cq1 . Then the projectors PL = Pa : L → L and PW 1 = Pa : W 1 → W 1 are connected by the identity PW 1 = U −1 PL U

or, equivalently,

PL = U PW 1 U −1 ,

152

III. STABILITY

where U is defined as in 2.3.4. −1 (b) Let L = Lpq and W −1 = Wpq , or L = Cq and W −1 = Cq−1 . Then the projectors PL = Pa : L → L and PW −1 = Pa : W −1 → W −1 are connected by the identity PW −1 = U PL U −1

or, equivalently,

PL = U −1 PW −1 U,

where U is defined as in 2.3.9. 1 Proof. (a) Assume that L = Lpq and W 1 = Wpq . It is more convenient to −1 prove the equivalent identity 1 − PW 1 = U (1 − PL )U . We take an arbitrary 1 x ∈ Wpq and denote by y ∈ Lpq the function U −1 (1 − PL )U x. Since (1 − PL )U x is equal to U x on (−∞, a) and U −1 is causal we conclude that y is equal to x on (−∞, a). Thus y coincides with (1 − PW 1 )x on (−∞, a). On the other hand, (1 − PL )U x is equal to zero on (a, +∞). Consequently y satisfies on (a, +∞) the differential equation U y = 0 or more detailed y˙ + y = 0. Therefore (see 2.3.5) y(t) = αe−t on (a, +∞) for some α ∈ E. Now assume that L = Cq and W 1 = Cq1 . Repeating the above argument we obtain that y = U −1 (1 − PL )U x satisfies on (a, +∞) the differential equation y(t) ˙ + y(t) = αe−t . Hence y(t) = (αt + β)e−t on (a, +∞). (b) Assume that L = Cq and W −1 = Cq−1 . This case can be interpreted as the variant of case (a) with L = Cq−1 and W 1 = Cq . In such an interpretation the proof is a word for word repetition of that argument. −1 For L = Lpq and W −1 = Wpq the assertion coincides with the definition of PW −1 , see 2.3.11. ¤

3.1.2. The representation of Xq as lq (Z, X♥i ). The spaces Lpq have been defined in 1.6.3 by virtue of the isomorphism with lq . The representation of Cq as lq has been discussed in 1.6.10 and 1.6.11. In this subsection we describe similar representations for the other functional spaces (3). Natural representations exist only for Lpq and Cq−1 (see 3.1.3 below). For other spaces we fix these representations in the way associated with our choice of the isomorphism U and projectors Pa . Let Xq , 1 ≤ q ≤ ∞ or q = 0, be one of the spaces (3). Let Pa : Xq → Xq , a ∈ R, be defined as in 3.1.1. We denote by X♥i the image of Pi − Pi+1 : Xq → Xq , i ∈ Z. We endow X♥i with the norm induced by the embedding in Xq . Proposition. (a) The image X♥i of Pi − Pi+1 : Xq → Xq does not depend on q. The norms on X♥i induced by the embeddings in Xq with different q are equivalent. (b) The mapping Υ : x 7→ { (Pi − Pi+1 )x : i ∈ Z } is a topological isomorphism from Xq onto lq = lq (Z, X♥i ). 1 (c) The operator Υ−1 U Υ : lq → lq , where U : Cq1 → Cq , U : Wpq → Lpq , −1 U : Cq → Cq−1 , or U : Lpq → Wpq , is induced by a main diagonal matrix. Clearly, if X = Cq the definitions of X♥i and Υ coincide with those of 1.6.11.

3.1. ALGEBRAIC PRELIMINARIES

153

Proof. (a) In the case Xq = Lpq the image of Pi − Pi+1 is the subspace of all x ∈ Lpq which are equal to zero outside (i, i + 1]. In other words, the image of Pi − Pi+1 on Lpq is naturally isometrically isomorphic to Lp ([i, i + 1], E) for all q. We turn to the case Xq = Cq . From the explicit formula for Pa in 3.1.1, it is easy to see that for all q the space Im(Pi −Pi+1 ) consists of all continuous functions x : R →¡E which are equal to zero on (−∞, i], to x(t) − x(i)e−(t−i) for t ∈ [i, i + 1], ¢ −(t−i) and to x(i + 1)e − x(i) e for t ∈ [i + 1, +∞). Thus Im(Pi − Pi+1 ) does not depend on q. Clearly, the norms induced by different q are equivalent. Now let Xq = Cq−1 . By 3.1.1 the isomorphism U : Cq → Cq−1 maps the image of Pi − Pi+1 : Cq → Cq onto the image of Pi − Pi+1 : Cq−1 → Cq−1 . The definition of U does not depend on q. (For example, this means that for all q the operator −1 .) Since the U : Cq → Cq−1 can be considered as a restriction of U : C∞ → C∞ image of Pi − Pi+1 : Cq → Cq on Cq does not depend on q, the image of Pi − Pi+1 : Cq−1 → Cq−1 on Cq−1 does not depend on q, too. The cases of other spaces are handled similarly. (b) The case Xq = Lpq is trivial. The case Xq = Cq−1 has been considered in 1.6.11. Assume X = Cq . Let Υ = ΥC : Cq → lq (Z, C♥i ) denote the isomorphism from 1.6.11. Clearly, ΥC U −1 : Cq−1 → lq (Z, C♥i ) is an isomorphism, too. From −1 3.1.1 it follows that U : Cq → Cq−1 maps C♥i onto C♥i . We define the iso−1 morphism U : lq (Z, C♥i ) → lq (Z, C♥i ) by the componentwise rule; specifically, −1 U maps {xi ∈ C♥i } to {U xi ∈ C♥i }. It is a straightforward verification that −1 −1 −1 the isomorphism UΥC U : Cq → lq (Z, C♥i ) coincides with the isomorphism −1 −1 Υ : Cq → lq (Z, C♥i ) described in the assertion. The cases of other spaces are handled similarly. (c) follows from the equality Υ−1 U Υ = U. ¤ −1 −1 3.1.3. The space C♥i . We recall from 3.1.2 that C♥i has been defined to be −1 −1 the image of the projector Pi − Pi+1 : Cq → Cq . The following proposition shows that the isomorphism Υ between Cq−1 and −1 lq (Z, C♥i ) defined in 3.1.2(b) can be considered as natural. −1 Proposition. The space C♥i consists of all functions f ∈ Cq−1 of the form −1 f = u˙ + v, where u, v ∈ Cq are supported in [i, i + 1]. The norm on C♥i induced −1 by the embedding in Cq is equivalent to the norm

kf k = inf{ kukCq + kvkCq : f = u˙ + v; supp u, supp v ⊆ [i, i + 1] }. Clearly, the last norm does not depend on q. Proof. Clearly, if u and v are supported in [i, i + 1] then f ∈ Im(Pi − Pi+1 ). Conversely, assume f ∈ Cq−1 is represented as f = u˙ + v, u, v ∈ Cq . Let w : R → R be a continuously differentiable function such that w(i) = −u(i), w(i + 1) = −u(i + 1), w0 (i) = v(i), and w0 (i + 1) = v(i + 1), cf. 2.3.8. We set u e(t) = u(t) + w(t), ve(t) = v(t) − w(t) ˙ for t ∈ [i, i + 1], and u e(t) = ve(t) = 0 for

154

III. STABILITY

t ∈ / [i, i + 1]. It is easy to verify that u e and ve are continuous and f = u˙ + v. Evidently, w can be chosen so that ke ukC , ke v kC ≤ K sup{ |u(t)|, |v(t)| : t ∈ [i, i + 1] }, where K does not depend on u and v. This estimate implies the equivalence of the norms. ¤ 3.1.4. The operators Qa , Ia , and Ra Let X be one of the spaces (3), and let a ∈ R. Let Pa : X → X be the projector onto Xa defined in 3.1.1. Thus Im Pa = Xa . We also set X a = Ker Pa . We have X = X a ⊕ Xa . Clearly, X a is isomorphic to the quotient space X/Xa , which we denote by X−∞/a , see 2.1.1. We recall that we denote the natural projection from X onto X−∞/a = X/Xa by Qa . Since Im Pa = Ker Qa we have Qa Pa = 0

and

Qa (1 − Pa ) = Qa .

We denote by Ia : X → Xa the operator of projection that differs from Pa : X → X only by the range of values, i.e., Ia x = Pa x for all x ∈ X. Clearly, Ia is causal in the natural sense. Let Ja : Xa → X be the natural embedding. Evidently, Ja Ia = Pa

and

Ia Ja = 1Xa ,

which, evidently, implies Pa Ja = Ja

and

Ia Pa = Ia .

Finally, for any causal operator L one has Ia LJa = La . We define the operator Ra : X−∞/a → X of continuation as the quotient operator induced by 1 − Pa (see 1.2.5), i.e., as the operator completing the diagram J

Xa −−−a−→   y

Qa

X −−−−→ X/Xa −−−−→ 0   1−P R a y y a

0 −−−−→ X

X

−−−−→ 0

to a commutative diagram (see 1.2.4); here Ja : Xa → X is the natural embedding. Clearly, Ra Qa = 1 − Pa .

3.1. ALGEBRAIC PRELIMINARIES

155

Multiplying this identity by Qa and utilizing the identity Qa (1 − Pa ) = Qa we obtain Qa Ra Qa = Qa , which, since Im Qa coincides with the whole X−∞/a , implies Qa Ra = 1X−∞/a , i.e., Ra is really an operator of continuation. 1 Let X be Cq , Lpq , Cq1 , or Wpq . Then we can identify the quotient space X−∞/a ¡ ¢ with the space X (−∞, a], E . Under this identification one can write an explicit formula for Ra . Namely, if X = Lpq , ½ ¡ ¢ z(t) for t < a, Ra z (t) = 0 for t ≥ a. 1 , If X = Cq or X = Wpq

¡ ¢ Ra z (t) =

½

z(t)

for t < a,

z(a)e−(t−a)

for t ≥ a.

If X = Cq1 , the function Ra z is equal to z on (−∞, a) and coincides with the function of the form (αt + β)e−t on (a, +∞). (Clearly, the coefficients α and β are determined uniquely by the values of z(a) and z(a).) ˙ 3.1.5. A representation of solutions of the initial value problem. Let (X, Y ) be one of the following pairs (Cq , Cq ),

(Lpq , Lpq ),

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ), (4)

and let L ∈ B+ (X, Y ). We consider the equation Lx = f.

(5)

−1 We recall from §2.4 that if Y 6= Wpq we define the initial value problem to be ¡ ¢ Lx (t) = f (t), a < t < b, (6)

x(t) = ϕ(t),

t < a.

−1 we define the initial value problem as And if Y = Wpq ¡ ¢ ¡ ¢ Pa Lx (t) = Pa f (t), t < b,

x(t) = ϕ(t),

t < a.

(7)

(8) (9)

And ¡ ¢if Y = Cq we consider only f and ϕ which satisfy the compatibility condition Lϕ (a) = f (a). We observe that equation (6) can be represented equivalently in the form (8), and thus (8), (9) can be considered as the most general form. Indeed, if Y = Lpq , (6) is trivially equivalent to (8). If Y = Cq−1 , (6) is equivalent to (8) by 2.3.8. And ¡ ¢ if Y = Cq , by the compatibility condition Lx (a) = f (a) we have ½ ¡ ¢ 0 for t ≤ a, ¢ Pa (Lx − f ) (t) = ¡ Lx − f (t) for t ≥ a. Thus (6) is equivalent to (8) in this case, too.

156

III. STABILITY

Proposition. Let (X, Y ) be one of the pairs (4), and let equation (5) be uniquely solvable on [a, b]. Then the solution x ∈ X−∞/b of the initial value problem (6), (7), or respectively (8), (9), can be represented in the form x = (Ja )−∞/b (La/b )−1 Qb Ia (f − LRa ϕ) + Qb Ra ϕ.

(10)

Here we assume that ϕ ∈ X−∞/a , f ∈ Y , and La/b acts from Xa/b into Ya/b . Finally, Ja : Xa → X is the natural embedding and (Ja )−∞/b : Xa/b → X−∞/b is its quotient operator. Proof. We shall interpret (6), (7) as a special case of (8), (9). So we restrict our consideration to the case (8), (9). First we verify that function (10) satisfies (8). We recall that we interpret the condition t < b in (8) as the equality of projections into Y−∞/b . Moreover, we assume that in (8) x ∈ X−∞/b , L acts from X−∞/b into Y−∞/b , and Pa means (Pa )−∞/b . Thus in detail, (8) means that (Pa )−∞/b L−∞/b x = Qb Pa f. Now we begin to verify (8) for the function (10). We have (Pa L)−∞/b x = (Pa L)−∞/b (Ja )−∞/b (La/b )−1 Qb Ia (f − LRa ϕ) + (Pa L)−∞/b Qb Ra ϕ = (Pa LJa )−∞/b (La/b )−1 (Ia )−∞/b Qb (f − LRa ϕ) + (Pa L)−∞/b Qb Ra ϕ. To simplify this formula, first we show that (Pa LJa )−∞/b (La/b )−1 (Ia )−∞/b = (Pa )−∞/b . We recall (see 3.1.4) that Pa = Ja Ia , Ia LJa = La , and Ia Ja = 1Xa . Therefore (Pa LJa )−∞/b (La/b )−1 (Ia )−∞/b = (Ja Ia LJa )−∞/b (La/b )−1 (Ia )−∞/b = (Ja La )−∞/b (La/b )−1 (Ia )−∞/b = (Ja )−∞/b La/b (La/b )−1 (Ia )−∞/b = (Ja )−∞/b 1a/b (Ia )−∞/b = (Ja )−∞/b (1Xa )−∞/b (Ia )−∞/b = (Ja )−∞/b (Ia Ja )−∞/b (Ia )−∞/b = (Ja Ia Ja Ia )−∞/b = (Pa Pa )−∞/b = (Pa )−∞/b .

3.2. INPUT–OUTPUT STABILITY: DISCRETE TIME

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We return to the verification of (8). Now we can write (Pa L)−∞/b x = (Pa )−∞/b Qb (f − LRa ϕ) + (Pa L)−∞/b Qb Ra ϕ = (Pa )−∞/b Qb f − (Pa )−∞/b Qb LRa ϕ + (Pa L)−∞/b Qb Ra ϕ = Qb Pa f − (Pa )−∞/b L−∞/b Qb Ra ϕ + (Pa L)−∞/b Qb Ra ϕ = Qb Pa f. Finally, we verify (9). Equality (9) means that Qa x = ϕ. Since Im Ja ⊆ Xa , function (10) satisfies the equality Qa x = 0 + (Qa )−∞/b Qb Ra ϕ = Qa Ra ϕ = ϕ. Thus function (10) satisfies (9).

¤

3.2. Input–output stability: discrete time We begin our discussion of stability with the discrete case. Below we shall see that the simplest case of discrete time can be interpreted as the principal case. Namely, all essential ideas appear in the discrete case and, moreover, the discussion of continuous time can to a considerable extent be reduced to the discussion of discrete time. The main result of this section is theorem 3.2.5 on the equivalence of input–output stability and causal invertibility. 3.2.1. The definition of local solubility. The theory of the initial value problem on Z is a trivial simplification of that on R. So we give only an outline of it. Let { Ei : i ∈ Z } be a family of Banach spaces. We consider the space X = lq = lq (Z, Ei ), 1 ≤ q ≤ ∞ or q = 0, see 1.5.7. We define the T -direction on lq to be the family of the subspaces Xt = { x ∈ lq : xi = 0 for i < t }, t ∈ R. Of course, Xt coincides with Xn , where n is the biggest integer less than t + 1. In particular, the quotient space Xn/m (see 2.1.1 for the definition) can be identified with the space of all families { xi ∈ Ei : n ≤ i < m } or with the corresponding subspace of lq . We denote intervals in Z by the usual symbols [n, m), (−∞, n], etc.. ¯ i : i ∈ Z } be families of Banach spaces. We set X = Let { Ei : i ∈ Z } and { E ¯ i ), 1 ≤ q ≤ ∞ or q = 0. Clearly, a causal operator lq (Z, Ei ) and Y = lq (Z, E T : X → Y has a triangular matrix (see 1.6.4 for the definition of a matrix). Namely, Tij = 0 if i < j. We recall (see the example in 1.6.4) that in the case q = ∞ an operator can be not restored by its matrix. Let T ∈ B+ (X, Y ). We consider the difference equation T x = f.

(1)

By the initial value problem for (1) on the interval [n, m) we mean the problem ¡ ¢ n ≤ i < m, T x i = fi , (2) xi = ϕi , i < n.

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Here f , ϕ, and x are families of the class lq defined for the corresponding sets of indices. To obtain a unique solution of (2) one must assume that x ∈ X−∞/m . The similar situation holds for both ϕ and f : the solution depends only on the projections of ϕ and f into X−∞/n and Yn/m , respectively. So we may assume either ϕ ∈ X and f ∈ Y , or ϕ ∈ X−∞/n and f ∈ Yn/m . We make the change x = z + ψ, where z is a new unknown sequence, and ψ = {ψi } ∈ X−∞/m coincides with ϕi for i < n and is defined to be zero for n ≤ i < m. We arrive at the initial value problem ¡ ¢ T z i = gi , n ≤ i < m, (3) zi = 0, i < n, where g = f − T ψ. We say that equation (1) is uniquely soluble on [n, m), −∞ < n < m < +∞, if the initial value problem (2) has a unique solution x ∈ X−∞/m for all ϕ and f (cf. §2.4). Clearly, (2) and (3) have a unique solutions for all f and ϕ or, respectively, for all g, simultaneously. Thus the unique solubility on [n, m) is equivalent to the invertibility of the operator Tn/m : Xn/m → Yn/m . We say that equation (1) is evolutionarily soluble on [n, m), −∞ < n < m < +∞, if it is uniquely soluble and the inverse of the operator Tn/m : Xn/m → Yn/m is causal (cf. §2.5). We say that equation (1) is locally soluble if it is evolutionarily soluble on all [n, m) (or, equivalently, — cf. 2.5.2 — uniquely soluble on all [n, m)), −∞ < n < m < +∞. Proposition. (a) The evolutionary solubility on [n, m) is equivalent to the invertibility of ¯ i ) for all n ≤ i < m. the main diagonal matrix entries Tii ∈ B(Ei , E (b) The local solubility is equivalent to the invertibility of the main diagonal ¯ i ) for all i ∈ Z. matrix entries Tii ∈ B(Ei , E Proof. The proof is evident, cf. 2.1.7, 2.6.2. ¤ Below in this section we always assume that equation (1) is locally soluble. 3.2.2. The extended space. We denote by Xe = le = le (Z, Ei ) the space of all collections { xi ∈ Ei : i ∈ Z } such that xi = 0, i < n, for some n ∈ Z depending on x. (There are no restrictions on the behaviour of xi at +∞.) The space le is called the extended space. We define the T -direction on le to be the family of the subspaces (le )t = { x ∈ le : xi = 0 for i < t }, t ∈ R. We stress that le is a linear space without norm. Nevertheless, some of our results and constructions associated with causal operators remain valid for le , since in §2.1 we have not assumed that X and Y are Banach. For example, one can consider the spaces (le )n/m , −∞ < n < m < +∞. Obviously, they are naturally isomorphic to (lq )n/m for all q. ¯ i ) in a similar way. We define the space Ye = le = le (Z, E

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¯ i ). We observe that any causal Let as above X = lq (Z, Ei ) and Y = lq (Z, E operator T : X → Y induces (see 2.1.1) the causal operators Tn/m : Xn/m → Yn/m , −∞ < n < m < +∞, which in turn induces naturally the causal operator T = Te : Xe → Ye . Namely, let us assume that x ∈ Xe and define (Te x)j . We pick n so that xi = 0 for i < n and take an arbitrary m > j. We set (Te x)j = (Tn/m Qm x)j , where Qm : X → X/Xm is the natural projection. Clearly, the definition of Te does not depend on n and m. (Another way to define Te is to use matrix representation.) Proposition. Let T ∈ B+ (X, Y ). The following assumptions are equivalent. (a) Equation (1) is locally soluble. (b) The operators Tn/m : Xn/m → Yn/m are (causally) invertible for all −∞ < n < m < +∞. (c) The operator Te : Xe → Ye is causally invertible. Proof. (a) ⇔ (b) follows from the definition of local solubility. (c) ⇒ (b) follows from 2.1.3. We prove (b) ⇒ (c). According to (b) we have a family of co-ordinated causal operators (Tn/m )−1 : Yn/m → Xn/m , −∞ < n < m < +∞. As above they induce a causal operator from Ye into Xe . It is easy to verify that this operator is the inverse of Te : Xe → Ye . ¤ 3.2.3. The definition of input–output stability. Below we assume that equation (1) is locally soluble. We say that equation (1) is input–output stable on Z if there exists K < ∞ such that for any interval [n, m), −∞ < n < m < +∞, and f ∈ Y and ϕ ∈ X, the solution x ∈ X−∞/m of (2) satisfies the estimate kxk ≤ K(kϕk + kf k).

(4)

Clearly, taking the infimum over all ϕ with the same projection into X−∞/n and over all f with the same projection into Yn/m , we can rewrite (4) equivalently as kxkX−∞/m ≤ K(kϕkX−∞/n + kf k[n,m) ), where kf k[n,m) = kfek and kϕkX−∞/n = kϕk e with ½ ½ fi for i ∈ [n, m), ϕi e fi = ϕ ei = 0 0 for i ∈ / [n, m);

for i < n, for i ≥ n.

Let τ ∈ Z be a fixed point, and let H ⊆ Z be either (−∞, τ ) or [τ, +∞). We say that equation (1) is input–output stable on H if there exists K < ∞ such that for any finite interval [n, m) ⊆ H, and f ∈ Y and ϕ ∈ X the solution x ∈ X−∞/m of (2) satisfies the estimate (4). Sometimes we shall specify that we consider stability in the pair (X, Y ). We note that input–output stability in the pairs (l0 , l0 ) and (l∞ , l∞ ) means the same provided T acts both from l0 to l0 and from l∞ to l∞ . Remark. Usually the stability on (−∞, τ ) is not investigated. But it also makes sense. It means that the output of a device which have been turned on long ago, remains bounded until the present moment (cf. the definitions of left solubility at a point in 2.5.5 and 2.5.6). Besides, the conjugate of a causal operator is anticausal (see the remark in 2.1.1). Thus the conjugate of an equation on [τ, +∞) possesses the properties of equation on (−∞, τ ).

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Proposition. Let equation (1) be locally soluble. Let H be one of the intervals (−∞, +∞), (−∞, τ ), or [τ, +∞). Equation (1) is input–output stable on H if and only if there exists K < ∞ such that k(Tn/m )−1 k ≤ K

for all [n, m) ⊆ H, −∞ < n < m < +∞.

(5)

Proof. Considering (2) with ϕ = 0, from the definition of input–output stability we obtain the estimate k(Tn/m )−1 k ≤ K. Now we prove the converse. Clearly, the solution z ∈ Xn/m of (3) can be represented as z = (Tn/m )−1 In/m g, where In/m : X → Xn/m is defined by the formula ½ gi for n ≤ i < m, (In/m g)i = 0 for i < n. Therefore the solution x of (2) can be represented in the form x = (Tn/m )−1 In/m (f − T ψ) + ψ. (We omit here some natural embeddings, cf. 3.1.5.) This formula shows that (5) implies (4). ¤ 3.2.4. Stability on semi-axes Proposition. Let equation (1) be locally soluble. (a) Equation (1) is input–output stable on (−∞, +∞) if and only if it is input– output stable both on (−∞, τ ) and [τ, +∞). (b) The definition of input–output stability on (−∞, τ ) and [τ, +∞) does not depend on the choice of the point τ . Proof. (a) We make use of 3.2.3. Clearly, the stability on (−∞, +∞) implies the stability both on (−∞, τ ) and [τ, +∞). Let us prove the converse. The following only is not completely evident. Assume that n < τ < m and T is invertible both on [n, τ ) and [τ, m). Then by 2.1.4(b) T is invertible on [n, m) and k(Tn/m )−1 k ≤ kT k · K 2 + 2K, where k(Tn/τ )−1 k, k(Tτ /m )−1 k ≤ K. Thus we have the uniform estimate for k(Tn/m )−1 k, −∞ < n < m < +∞, which implies the stability on (−∞, +∞). (b) follows from 2.1.4(b) in a similar way. ¤ 3.2.5. Input–output stability and causal invertibility Theorem. Let equation (1) be locally soluble. (a) Let q 6= 0. Equation (1) is input–output stable on [τ, +∞) in the pair (lq , lq ) if and only if the operator T : (lq )τ /+∞ → (lq )τ /+∞ is causally invertible. ¡ ¢ An operator T ∈ B+ (l0 )τ /+∞ , (l0 )τ /+∞ can be extended naturally to the causal operator T : (l∞ )τ /+∞ → (l∞ )τ /+∞ . Equation (1) is input– output stable on [τ, +∞) in the pair (l0 , l0 ) if and only if the operator T : (l∞ )τ /+∞ → (l∞ )τ /+∞ is causally invertible.

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(b) Let q 6= ∞. Equation (1) is input–output stable on (−∞, τ ) in the pair (lq , lq ) if and only if the operator T : (lq )−∞/τ → (lq )−∞/τ is causally invertible. ¡ ¢ An operator T ∈ B+ (l∞ )−∞/τ , (l∞ )−∞/τ maps the subspace (l0 )−∞/τ into (l0 )−∞/τ . Equation (1) is input–output stable on (−∞, τ ) in the pair (l∞ , l∞ ) if and only if the operator T : (l0 )−∞/τ → (l0 )−∞/τ is causally invertible. (c) Let q 6= 0, ∞. Equation (1) is input–output stable on (−∞, +∞) in the pair (lq , lq ) if and only if the operator T : lq → lq is causally invertible. Proof. Assume T is causally invertible. Then by 1.2.2 k(Tn/m )−1 k ≤ kT −1 k, and stability follows from 3.2.3. We turn to the converse. (a) We denote briefly the spaces (lq )τ /+∞ and (le )τ /+∞ by lq and le , respectively. We consider the case q 6= 0. By 3.2.2 the operator Te : le → le is causally invertible. Clearly, it suffices to show that (Te )−1 maps lq ⊆ le into lq . We take any f ∈ lq . Let x ∈ le be the solution of the equation Te x = f . Clearly, the natural projection xτ /m , τ < m < +∞, of x into (le )τ /m satisfies the equality Tτ /m xτ /m = fτ /m . By (5) we have kxτ /m klq ≤ K kfτ /m klq ≤ K kf klq . Consequently kxklq ≤ K kf klq , i.e., x ∈ lq . Now we consider the case q = 0. It is trivial that since T is causal its restriction T : (l0 )τ /+∞ → (l0 )τ /+∞ is determined by its matrix in the sense of 1.6.4. Clearly, this matrix induces the operator T : (l∞ )τ /+∞ → (l∞ )τ /+∞ . The same reasoning as above shows that f ∈ (l∞ )τ /+∞ implies x ∈ (l∞ )τ /+∞ . Thus the operator T : (l∞ )τ /+∞ → (l∞ )τ / + ∞ is causally invertible. (b) We denote briefly the spaces (lq )−∞/τ and (le )−∞/τ by lq and le , respectively. Assume q 6= ∞. We observe that le is the union of the spaces (lq )n/τ , n > −∞. From 3.2.3 and 3.2.2 it follows that T = Te : le → le is causally invertible and kT −1 f klq ≤ Kkf klq for all f ∈ le . It remains to recall that le is dense in lq , provided q 6= ∞. Assume q = ∞. Since T is causal it maps (l∞ )n/τ into (l∞ )n/τ . As it has been mentioned above, the union of the spaces (l∞ )n/τ , n > −∞, coincides with (le )−∞/τ . Clearly, the closure of (le )−∞/τ in (l∞ )−∞/τ is (l0 )−∞/τ . Hence T maps (l0 )−∞/τ into (l0 )−∞/τ . The same reasoning as above shows that the operator T : (l0 )−∞/τ → (l0 )−∞/τ is invertible. (c) follows from (a) and (b), and 3.2.4 and 2.1.6. ¤ 3.2.6. Invertibility in (l∞ , l∞ ) implies that in (l0 , l0 ). Let I be a set of ¯ i : i ∈ I } be collections of Banach spaces. We indices, and let { Ei : i ∈ I } and { E ¯ i ). consider the spaces lq = lq (I, Ei ) and lq = lq (I, E ¯ i ) be a bounded linear operator. AsProposition. Let T : l∞ (I, Ei ) → l∞ (I, E sume that T maps l0 into l0 . If T∞ : l∞ → l∞ is invertible then T0 : l0 → l0 is also invertible.

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0 0 0 Proof. We make use of 1.8.3. We consider the operator T∞ : l∞ → l∞ . By 0 0 0 ⊥ 1.8.3 l∞ can be represented as l∞ ' l0 ⊕ l0 . Since T∞ maps l0 into l0 , from 1.2.9 0 0 it follows that T∞ maps l0⊥ into l0⊥ . By virtue of the isomorphism l∞ ' l00 ⊕ l0⊥ the 0 0 space l0 is identified with a subspace of l∞ . Therefore the conjugate T00 : l00 → l00 0 0 0 to T0 : l0 → l0 can be considered as a restriction of T∞ : l∞ → l∞ . Consequently 0 0 0 0 0 0 0 0 0 (since T0 maps l0 into l0 ) T∞ maps l0 ⊆ l∞ into l0 ⊆ l∞ . Thus the matrix of T∞ 0 0 associated with the decomposition l∞ ' l00 ⊕ l0⊥ is main diagonal. Since T∞ is 0 0 0 invertible its main diagonal entry T11 ' T0 : l0 → l0 is also invertible. Hence T0 is invertible, too. ¤ ¡ ¢ Example (cf. 1.6.4). Let us consider the spaces l = l N ∩ (−∞, τ ), C and 0 0 ¡ ¢ l∞ = l∞ N∩(−∞, τ ), C . We show that the invertibility in (l0 , l0 ) does not always imply the invertibility in (l∞ , l∞ ). We consider a bounded functional f : l∞ → C such that f is equal to zero on l0 and f (u) = 1, where u = { ui = 1 : i < τ } ∈ l∞ is the constant sequence consisting of unities. We define the operator T : l∞ → l∞ by the formula (T x)i = xi − f (x). Clearly, T u = 0. Therefore T is not invertible in (l∞ , l∞ ). Nevertheless, the restriction of T to l0 coincides with the identity operator. Hence T is invertible in (l0 , l0 ). In view of this example it is natural to extend the T -direction on l∞ by the additional subspace l0 . Note that l0 is not complemented in l∞ .

Remark. In 3.4.3 and 5.5.6 we shall see that for operators T of the classes e and t the invertibility in (l0 , l0 ) implies the invertibility in (l∞ , l∞ ). 3.2.7. Input–output stability and causal invertibility in (l0 , l0 ). We return to the equation (1). Corollary. Let equation (1) be locally soluble. Let H be one of the intervals (−∞, +∞), (−∞, τ ), or [τ, +∞). Equation (1) is input–output stable on H in the pair (l0 , l0 ) if and only if the operator T : l0 → l0 is causally invertible on H. Proof. Clearly, causal invertibility implies input–output stability. We prove the converse. The case H = (−∞, τ ) has been considered in 3.2.5(b). We turn to the case H = [τ, +∞). Let equation (1) be input–output stable on [τ, +∞). Then by 3.2.5 the operator T : (l∞ )τ /+∞ → (l∞ )τ /+∞ is causally invertible. By 3.2.6 this implies the invertibility of T : (l0 )τ /+∞ → (l0 )τ /+∞ . We recall that by 3.2.2 the operator T is invertible on all finite intervals. From these and 2.1.5 it follows that T : (l0 )τ /+∞ → (l0 )τ /+∞ is causally invertible. We complete with the case H = (−∞, +∞). From 3.2.4 it follows that the equation is input–output stable both on (−∞, τ ) and [τ, +∞). Then by what has been proved, T : l0 → l0 is causally invertible on [τ, +∞). On the other hand, T : l0 → l0 is causally invertible on (−∞, τ ). Thus by 2.1.6 T : l0 → l0 is causally invertible on (−∞, +∞). ¤

3.3. Input–output stability: continuous time In this section we generalize the results of the previous section to the case of functional differential equations and functional equations on R. The main result

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of this section is theorem 3.3.6 on the equivalence of input–output stability and causal invertibility. 3.3.1. The definition of local solubility. Let E be a Banach space. We consider the spaces Cq = Cq (R, E),

Cq1 = Cq1 (R, E),

Cq−1 = Cq−1 (R, E),

Lpq = Lpq (R, E),

1 1 Wpq = Wpq (R, E),

−1 −1 Wpq = Wpq (R, E),

(1)

where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. In this section we assume that (X, Y ) is one of the following pairs (cf. 2.4.1): (Cq , Cq ),

(Lpq , Lpq ),

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ). (2)

Let L ∈ B+ (X, Y ). We study the equation Lx = f.

(3)

We say that equation (3) is locally soluble if it is evolutionarily soluble on [a, b] (equivalently, — see 2.5.2 — uniquely soluble on [a, b]) for all −∞ < a < b < +∞. In this section we always assume that equation (3) is locally soluble. 3.3.2. Local spaces. Let X = Xq be one of the spaces (1). We denote by Xloc the space of all distributions x ∈ D0 such that for any −∞ < a < b < +∞ there exists y ∈ X coinciding with x on (a, b) in the sense of distributions. Since the spaces Xq with different q coincide locally, Xloc does not depend on the index q. −1 1 So we shall employ notation Cloc , Cloc , Cloc , Lp loc , Wp1 loc and Wp−1loc . Another definitions of Lp loc and Wp1 loc were given earlier in 2.3.1 and 2.3.3. The following proposition shows that the new definitions are in agreement with the old ones. Proposition. (a) x ∈ Cloc if and only if x : R → E is a continuous function. And x ∈ Lp loc (in the sense of the last definition) if and only if x : R → E is a measurable function and the restriction of x to any interval (a, b), −∞ < a < b + ∞, is the function of the class Lp . 1 (b) x ∈ Cloc if and only if x˙ ∈ Cloc . And x ∈ Wp1 loc (in the sense of the last definition) if and only if x˙ ∈ Lp loc . −1 (c) f ∈ Cloc if and only if f = F˙ , where F ∈ Cloc . And f ∈ Wp−1loc if and only if f = F˙ , where F ∈ Lp loc . Proof. (a) Let y(a,b) be a continuous (respectively, measurable) function which coincides with x on (a, b) in the sense of distributions. Clearly, y(a,b) (t) = y(c,d) (t) for t ∈ (a, b) ∩ (c, d) in the pointwise (respectively, almost everywhere) sense. Thus we can define correctly a continuous (measurable) function y : R → E which coincides with y(a,b) on (a, b) for all (a, b).

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1 (b) Let x ∈ Cloc . Then by (a) x is continuous. Since x coincides with a continuously differentiable function on each finite interval (a, b) we obtain that x is continuously differentiable on the whole of R. Conversely, assume x˙ ∈ Cloc . Then by 2.3.5 and 2.3.1 x is continuously differ1 entiable. Clearly, x ∈ Cloc . 1 The case of Wp loc is handled similarly. −1 (c) Assume f ∈ Cloc . Let F be a primitive of f . We consider a function y ∈ Cq−1 which coincides with f on (a, b). By 2.3.5(b) F differs from a primitive of y on (a, b) by a constant. Thus F ∈ Cloc . Conversely, assume f = F˙ with F ∈ Cloc . Let y be a compactly supported continuous function which coincides with F on (a, b). Then y ∈ Cq for all q and f = y˙ on (a, b). The case of Wp−1loc is handled similarly. ¤ We define the T -direction on Xloc to be the family of the subspaces (Xloc )t = { x ∈ Xloc : x(s) = 0 for s < t }, t ∈ R. Note that Xloc is a linear space without norm. Clearly, Xt ⊆ (Xloc )t and, moreover,

(Xloc )t ∩ X = Xt . The last identity allows one to consider (X)−∞/t = X/Xt as a subspace of (Xloc )−∞/t = Xloc /(Xloc )t . Actually, we identify an element x + Xt , x ∈ X, with x + (Xloc )t . This embedding is injective. Namely, for x1 , x2 ∈ X if x1 + (Xloc )t = x2 + (Xloc )t , i.e., x1 − x2 ∈ (Xloc )t then x1 − x2 ∈ Xt and x1 + Xt = x2 + Xt . Consequently the spaces (Xloc )a/b , −∞ < a < b < +∞, are naturally isomorphic to Xa/b . 3.3.3. The extended space. Let X be one of the spaces (1). We denote by Xe the space of all functions (distributions) x ∈ Xloc such that x(t) = 0, t < a, for some a ∈ R depending on x (there are no any restrictions on the behaviour of x at +∞). The space Xe is called the extended space. We define the T -direction on Xe to be the family of the subspaces (Xe )t = { x ∈ Xe : x(s) = 0 for s < t }, t ∈ R. We note that Xe is a linear space without norm. It is easy to show that the spaces (Xe )a/b , −∞ < a < b < +∞, are naturally isomorphic to Xa/b . Let (X, Y ) be one of the pairs (2). Any causal operator L : X → Y induces the causal operators La/b : Xa/b → Ya/b , −∞ < a < b < +∞, (see 2.1.1) and Le : Xe → Ye . Namely, we take an arbitrary x ∈ Xe . By the definition of Xe we can choose a ∈ R such that x(t) = 0, t < a. Then we take an arbitrary b > a and define the projection of Le x into Ya/b to be La/b x ˜, where x ˜ ∈ Xa/b is the natural projection of x into Xa/b . It is easy to verify that this definition determines the function Le x ∈ (Ye )a ⊆ Ye correctly. Proposition. Let L ∈ B+ (X, Y ). The following assumptions are equivalent. (a) Equation (3) is locally soluble. (b) The operators La/b : Xa/b → Ya/b are (causally) invertible for all −∞ < a < b < +∞. (c) The operator Le : Xe → Ye is causally invertible.

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165

Proof. (a) ⇔ (b) follows from the definition of local solubility and the results of §2.4. (c) ⇒ (b) follows from 2.1.3. We prove (b) ⇒ (c). In concern with (b), we have a family of co-ordinated causal operators (La/b )−1 : Ya/b → Xa/b , −∞ < a < b < +∞. As above they induce a causal operator from Ye into Xe . It is straightforward to verify that this operator is the inverse of Le : Xe → Ye . ¤ 3.3.4. The definition of input–output stability. Let (X, Y ) be one of the pairs (2). We consider the equation (3). We recall that equation (3) is assumed to be locally soluble. Let τ ∈ R be a fixed point, and let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Assume Y = Lpq or Y = Cq−1 . We say that equation (3) is input–output stable on H (in the pair (X, Y )) if there exists K < ∞ such that for any segment [a, b] ⊆ H, and f ∈ Y and ϕ ∈ X the solution x ∈ X−∞/b of the initial value problem ¡ ¢ Lx (t) = f (t), a < t < b, (4) x(t) = ϕ(t), t < a, satisfies the estimate kxk ≤ K(kϕk + kf k),

(5)

where kxk means the norm on X−∞/b . Taking the infimum over all ϕ with the same projection into X−∞/b and over all f with the same values on (a, b) we can rewrite (5) equivalently as kxkX−∞/b ≤ K(kϕkX−∞/a + kf k[a,b) ),

(6)

where kf k[a,b) = inf{ kgk : g ∈ Y and g(t) = f (t) for t ∈ (a, b) } and kϕkX−∞/a = kQa ϕk. Clearly, kf k[a,b) = k(Pa − Pb )f k if Y = Lpq . And if Y = Cq−1 the seminorm kf k[a,b) is equivalent to both k(Pa − Pb )f k and kQb Pa f k. Thus in the last estimate kf k[a,b) can be replaced by any of these quantities. (We recall that Pa and Qb are defined in 3.1.1 and 3.1.4, respectively.) Assume Y = Cq . We say that equation (3) is input–output stable on H (in the pair (X, Y )) if there exists K < ∞ such that for any segment ¡ ¢ [a, b] ⊆ H, and f ∈ Y and ϕ ∈ X satisfying the compatibility condition Lϕ (a) = f (a) the solution x ∈ X−∞/b of (4) satisfies the estimate (5) which can be rewritten equivalently as (6), where kf k[a,b) = inf{ kgk : g ∈ Y and g(t) = f (t) for t ∈ (a, b) } and kϕkX−∞/a = kQa ϕk. In this case kf k[a,b) is not equivalent to kQb Pa f k; we have only the estimate kQb Pa f k ≤ K1 kf k[a,b) for some K1 . Nevertheless, since, by 3.1.5, the solution x depends only on Ia f = Pa f we can also replace kf k[a,b) in (6) by kQb Pa f k. −1 Assume Y = Wpq . We say that equation (3) is input–output stable on H (in the pair (X, Y )) if there exists K < ∞ such that for any segment [a, b] ⊆ H, and f ∈ Y and ϕ ∈ X the solution x ∈ X−∞/b of the regularized initial value problem (see 2.4.5) ¡ ¢ ¡ ¢ Pa Lx (t) = Pa f (t), t < b, (7) x(t) = ϕ(t), t < a,

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satisfies the estimate (5) or, equivalently, (6), where kϕkX−∞/a = kQa ϕk and kf k[a,b) means kQb Pa f k. We note that kf k[a,b) depends on the choice of Pa , see the discussion in 2.3.11. Nevertheless, the following proposition shows that the −1 definition of the input–output stability in the pair (Lpq , Wpq ) does not depend on the choice of Pa . Proposition. Let (X, Y ) be one of the pairs (2). Let equation (3) be locally soluble in the pair (X, Y ). Let H be one of the intervals (−∞, +∞), (−∞, τ ], or [τ, +∞). Equation (3) is input–output stable on H if and only if there exists K < ∞ such that k(La/b )−1 k ≤ K

for all [a, b] ⊆ H, −∞ < a < b < +∞.

(8)

Proof. (Cf. 3.2.3). We consider the initial value problem with ϕ = 0 and f ∈ Ya . Directly from the definition of input–output stability we obtain the estimate k(La/b )−1 k ≤ K. We make a special additional remark concerning the −1 case Y = Wpq , cf. 2.4.5. Suppose that in (7) f ∈ Ya and ϕ = 0. Then Pa f = f and x ∈ Xa . Consequently in this case the problem (7) also can be rewritten equivalently as the equation Lx = f with L : Xa/b → Ya/b . To prove the converse we recall from 3.1.5 that the solution x of the initial value problem can be represented in the form x = (Ja )−∞/b (La/b )−1 Qb Ia (f − LRa ϕ) + Qb Ra ϕ. This explicit formula for x shows that (8) implies the input–output stability. We note only that (we pay special attention to the case Y = Cq ) kIa (f − LRa ϕ)k = kPa (f − LRa ϕ)k ≤ kPa f k + kPa k · kLk · kRa k · kϕk. ¤ 3.3.5. Stability on semi-axes Proposition. Let equation (3) be locally soluble. (a) Equation (3) is input–output stable on (−∞, +∞) if and only if it is input– output stable both on (−∞, τ ] and [τ, +∞). (b) The definition of input–output stability on (−∞, τ ] and [τ, +∞) does not depend on the choice of the point τ ∈ R. Proof. The proof is essentially a word for word repetition of that of 3.2.4.

¤

3.3.6. Input–output stability and causal invertibility. Let (X, Y ) be one of the pairs (2). When we need to emphasize the concrete value of q, we shall employ the notation (Xq , Yq ). Note that input–output stability in the pairs (X0 , Y0 ) and (X∞ , Y∞ ) means the same provided T acts in both pairs. Theorem. Let (Xq , Yq ) be one of the pairs (2), and let τ ∈ R. Let equation (3) be locally soluble in the pair (Xq , Yq ). (a) Let q 6= 0. Equation (3) is input–output stable on [τ, +∞) in the pair (Xq , Yq ) if and only if the operator L : (Xq )τ /+∞ → (Yq )τ /+∞ is causally invertible.

3.3. INPUT–OUTPUT STABILITY: CONTINUOUS TIME

167

¡ ¢ An operator L ∈ B+ (X0 )τ /+∞ , (Y0 )τ /+∞ can be extended naturally to the operator L : (X∞ )τ /+∞ → (Y∞ )τ /+∞ . The equation (2) is input– output stable on [τ, +∞) in the pair (X0 , Y0 ) if and only if the operator L : (X∞ )τ /+∞ → (Y∞ )τ /+∞ is causally invertible. (b) Let q 6= ∞. Equation (3) is input–output stable on (−∞, τ ] in the pair (Xq , Yq ) if and only if the operator L : (Xq )−∞/τ → (Yq )−∞/τ is causally invertible. ¡ ¢ An operator L ∈ B+ (X∞ )−∞/τ , (Y∞ )−∞/τ maps the subspace (X0 )−∞/τ into (Y0 )−∞/τ . Equation (3) is input–output stable on (−∞, τ ] in the pair (X∞ , Y∞ ) if and only if the operator L : (X0 )−∞/τ → (Y0 )−∞/τ is causally invertible. (c) Let q 6= 0, ∞. Equation (3) is input–output stable on (−∞, +∞) in the pair (Xq , Yq ) if and only if the operator L : Xq → Yq is causally invertible. d Example. The operator U = dt + 1 is causally invertible, but the inverse of d 0 U = − dt + 1 is anticausal, see 2.3.12. Respectively, the equation x˙ + x = f is stable, but the equation −x˙ + x = f is not.

Proof. Assume that L is causally invertible on R. Then from 1.2.2 it follows the estimate k(La/b )−1 k ≤ kL−1 k for all −∞ < a < b < +∞, which by 3.3.4 implies input–output stability. The cases of semi-axes are handled similarly. We prove the converse. Let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Assume that equation (3) is input–output stable on H. By 3.3.5, without loss of generality we may assume that τ = 0. We begin with the case (X, Y ) = (Lpq , Lpq ). We make use of 3.2.5. We consider the representation of Lpq as lq and the corresponding matrix of L. By 3.3.4, for all [n, m] ⊆ H, −∞ < n < m < +∞, the operators Ln/m are invertible and k(Ln/m )−1 k ≤ K. By 3.2.3 this implies the input–output stability of the corresponding difference equation on H ⊆ R, which in turn by 3.2.5 implies the causal invertibility of L on H. After that, we turn to the case (X, Y ) = (Cq , Cq ). Let us consider the space lq = lq (I, C♥i ), the isomorphism Υ : Cq → lq (I, C♥i ) from 3.1.2, and the operator T = ΥLΥ−1 : lq → lq . Since the isomorphisms Υ and Υ−1 are causal in the discrete case (i.e., Υ(Cq )n ⊆ (lq )n and Υ−1 (lq )n ⊆ (Cq )n , n ∈ Z), the operators Tn/m = Υn/m Ln/m (Υ−1 )n/m , [n, m] ⊆ H, −∞ < n < m < +∞, are uniformly invertible. Indeed, (Tn/m )−1 = Υn/m (Ln/m )−1 (Υ−1 )n/m ; hence k(Tn/m )−1 k ≤ kΥn/m k · k(Ln/m )−1 k · k(Υ−1 )n/m k ≤ kΥk · K · kΥ−1 k for all [n, m] ⊆ H, −∞ < n < m < +∞, where K is the constant from 3.3.4. By 3.2.3 and 3.2.5 this implies the causal invertibility of T on H. Then from the causal invertibility of T and the identity L = Υ−1 T Υ it follows the causal invertibility of L on H. Now we assume that (X, Y ) = (Cq , Cq−1 ). Let us consider the isomorphism U : Cq → Cq−1 defined in 2.3.9 and the operator T = U −1 L : Cq → Cq . Since U and

168

III. STABILITY

U −1 are causal the operators Ta/b = (U −1 )a/b La/b , [a, b] ⊆ H, −∞ < a < b < +∞, are uniformly invertible. Indeed, (Ta/b )−1 = (La/b )−1 Ua/b and therefore k(Ta/b )−1 k ≤ k(La/b )−1 k · kUa/b k ≤ KkU k for all [a, b] ⊆ H, −∞ < a < b < +∞. By what has been proved in the previous paragraph, it follows the causal invertibility of T on H. This and the identity L = U T imply the causal invertibility of L on H. −1 1 The cases of the pairs (Lpq , Wpq ), (Cq1 , Cq ), and (Wpq , Lpq ) are handled sim1 1 , Lpq ) one ilarly. We only note that in the cases of the pairs (Cq , Cq ) and (Wpq −1 should use 2.3.4 instead of 2.3.9 and consider the operator T = LU . ¤ 3.3.7. Invertibility in (X∞ , Y∞ ) implies that in (X0 , Y0 ) Proposition. Let L : (X∞ )τ /+∞ → (Y∞ )τ /+∞ be a bounded linear operator. Assume that L maps (X0 )τ /+∞ into (Y0 )τ /+∞ . If L : (X∞ )τ /+∞ → (Y∞ )τ /+∞ is invertible then L : (X0 )τ /+∞ → (Y0 )τ /+∞ is also invertible. Proof. Clearly, without loss of generality we may assume that τ = 0. First we consider the case (X, Y ) = (C, C). By 3.1.2 C−∞/τ ' l∞ (N0 , C♥i ) and (C0 )−∞/τ ' l0 (N0 , C♥i ), where N0 = { 0, 1, 2, . . . }. Thus the proof is reduced to 3.2.6. Assume that (X, Y ) = (C, C −1 ). We consider the operator T = U −1 L : C → C, where U is defined as in 2.3.9. Because of U and U −1 are causal the restrictions U : C−∞/τ → (C −1 )−∞/τ and U : (C0 )−∞/τ → (C0−1 )−∞/τ are isomorphisms. Therefore the invertibility of L : C−∞/τ → (C −1 )−∞/τ is equivalent to the invertibility of T : C−∞/τ → C−∞/τ , and the invertibility of L : (C0 )−∞/τ → (C0−1 )−∞/τ is equivalent to the invertibility of T : (C0 )−∞/τ → (C0 )−∞/τ . Thus we have reduced this case to the previous one. The cases of other spaces are handled similarly. ¤ Remark. In 3.5.4 and 5.5.6 (5.5.11) we shall see that for operators T of the wide enough classes e and t the invertibility in (X0 , Y0 ) implies the invertibility in (X∞ , Y∞ ). See also 6.2.7. 3.3.8. Input–output stability and causal invertibility in (X0 , Y0 ) Corollary. Let equation (3) be locally soluble. Let H be one of the intervals (−∞, +∞), (−∞, τ ), or [τ, +∞). Equation (1) is input–output stable on H in the pair (X0 , Y0 ) if and only if the operator T : X0 → Y0 is causally invertible on H. Proof. The proof is similar to that of 3.2.7. ¤ 3.3.9. Small perturbations preserve input–output stability. Sometimes the following corollary can be used as an alternative to Lyapunov’s second method.

3.3. INPUT–OUTPUT STABILITY: CONTINUOUS TIME

169

Corollary. Let (X, Y ) be one of the pairs (2), and let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Let L, N ∈ B+ (X, Y ). If equation (3) is locally soluble and input–output stable on H, and kL−1 k · kN k < 1 then the perturbed equation (L − N ) x = f is also locally soluble and input–output stable on H. Proof. If q 6= 0, ∞, this is an immediate consequence of 3.3.6 (3.3.8), 2.2.12, 2.1.3, 3.3.3, and 3.3.4, see also 2.5.7. In order to consider the cases q = 0 and q = ∞, first we observe that by 3.3.5(a) and 2.1.6 it suffices to restrict ourselves to the cases H = (−∞, τ ] and H = [τ, +∞). Then it remains the only not completely evident case q = ∞ and H = (−∞, τ ]. In this case we consider L and N in (X0 , Y0 ) instead of (X∞ , Y∞ ) (see 3.3.6(b)) and repeat the previous argument. ¤ Remark. (a) Of course, a similar fact holds for equations on Z. (b) We stress that the proof of the corollary is not so trivial within the framework of the traditional exposition of the theory of ordinary differential equations. Example. We consider the ordinary differential operator ¡ ¢ Lx (t) = x(t) ˙ + a(t)x(t) ¡ ¢ with a : R → B(E). Clearly, if a ∈ C R, B(E) then the operator of multiplication ¡ ¢ Ax (t) = a(t)x(t) acts continuously on Cq for all q and therefore ¡the operator L ¢ acts continuously both in (Cq1 , Cq ) and (Cq , Cq−1 ). And if a ∈ L∞ R, B(E) then the operator A acts continuously on Lpq for all p and q and therefore the operator 1 −1 L acts continuously both in (Wpq , Lpq ) and (Lpq , Wpq ). We assume that L is causally invertible and thus the equation Lx = f is input– output stable. (In 6.3.10 and 6.3.12 we shall show that these properties do not depend on the pair of spaces in which the operator L is considered.) We consider the simplest perturbation ¡ ¢ N x (t) = b(t)x(t). Clearly, if kbkL∞ is small then the norm on N is small, and by the corollary the stability of the equation Lx = f implies the stability of the equation (L−N )x = f . We show that this estimate of kN k can be improved. 1 1 ⊆ Cq ⊆ L∞q . Therefore if x ∈ W1q (a) ([DaK], [MaS]) It is easy to see that W1q 1 and b ∈ L1∞ then bx ∈ L1q . Thus N acts from W1q to L1q provided b ∈ L1∞ . It 1 is easy to show that kN : W1q → L1q k is small if kbkL1∞ is small. (b) Let us consider the series L−1 + L−1 N L−1 + L−1 N L−1 N L−1 + . . . .

170

III. STABILITY

It is easy to show (cf. 1.4.2) that if this series converges in norm, its sum is the (causal) inverse of L − N . We represent the product N L−1 N L−1 as the L−1

N

L−1

N

composition C −−→ C 1 −→ C −1 −−→ C −→ C. If the norm kN : C 1 → C −1 k is small then the norm kN L−1 N L−1 : C → Ck is also small, which implies the convergence of the series. It can be shown (see [Z7 ]) that kN : C 1 → C −1 k is small if the coefficient b ∈ C is rapidly oscillating, e.g., b(t) = eiωt with a big ω ∈ R. Similar reasoning with C replaced by L2 , allow one to estimate kN : L2 → W2−1 k effectively (not only to say that it is small), see [Ku16 ]. 3.3.10. Initial value problem on semi-axis. Often stability is defined in terms of an initial value problem on [a, +∞). In this subsection we give relevant formulations. Let (X, Y ) be one of the pairs (2). We consider equation (3). We take a ∈ R, ϕ ∈ X and f ∈ Yloc ; see 3.3.2 for the definition ¡ ¢ of Yloc . (If Y = Cq , as usual we assume that the compatibility condition Lϕ (a) = f (a) is satisfied.) By the −1 solution of the initial value problem (Y 6= Wpq ) ¡ ¢ Lx (t) = f (t), t > a, (9) x(t) = ϕ(t), t < a, −1 or, respectively, (Y = Wpq ) ¡ ¢ ¡ ¢ Pa Lx (t) = Pa f (t),

x(t) = ϕ(t),

t ∈ R, t < a,

(10)

we mean a function x ∈ Xloc such that for all b ∈ (a, +∞) the natural projection of x into (Xloc )−∞/b lies in X−∞/b (see the end of 3.3.2 for the discussion of the embedding X−∞/b ⊆ (Xloc )−∞/b ) and satisfies the initial condition and the equality ¡ ¢ Lx (t) = f (t), a < t < b, or, respectively,

¡

¢ ¡ ¢ Pa Lx (t) = Pa f (t),

a < t < b.

Clearly, if equation (3) is locally soluble the initial value problem (9) or (10) has a unique solution. Proposition. Let (Xq , Yq ) be one of the pairs (2). Let L ∈ B+ (Xq , Yq ) and q 6= 0. Let equation (3) be locally soluble. Let a ∈ R. Then equation (3) is input– output stable on [a, +∞) if and only if for all ϕ ∈ X and f ∈ Y (satisfying the compatibility condition if Y = C) the solution x of the initial value problem (9) or (10) lies in X and satisfies the estimate kxkX ≤ K(kϕk + kf k), where K does not depend on ϕ and f . Proof. The proof is evident, cf. the proof of 3.2.5(a). ¤ We leave the consideration of the cases H = (−∞, a] and H = (−∞, +∞), and q 6= 0 to the reader.

3.4. EXPONENTIAL STABILITY: DISCRETE TIME

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3.4. Exponential stability: discrete time We begin our consideration of exponential stability with the discrete case, cf. §3.2. This allows us to separate the fundamental ideas and secondary details. The main result of this section is theorem 3.4.7 on the equivalence of exponential stability and causal invertibility. We discuss exponential stability only for equations with exponentially fading memory, i.e., for equations of the form T x = f with T ∈ e, see 1.6.9 for the definition of e. As we shall see in the remark in 3.4.7, this is substantially the widest class of equations for which exponential stability can occur. Our proof of theorem 3.4.7 is based on another important fact — theorem 3.4.2 — which states that the inverse of an operator belonging to the class e also belongs to e. For completeness, we prove some assertions for the general case of Zn instead of Z. 3.4.1. Exponential weights and the class e. Let n be a positive integer, and let { Ei : i ∈ Zn } be a family of Banach spaces. We consider the spaces lq = lq (Zn , Ei ), 1 ≤ q ≤ ∞ or q = 0, see 1.5.7 for the definition. A function β : Zn → R is called a Lipschitz function if it satisfies the Lipschitz condition |β(i) − β(j)| ≤ γki − jk1 , where khk1 = |h1 | + · · · + |hn |. The constant γ ≥ 0 is called the Lipschitz constant. We call the function i 7→ eβ(i) an exponential weight. For an arbitrary collection x = { xi ∈ Ei : i ∈ Zn } we set (Ψβ x)i = eβ(i) xi . We denote by lq,β = lq,β (Zn , Ei ), 1 ≤ q ≤ ∞ or q = 0, the space of all collections x = { xi ∈ Ei : i ∈ Zn } such that Ψ−β x ∈ lq with the norm kxk = kxklq,β = kΨ−β xklq . Thus by definition Ψβ : lq → lq,β is an isometric isomorphism. ¯ i : i ∈ Zn }. Now assume that we have a pair of families { Ei : i ∈ Zn } and { E n n ¯ We consider the spaces lq,β = lq,β (Z , Ei ), and lq,β = lq,β (Z , Ei ), 1 ≤ q ≤ ∞ or q = 0. We say that an operator Tβ ∈ B(lq,β , lq,β ) is a continuation of an operator T ∈ B(lq , lq ) if T x = Tβ x for all x ∈ lq ∩ lq,β . We say that T ∈ B(lq , lq ) admits a continuation to an operator Tβ ∈ B(lq,β , lq,β ) if a continuation Tβ ∈ B(lq,β , lq,β ) exists. ¯ i ) : i, j ∈ Zn } induces the operator We say that a matrix {Tij } = { Tij ∈ B(Ej , E ¯ i ) if Tβ : lq,β (Zn , Ei ) → lq,β (Zn , E (Tβ x)i =

X

Tij xj

(1)

j∈Zn

with the series converges unconditionally, cf. 1.6.4. We note the following trivial fact. Assume a matrix {Tij } induces an operator T ∈ B(lq , lq ) as well as an operator Tβ ∈ B(lq,β , lq,β ). Then the operator Tβ is a continuation of the operator T .

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III. STABILITY

Mostly we use the two following examples of weights. For any ϑ ∈ Rn we set (Ψϑ x)i = ehϑ,ii xi

and

(Ψϑ,|·| x)i = ehϑ,|i|i xi ,

where hϑ, ii = ϑ1 i1 + · · · + ϑn in and |i| = (|i1 |, . . . , |in |), and thus hϑ, |i|i = ϑ1 |i1 | + · · ·+ϑn |in |. We denote the corresponding spaces lq,β by lq,ϑ and lq,ϑ,|·| , respectively. We observe that the Lipschitz constant γ for these cases is equal to kϑk∞ = max{|ϑ1 |, . . . , |ϑn |}. Clearly, lq,0 and lq,0,|·| coincide with lq . ¡ ¢ ¯ i denotes the set of all matrices We recall from 1.6.9 that e = e Zn , Ej , E ¯ i ) : i, j ∈ Zn } satisfying the following equivalent conditions { Tij ∈ B(Ej , E kTij k ≤ N e−νki−jk1 for some N < ∞ and ν > 0, X eγkhk1 sup kTij k < ∞ for some γ > 0. {Tij } γ = h∈Zn

(2) (3)

i−j=h

We call the matrix of the class e a matrix with exponential ¢memory. Every matrix ¡ n n ¯ {Tij } ∈ e induces an operator ¡ T n∈ B lq (Z n, Ej ), ¢lq (Z , Ei ) . We denote the set of ¯ i ) , and call the operator T ∈ e an all such operators by e = e lq (Z , Ej ), lq (Z , E operator with exponential memory. Proposition. Let 1 ≤ q ≤ ∞ or q = 0. (a) Let {Tij } ∈ e. Let β be a Lipschitz function, and let the Lipschitz constant γ of β satisfy (3). Then {Tij } induces the operator Tβ ∈ B(lq,β , lq,β ) with kTβ : lq,β → lq,β k ≤ {Tij } γ . In particular, {Tij } induces the operators Tϑ ∈ B(lq,ϑ , lq,ϑ ) and Tϑ,|·| ∈ B(lq,ϑ,|·| , lq,ϑ,|·| ) if γ = kϑk∞ satisfies (3). (b) Assume T ∈ B(lq , lq ) and there exists γ > 0 such that T admits a continuation to an operator Tϑ ∈ B(lq,ϑ , lq,ϑ ) for all ϑ ∈ Rn such that kϑk∞ ≤ γ. Then T ∈ e. (c) Assume T ∈ B(lq , lq ) and there exists γ > 0 such that T admits a continuation to an operator Tϑ,|·| ∈ B(lq,ϑ,|·| , lq,ϑ,|·| ) for all ϑ ∈ Rn such that kϑk∞ ≤ γ. Then T ∈ e. Proof. (a) We consider the auxiliary matrix {Rij } = {eβ(j)−β(i) Tij }. (It can be interpreted as the composition of matrices of Ψ−β , T , and Ψβ . We recall that the isomorphism Ψβ : lq → lq,β is induced by the main diagonal matrix with the entries (Ψβ )ii = eβ(i) .) By (3) and 1.6.8(c), {Rij } induces an operator, say R, of the class s acting from lq into lq with kRk ≤ {Tij } γ . Let us consider the operator Tβ = Ψβ RΨ−β : lq,β → lq,β . Since R is induced by

3.4. EXPONENTIAL STABILITY: DISCRETE TIME

173

the matrix {Rij } we have (Tβ x)i = eβ(i) (RΨ−β x)i X = eβ(i) Rij (Ψ−β x)j j∈Zn

= eβ(i) =

X

X

Rij e−β(j) xj

j∈Zn

eβ(i)−β(j) Rij xj

j∈Zn

=

X

Tij xj

j∈Zn

with the series convergent absolutely. Therefore the operator Tβ is induced by the matrix {Tij }. It remains to note that since Ψ−β and Ψβ are isometric it follows that kTβ k = kΨ−β RΨβ : lq,β → lq,β k = kR : lq → lq k. (b), (c) Let β be a Lipschitz function, and let R ∈ B(lq,β , lq,β ) be an arbitrary operator. We define the matrix entries Rij of the operator R by the usual rule (see 1.6.4) Rij = Qi RJj , where Jj : Ej → lq,β is the natural embedding, and ¯ i is the natural projection. Qi : lq,β → E We estimate kRij k. Let l{j} , j ∈ Zn , denote the space of all x = {xi } such that xi = 0 for i 6= j. Clearly, l{j} ⊆ lq,β for all q and β. We take an arbitrary e ∈ Ej and consider e˜ ∈ l{j} with (˜ e)j = e. By definition k˜ eklq,β = e−β(j) kek. Next, we have kRij ekE¯ i = k(R˜ e)i kE¯ i ≤ eβ(i) kR˜ eklq,β ≤ eβ(i) kR : lq,β → lq,β k · k˜ eklq,β ≤ eβ(i)−β(j) kR : lq,β → lq,β k · kek. Taking here the supremum over all e ∈ Ej , kek ≤ 1, we obtain kRij k ≤ eβ(i)−β(j) kR : lq,β → lq,β k. Now let R be either Tϑ or Tϑ,|·| , see assertions (b) and (c). Since, by assumption, Tϑ and Tϑ,|·| coincide with T : lq → lq on l{j} the matrix entries Tij do not depend on ϑ. In particular, the preceding estimate for kRij k applied to k(Tϑ )ij k and k(Tϑ,|·| )ij k holds for the matrix entries of the operator T : lq → lq . If β(i) = hϑ, ii, which corresponds to case (b), the last estimate implies that kTij k ≤ ehϑ,i−ji kTq,ϑ : lq,ϑ → lq,ϑ k.

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If β(i) = hϑ, |i|i, which corresponds to case (c), the previous estimate implies that kTij k ≤ ehϑ,|i|−|j|i kTq,ϑ,|·| : lq,ϑ,|·| → lq,ϑ,|·| k. Let Θ = Θγ denote the set of all ϑ ∈ Rn such that |ϑi | = γ for all i. Clearly, Θ is finite. Taking in the preceding estimates the minimum over all ϑ ∈ Θ we obtain kTij k ≤ e−γki−jk1 max{ kT : lq,ϑ → lq,ϑ k : ϑ ∈ Θ } or, respectively, kTij k ≤ e−γki−jk1 max{ kT : lq,ϑ,|·| → lq,ϑ,|·| k : ϑ ∈ Θ }, i.e., the estimate (2). Thus {Tij } is a matrix of the class e. To prove that T is an operator of the class e we should show that T is induced by its matrix. If q 6= ∞ this follows from 1.6.4(b). Consider the case q = ∞. We observe that l∞ is contained in the sum of the spaces l0,ϑ (respectively, l0,ϑ,|·| ) over all ϑ ∈ Θ. By (a) the operator T on each l0,ϑ (respectively, on l0,ϑ,|·| ) is induced by the matrix rule (1). Hence it is defined by the same rule on l∞ . ¤ 3.4.2. The class e is full

¡ ¢ ¯ i ) , 1 ≤ q ≤ ∞ or q = 0, Theorem. Let an operator T ∈ e lq (Zn , Ej ), lq (Zn , E be invertible. Then T −1 is also an operator of the class e. Remark. In 5.2.12 we shall prove a slightly more general statement. Proof. By 3.4.1(a), for any ϑ, kϑk∞ ≤ γ, the matrix of the operator T induces an operator Tϑ,|·| ∈ B(lq,ϑ,|·| , lq,ϑ,|·| ). We consider the operators T[ϑ] = Ψ−ϑ,|·| Tϑ,|·| Ψϑ,|·| : lq → lq . Evidently (cf. the proof of 3.4.1(a)), T[ϑ] is induced by the matrix (T[ϑ] )ij = ehϑ,|j|−|i|i Tij . Clearly, T[ϑ] depends on ϑ continuously in the norm · defined in 1.6.8. Hence by 1.6.8(c) T[ϑ] depends on ϑ continuously in the ordinary norm k · k. Since T = T[0] is invertible, by 1.3.6(c) it follows that T[ϑ] is also invertible for all ϑ in a neighbourhood of zero, i.e., for all ϑ satisfying kϑk∞ ≤ ν with ν > 0. Consequently the operators Tϑ,|·| = Ψϑ,|·| T[ϑ] Ψ−ϑ,|·| are invertible for the same ϑ. We consider ξ = (ν, . . . , ν), where ν > 0 is as above. Clearly, lq,ξ,|·| is the widest of the spaces lq,ϑ,|·| , kϑk∞ ≤ ν. Thus all the operators Tϑ,|·| , kϑk∞ ≤ ν, are restrictions of the widest operator Tξ,|·| . Consequently the inverses of Tϑ,|·| , kϑk∞ ≤ ν, coincide with the restrictions of the inverse of Tξ,|·| . Thus the operator T −1 admits a continuation to the operator (Tϑ,|·| )−1 ∈ B(lq,ϑ,|·| , lq,ϑ,|·| ) for all ϑ, kϑk∞ ≤ ν. By 3.4.1(c) this implies T −1 ∈ e. ¤

3.4. EXPONENTIAL STABILITY: DISCRETE TIME

175

3.4.3. The independence of invertibility from q for T ∈ e. We observe that the formulation of assertion 3.2.5(b) can be simplified (cf. 3.2.7) if one can prove that the invertibility in (l∞ , l∞ ) implies the invertibility in (l0 , l0 ). This is one of possible applications of statements of the following kind. ¡ ¢ ¯ i ) , 1 ≤ q ≤ ∞ or Corollary. Let an operator T = Tq ∈ e lq (Zn , Ej ), lq (Zn , E q = 0, be invertible. Then for any 1 ≤ r ≤ ∞ or r = 0 the operator T = Tr : lr → lr induced by the matrix of Tq is also invertible. If n = 1, and Tq and (Tq )−1 are causal then Tr and (Tr )−1 are causal, too. Proof. By the theorem T −1 : lq → lq is an operator of the class e, i.e., its matrix {(T −1 )ij } belongs to the class e, and T −1 is induced by its matrix. We consider the product {Rik } of the matrices {Tij } and {(T −1 )jk } (or, respectively, {(T −1 )ij } and {Tjk }). By 1.6.8(b) it is well defined. Clearly, {Rik } induces the identity operator. Hence {Rik } is the identity matrix. It follows that the operator from lr to lr induced by {(T −1 )ij } is the inverse of T : lr → lr . Clearly, if T is causal its matrix is lower-triangular. And conversely, if {Tij } is lower-triangular, T is causal. Thus the last assertion is evident. ¤ 3.4.4. The case of semi-axes. Below in this section we assume that Zn = Z. Let τ ∈ Z be a fixed point, and let H ⊆ Z be either (−∞, τ ) or [τ, +∞). One can easily define β on H, spaces¢ lq,β (H, Ej ), classes of ¡ ¢ a Lipschitz function ¡ ¯ ¯ i ) . The simplest way of matrices e H, Ej , Ei , and operators e lq (H, Ej ), lq (H, E ¯ i are defined to be zero ones for doing these is to assume that the spaces Ei and E i∈ / H. Then the Lipschitz function β can be extended to Z \ H as a constant, collections { xi : i ∈ H } can be identified with { xi : i ∈ Z } and so on. Thus we arrive at the case considered above. In particular, statements 3.4.1–3 take place on H. In the following corollary we formulate those of them we need for the further references. Corollary. Let 1 ≤ q ≤ ∞ or q = ¡0. Let τ ∈ Z be a fixed point, and let H be ¢ ¯ either (−∞, τ ) or [τ, +∞). Let T ∈ e lq (H, Ej ), lq (H, Ei ) . (a) The matrix of T induces a bounded operator from lq,ϑ to lq,ϑ for all ϑ ∈ R close enough to zero. Here lq,ϑ corresponds to the function β(i) = ϑi. (b) If the operator T is invertible then T −1 is also an operator of the class e. (c) Let the operator T be invertible. Then for any 1 ≤ r ≤ ∞ or r = 0 ¯ i ) induced by the matrix of T is also the operator Tr : lr (H, Ej ) → lr (H, E invertible. If T and T −1 are causal then Tr and (Tr )−1 are causal, too. Proof. The proof is plain. ¤ We define the T -direction on X = lq,ϑ to be the family of the subspaces Xt = { x ∈ lq,ϑ : xi = 0 for i < t }, t ∈ R. Clearly, (lq,ϑ )−∞/τ and (lq,ϑ )τ /+∞ are isomorphic to lq,ϑ (H, Ej ) with a relevant H. Thus an analogue of the proposition holds for these spaces, too.

176

III. STABILITY

3.4.5. The definition of uniform solubility. We return to the notation of 3.2.1. Let T ∈ B+ (X, Y ). We consider the difference equation T x = f.

(4)

We say that equation (4) is uniformly soluble if it is locally soluble and there exist δ ≥ 1 and K < ∞ such that for any [n, m), m − n ≤ δ, and f and ϕ the solution x ∈ X−∞/m of the initial value problem ¡ ¢ T x i = fi ,

n ≤ i < m,

xi = ϕi ,

i < n,

(5)

satisfies the estimate kxk ≤ K(kϕk + kf k), where kxk means the norm on X−∞/m . Arguing as in 3.2.3 we can rewrite this estimate equivalently as kxkX−∞/m ≤ K(kϕkX−∞/n + kf k[n,m) ), where kf k[n,m) and kϕkX−∞/n are defined as in 3.2.3. Proposition. (a) The uniform solubility of (4) is equivalent to the uniform invertibility of operators Tn/m , m − n ≤ δ, for some δ ≥ 1. (b) The uniform solubility of (4) is equivalent to the uniform invertibility of ¯ i ) for all i ∈ Z. In particular, the main diagonal matrix entries Tii ∈ B(Ei , E the definition of uniform solubility does not depend on δ. Proof. (a) follows from the formula x = (Tn/m )−1 In/m (f − T ψ) + ψ for the solution of (5), see the proof of 3.2.3. (b) Considering m − n = 1 we see from (a) that input–output stability implies the uniform invertibility of Tii . Conversely, assume the main diagonal entries Tii are uniformly invertible, i.e., k(Tii )−1 k ≤ M for all i. We represent the matrix of Tn/m as the sum of the main diagonal (i.e., (Td )ij = 0 if i 6= j) and the strongly lower triangular (i.e., (Tν )ij = 0 if i ≤ j) matrices, and denote the corresponding operators by Td and Tν , respectively. Thus T = Td +Tν . Clearly, k(Td )−1 k ≤ M and kTν k ≤ kT −Td k ≤ kT k + kTd k ≤ 2kT k. We consider the representation (Td + Tν )−1 = (Td )−1 − (Td )−1 Tν (Td )−1 + (Td )−1 Tν (Td )−1 Tν (Td )−1 − . . . . We observe that the number of non-zero members of this series is less than or equal to m − n. Therefore k(Tn/m )−1 k possesses the uniform estimate. ¤

3.4. EXPONENTIAL STABILITY: DISCRETE TIME

177

3.4.6. The definition of exponential stability. For ϑ ∈ R and τ ∈ Z we set (Ψϑ,τ x)i = eϑ(i−τ ) xi , i ∈ Z. Clearly, β(i) = ϑ(i − τ ) is a Lipschitz function, see 3.4.1 for the definition. We denote by lq,ϑ,τ the space of all collections x such that Ψ−ϑ,τ x ∈ lq with the norm kxklq,ϑ,τ = kΨ−ϑ,τ xklq . We note that lq,ϑ,τ differs from spaces lq,ϑ defined in 3.4.1 only by the norm; moreover, kxklq,ϑ,τ = Ckxklq,ϑ , where C = eϑτ . Hence any operator T has the same norm as acting from lq,ϑ to lq,ϑ and as acting from lq,ϑ,τ to lq,ϑ,τ . We define the T -direction on X = lq,ϑ,τ by virtue of the isomorphism Ψϑ,τ , i.e., as the family of the subspaces Xt = { x ∈ lq,ϑ,τ : xi = 0 for i < t }, t ∈ R. Assume that equation (4) is uniformly soluble. Let τ ∈ Z be a fixed point, and let H ⊆ Z be (−∞, +∞), (−∞, τ ), or [τ, +∞). We say that equation (4) is exponentially stable on H if there exist ν > 0 and N < ∞ such that for any interval [n, m) ⊆ H, −∞ < n < m < +∞, and ϕ ∈ X the solution x ∈ X−∞/m of the homogeneous initial value problem ¡ ¢ T x i = 0, xi = ϕi ,

n ≤ i < m, i < n,

(6)

satisfies the estimate kxklq,−ν,n ≤ N kϕk.

(7)

Here as usual by kxklq,−ν,n we mean the norm of x in (lq,−ν,n )−∞/m (note that (lq )−∞/m ⊆ (lq,−ν,m )−∞/m ). Taking the infimum over all ϕ with the same projection into X−∞/n we can rewrite (7) equivalently as kxk(lq,−ν,n )−∞/m ≤ N kϕkX−∞/n . Sometimes we shall specify that we consider exponential stability in the pair (X, Y ). We note that for the pairs (l0 , l0 ) and (l∞ , l∞ ) the estimate (7) can be rewritten as |xi | ≤ N e−ν(i−n) kϕkl∞ , n ≤ i < m. In particular, exponential stability in the pairs (l0 , l0 ) and (l∞ , l∞ ) means the same provided T acts both in (l0 , l0 ) and (l∞ , l∞ ). 3.4.7. Exponential stability and causal invertibility. We denote by e+ the intersection of e and B+ . Theorem. Let 1 ≤ q ≤ ∞ or q = 0. Let T ∈ e+ (lq , lq ) and equation (4) be uniformly soluble. Let τ ∈ Z, and let H ⊆ Z be (−∞, +∞), (−∞, τ ), or [τ, +∞). Then the following assumptions are equivalent. (a) Equation (4) is exponentially stable on H. (b) Equation (4) is input–output stable on H. (c) The operator T : lq → lq is causally invertible on H.

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III. STABILITY

Proof. (b) ⇒ (c) follows from 3.2.5 except the cases q = 0, ∞. The case q = 0 follows from 3.2.7. We assume that q = ∞. The case H = [τ, +∞) follows from 3.2.5(a). The case H = (−∞, τ ) follows from 3.2.5(b) and 3.4.4(c). The case H = (−∞, +∞) is reduced to the cases of semi-axes by means of 3.2.4 and 2.1.6. (c) ⇒ (a). Let ψ be defined as in 3.2.1. We make use of the formula x = −(Tn/m )−1 In/m T ψ + ψ for the solution of (6), see the proof of 3.2.3. Clearly, ψ ∈ (lq,−ν,n )−∞/m for all ν ≥ 0. Moreover, kψk(lq,−ν,n )−∞/m ≤ kψk(lq )−∞/m = kϕk(lq )−∞/n . By 3.4.1(a) T ψ belongs to lq,−ν,n for sufficiently small ν > 0 and kT ψk(lq,−ν,n )−∞/m ≤ kT : lq,−ν,n → lq,−ν,n k · kψk(lq,−ν,n )−∞/m . Clearly, kIn/m : lq,−ν,n → (lq,−ν,n )n/m k ≤ 1. By 3.4.2 and 3.4.1(a) (or 3.4.4) the operator T −1 admits a continuation to a bounded operator from lq,−ν,n to lq,−ν,n for ν > 0 small enough. Thus from the formula for solution we obtain the estimate (7). (a) ⇒ (b). We consider the initial value problem (5). We represent f as the sum Pm−1 f = k=n f k , where (f k )i = 0 for i 6= k, and (f k )k = fk . Clearly, kf k klq ≤ kf klq . Let xn be the solution of the initial value problem ¡ n¢ T x i = (f n )i , n ≤ i < m, (xn )i = ϕi ,

i < n.

And let xk , k = n + 1, . . . , m − 1, be the solution of the initial value problem ¡ k¢ T x i = (f k )i , k ≤ i < m, (xk )i = 0,

i < k.

Pm−1 Clearly, the solution x of (5) coincides with the sum k=n xk . Since equation (4) is uniformly soluble, for all k = n, n + 1, . . . , m − 1 we have k(xk )k k ≤ K(kϕk + kf k). We observe that by the choice of f k we have (T xk )i = 0 for k + 1 ≤ i < m. Thus xk can be interpreted as the solution of the homogeneous initial value problem with the initial point k + 1. Therefore by the definition of exponential stability xk satisfies the estimate kxk klq,−ν,k+1 ≤ N kxk k(−∞,k+1) ≤ N K(kϕk + kf k). Consequently |(xk )i | ≤ e−ν(i−k−1) N K(kϕk + kf k)

for i > k.

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179

Summing up these estimates and taking into account that xk = 0 on (−∞, k) for k = n + 1, . . . , m − 1, we obtain the estimate for x from the definition of input–output stability. ¤ Remark. The analysis of the proof shows that its part (a) ⇒ (b) ⇒ (c) does not use the assumption T ∈ e. (To be precise, there is one exception: if q = ∞, in the part (a) ⇒ (b) of the proof exponential stability implies only the causal invertibility of T as acting from l0 into l0 .) Moreover, the exponential estimates of xk means that the matrix of T −1 belongs to e. Consequently by 1.6.4(b) the operator T −1 belongs to e (provided q 6= ∞). By 3.4.2 this implies that T ∈ e. Thus equation (4) can be exponentially stable only if T ∈ e (with the above specifications for the case q = ∞). 3.4.8. The independence of stability from q for T ∈ e Corollary. Let 1 ≤ q ≤ ∞ or q = 0, and let 1 ≤ r ≤ ∞ or r = 0. Let τ ∈ Z, and let H ⊆ Z be (−∞, +∞), (−∞, τ ), or [τ, +∞). Let T ∈ e+ (lq , lq ), and let equation (4) be uniformly soluble and exponentially or input–output stable on H in the pair (lq , lq ). Then equation (4) is uniformly soluble, and exponentially and input–output stable on H in the pair (lr , lr ). Proof. By 3.4.5(b) the uniform solubility is independent of q. By 3.4.7 both exponential and input–output stability in the pair (lq , lq ) are equivalent to the causal invertibility of L in (lq , lq ), which by 3.4.3 and 3.4.4(c) does not depend on q. Thus stability does not depend on q, too. ¤

3.5. Exponential stability: continuous time In this section we extend the results of the previous section to the case of functional differential and functional equations on R. The main result is theorem 3.5.9 on equivalence of exponential stability and causal invertibility. The class e on Z was defined in terms of matrix representation. In the case of continuous time this approach involves some difficulties. First, the representation 1 −1 of the spaces Cq1 , Wpq , Cq−1 , and Wpq as lq is rather complicated, see 3.1.2. Because of this we try to avoid the usage of matrix representations for differential operators and their inverses when it is possible. Secondly, the continuous analogue of matrix should be treated as an explicit formula for an operator. But for many spaces, there is no effective general formula for linear bounded operator. So we prefer to take for the main definition of the membership in the class e on R the action in the pair of spaces with exponential weights. 3.5.1. Exponential weights on R. Let E be a Banach space, and let D0 = D0 (R, E) be the space of all distributions, see 2.3.5. For any ϑ ∈ R we define the mapping Ψϑ : D0 → D0 by the rule ¡ ¢ Ψϑ x (t) = eϑt x(t). Since the function t 7→ eϑt is infinitely many times differentiable this definition makes sense. Clearly, Ψϑ is invertible and (Ψϑ )−1 = Ψ−ϑ .

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III. STABILITY

Let X = Xq be one of the spaces Cq = Cq (R, E),

Cq1 = Cq1 (R, E),

Cq−1 = Cq−1 (R, E),

Lpq = Lpq (R, E),

1 1 Wpq = Wpq (R, E),

−1 −1 Wpq = Wpq (R, E),

(1)

where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by Xϑ = Xq,ϑ the space of all distributions of the form Ψϑ x ∈ D0 , where x runs over X. We endow Xϑ with the norm kyk = kykXϑ = kΨ−ϑ ykX , cf. 3.4.1. Thus by definition Ψϑ : X → Xϑ is an isometric isomorphism. The detailed notation for Xϑ will appear as Lpq,ϑ , or −1 1 Wpq,ϑ (R, E), or Cq,ϑ , or Cq,ϑ (R, E) and so on. 1 Proposition. Let L stand for Cq or Lpq , let W 1 stand for Cq1 or Wpq , and let −1 −1 −1 W stand for Cq or Wpq , respectively.

(a) For all ϑ ∈ R the space Lϑ consists of all y ∈ Lloc bounded by the norm kΨ−ϑ ykL . See 3.3.2 for the definition of Lloc . (b) For all ϑ ∈ R the space Wϑ1 consists all y ∈ Lϑ such that its distribution derivative y˙ also belongs to Lϑ . The norm kyk = kykLϑ + kyk ˙ Lϑ on Wϑ1 is equivalent to the initial norm. (We recall from 2.3.5 and 2.3.1 that if y˙ ∈ Lpq,ϑ the derivative can be interpreted as the Lebesgue derivative, and if y˙ ∈ Cq,ϑ it can be interpreted as the classical derivative.) (c) For all ϑ ∈ R the space Wϑ−1 consists of all distributions g of the form g = u˙ + v, where u, v ∈ Lϑ , cf. 2.3.6. The norm kgk = inf{ kukLϑ + kvkLϑ : g = u˙ + v; u, v ∈ Lϑ } on Wϑ−1 is equivalent to the initial norm. (d) For all ϑ > −1 the operator U x = x+x ˙ is a topological causal isomorphism from Wϑ1 onto Lϑ , and the operator U −1 is defined by the formula ¡

U

−1

¢ f (t) =

Z

+∞

e−s f (t − s) ds.

0

(e) For all ϑ > −1 the operator U x = x+x ˙ is a topological causal isomorphism −1 from Lϑ onto Wϑ , and the operator U −1 is defined by the formula ¡

U

−1

¢ f (t) = u(t) +

Z

+∞

¡ ¢ e−s v(t − s) − u(t − s) ds,

f = u˙ + v.

0

Proof. (a) is evident. (b) Assume y ∈ Wϑ1 , i.e., there exists x ∈ W 1 such that y = Ψϑ x. Clearly, y ∈ Lϑ . And from the formula (see 2.3.5(d)) ¡ ¢. y(t) ˙ = eϑt x(t) = ϑeϑt x(t) + eϑt x(t) ˙

3.5. EXPONENTIAL STABILITY: CONTINUOUS TIME

181

it follows that y˙ ∈ Lϑ , too. The same formula implies also that kykLϑ + kyk ˙ Lϑ = kxkL + kΨ−ϑ yk ˙ L ≤ kxkL + ϑkxkL + kxk ˙ L ≤ (1 + ϑ)kxkW 1 = (1 + ϑ)kykWϑ1 . 1 1 In a similar fashion¡ one proves that ¢ y, y˙ ∈ Lϑ implies Ψ−ϑ y ∈ W , i.e., y ∈ Wϑ and kykWϑ1 ≤ (1 + ϑ) kykLϑ + kyk ˙ Lϑ . Thus the norms are equivalent. −1 (c) Assume g ∈ Wϑ , i.e., there exists f ∈ W −1 such that g = Ψϑ f . Let f = u˙ + v, where u, v ∈ L. Then

g = Ψϑ u˙ + Ψϑ v. From the formula (which is substantially proved above) . Ψϑ u˙ = (Ψϑ u) − ϑΨϑ u we obtain that g possesses the representation g = (Ψϑ u). + Ψϑ (v − ϑu), where Ψϑ u, Ψϑ (v − ϑu) ∈ Lϑ . Clearly, kΨϑ ukLϑ + kΨϑ (v − ϑu)kLϑ = kukL + kv − ϑukL ¡ ¢ ≤ (1 + ϑ) kukL + kvkL . Taking here the infimum over all representations f = u˙ + v, u, v ∈ L, we obtain the inequality between the norms. In a similar fashion one proves that¡ if g = u˙ + v, u,¢v ∈ Lϑ , then Ψ−ϑ g ∈ W −1 , i.e., g ∈ Wϑ−1 and kgkW −1 ≤ (1 + ϑ) kukLϑ + kukLϑ . Taking here the infimum ϑ over all representations g = u˙ + v, u, v ∈ Lϑ , we obtain the opposite inequality between the norms. (d) By (b) the operator U x = x˙ + x acts from Wϑ1 into Lϑ for all ϑ. We consider 1 the operator Uϑ = Ψ−1 ϑ U Ψϑ : W → L. Straightforward calculations yields Uϑ x = x˙ + (1 + ϑ)x. A word for word repetition of the proof of 2.3.4 shows that ¡ −1 ¢ Uϑ f (t) =

Z

+∞

e−(1+ϑ)s f (t − s) ds

0

provided 1 + ϑ > 0. Consequently the operator Ψϑ Uϑ−1 Ψ−1 is the inverse of ϑ 1 U : Wϑ → Lϑ . Simple calculations show that for all ϑ > −1 ¡

U

−1

¢ f (t) =

Z

+∞ 0

e−s f (t − s) ds.

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III. STABILITY

(e) By (c) the operator U x = x˙ + x acts from Lϑ into Wϑ−1 for all ϑ. We −1 consider the operator Uϑ = Ψ−1 . Clearly, ϑ U Ψϑ : L → W Uϑ x = x˙ + (1 + ϑ)x. A word for word repetition of the proof of 2.3.9 shows that ¡ −1 ¢ Uϑ f (t) = u(t) +

Z

+∞

¡ ¢ e−(1+ϑ)s v(t − s) − u(t − s) ds

0

for f = u˙ + v provided 1 + ϑ > 0. Consequently the operator Ψϑ Uϑ−1 Ψ−1 ϑ is the −1 inverse of U : Lϑ → Wϑ . Simple calculations show that ¡

U

−1

¢ f (t) = u(t) +

Z

+∞

¡ ¢ e−s v(t − s) − u(t − s) ds,

f = u˙ + v,

0

for all ϑ > −1.

¤

3.5.2. The class e (continuous time). Let (X, Y ) be one of the pairs (Cq , Cq ),

(Lpq , Lpq ),

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ). (2)

For ϑ ∈ R we denote by (Xϑ , Yϑ ) the corresponding pair of weighted spaces. We say that an operator Lϑ ∈ B(Xϑ , Yϑ ) is a continuation of an operator L ∈ B(X, Y ) if Lx = Lϑ x for all x ∈ X ∩ Xϑ . (Here, by the equality of x ∈ X and z ∈ Xϑ we mean the equality of x and z as elements of D0 . By 3.5.1(a) 1 if X = Cq , X = Cq1 , or X = Wpq this is equivalent to the pointwise equality; and if X = Lpq this is equivalent to the equality almost everywhere.) We say that L ∈ B(X, Y ) admits a continuation to an operator Lϑ ∈ B(Xϑ , Yϑ ) if a continuation Lϑ ∈ B(Xϑ , Yϑ ) exists. We say that an operator L ∈ B(X, Y ) has exponential memory if it admits a continuation to an operator Lϑ ∈ B(Xϑ , Yϑ ) for all ϑ ∈ R close enough to 0. We denote by e = e(X, Y ) the class of all operators with exponential memory. If X = Y we employ the brief notation e(X). Clearly, the sum and the composition of operators of the class e is also an operator of the class e. 1 Example. Let L stand for Cq or Lpq , let W 1 stand for Cq1 or Wpq , and let −1 −1 −1 W stand for Cq or Wpq , respectively. (a) The operators of embedding I and differentiation d/dt belong to the classes e(W 1 , L) and e(L, W −1 ). (b) Assume D, B ∈ e(L). Then the operators D d/dt + BI : W 1 → L and d/dt D + IB : L → W −1 belong to the classes e(W 1 , L) and e(L, W −1 ), respectively. In particular, the operator U belongs to e(W 1 , L) and e(L, W −1 ). (c) From 3.5.1 it follows that the operator U −1 also belongs to the classes e(L, W 1 ) and e(W −1 , L).

The following proposition is an analogue of 3.1.2 for the spaces with exponential weights. Specifically it shows that our new definition of the class e is in agreement with that in 1.6.9.

3.5. EXPONENTIAL STABILITY: CONTINUOUS TIME

183

Proposition. Let Pa be defined as in 3.1.1 and X♥i be defined as in 3.1.2. (a) Let ϑ ∈ R and X be either Lpq or Cq−1 . The operator Pa : X → X admits a continuation to Pa : Xϑ → Xϑ . The mapping Υ : x 7→ {(Pi − Pi+1 )x} is a topological isomorphism from Xϑ onto lq,ϑ (Z, X♥i ). The class e(X) consists of operators T ∈ B(X) induced by matrices { Tij ∈ B(X♥j , X♥i ) } of the class e by virtue of this isomorphism. 1 −1 (b) Let ϑ > −1 and X be one of the spaces Cq , Cq1 , Wpq , or Wpq . The operator Pa : X → X admits a continuation to Pa : Xϑ → Xϑ . The mapping Υ : x 7→ {(Pi − Pi+1 )x} is a topological isomorphism from Xϑ onto lq,ϑ (Z, X♥i ). (c) Let (X, Y ) be one of the pairs (2). Then the class e(X, Y ) consists of operators L ∈ B(X, Y ) induced by matrices { Lij ∈ B(X♥j , Y♥i ) } of the class e. ¡ (d) Let (X, Y ) be one of the pairs (2). If an operator L ∈ e X, Y ) is invertible then L−1 is also an operator of the class e. Proof. (a) We begin with the case X = Lpq . Let as usual (see 3.3.2) Lp loc = Lp loc (R, E), 1 ≤ p ≤ ∞, be the linear space of all measurable functions x : R → E whose restrictions to any segment [a, b], −∞ < a < b + ∞, belong to Lp [a, b]. Clearly, for any a ∈ R the formula ¡

¢ Pa x (t) =

½

0

for t < a,

x(t)

for t ≥ a

defines the projector Pa : Lp loc → Lp loc , and Pa maps Lpq,ϑ into Lpq,ϑ . It is easy to see that for any x ∈ Lp loc the function xi = (Pi − Pi+1 )x is defined by the rule: xi (t) = x(t) for t ∈ (i, i + 1], and xi (t) = 0 for t ∈ / (i, i + 1]. Clearly, xi ∈ Lp♥i . Let lloc = lloc (Z, Lp♥i ) denote the linear space of all families { xi ∈ Lp♥i : i ∈ Z }. Clearly, the correspondence Υ : x 7→ {xi } is an isomorphism from Lp loc onto lloc . We note that X ¡ −1 ¢ Υ {xi } (t) = xi (t). i∈Z

Here, by the convergence of the series we mean the convergence in the sense of distributions (see 2.3.5 for the definition). Equivalently, one may mean that the series converges almost everywhere or in the norm of Lp on bounded segments. It is a simple and straightforward verification that Υ establishes an isomorphism from Lpq,ϑ onto lq,ϑ = lq,ϑ (Z, Lp♥i ) for all ϑ. Since Υ does not depend on q we have that Υ maps Lpq ∩ Lpq,ϑ onto lq ∩ lq,ϑ . Therefore T ∈ B(Lpq ) admits a continuation to Tϑ ∈ B(Lpq,ϑ ) if and only if Te = ΥT Υ−1 ∈ B(lq ) admits a continuation to Teϑ ∈ B(lq,ϑ ). By the definition of e(Lpq ) and 3.4.1 this means that T ∈ e if and only if Te ∈ e. It remains to recall that by definition 3.4.1 Te ∈ e if and only if Te is induced by a matrix of the class e. −1 Next, we consider the case X = Cq−1 . Let Cloc be defined as in 3.3.2. We −1 −1 define the projector Pa : Cloc → Cloc by the same rule as in 2.3.8. Namely, for any −1 f ∈ Cloc we pick a representation f = u+v, ˙ u, v ∈ Cloc , such that u(a) = v(a) = 0;

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then we set Pa f = u˙ 1 + v1 , where ½ u1 (t) =

0

for t ≤ a,

u(t) for t ≥ a;

½ v1 (t) =

0

for t ≤ a,

v(t) for t ≥ a.

The proof of the independence of the definition of Pa f from the choice of the representation f = u˙ + v with u(a) = v(a) = 0 is a word for word repetition of that in 2.3.8. −1 We prove that Υ : x 7→ {(Pi − Pi+1 )x} is an isomorphism from Cloc onto −1 −1 −1 onto lq,ϑ (Z, C♥i ). To do this we discuss lloc (Z, C♥i ) and an isomorphism from Cq,ϑ the definition of Υ more detailed. −1 For any f ∈ Cloc of the form f = u˙ + v, u, v ∈ Cloc , we define the function w : R → E by the rule ¡ ¢¡ ¢2 w(t) = v(i)t − u(i) 4|t − i|2 − 1

for |t − i| ≤ 1/2,

cf. 2.3.8. It is straightforward to verify that w is continuously differentiable, and w(i) = −u(i) and w(i) ˙ = v(i) for all i ∈ Z. We consider u e = u + w and ve = v − w. ˙ ˙ ˙ Clearly, f = u e + ve. We observe that in the representation f = u e + ve the functions u e and ve has the property u e(i) = ve(i) = 0 for all i ∈ Z. We note that if u, v ∈ Cq,ϑ then w ∈ Cq,ϑ and consequently u e, ve ∈ Cq,ϑ ; moreover, there exists K < ∞ independent of u and v such that kwkCq,ϑ , kwk ˙ Cq,ϑ , ke ukCq,ϑ , ke v kCq,ϑ ≤ K(kukCq,ϑ + kvkCq,ϑ ).

(3)

−1 −1 For any f ∈ Cloc we define fi ∈ Cloc , i ∈ Z, as follows. We take a representation f = u˙ + v with u(i) = v(i) = 0 for all i ∈ Z. Then we set fi = u˙ i + vi , where

½ ui (t) =

u(t) for t ∈ [i, i + 1], 0 for t ∈ / [i, i + 1];

½ vi (t) =

v(t) 0

for t ∈ [i, i + 1], for t ∈ / [i, i + 1].

−1 −1 Clearly, fi = (Pi − Pi+1 )f . By 3.1.3 fi ∈ C♥i . Thus Υf = {fi } ∈ lloc (Z, C♥i ). It is evident that X ¡ −1 ¢ Υ {fi } (t) = fi (t), i∈Z

where by the convergence of the series we mean the convergence in the sense of distributions. −1 From the estimate (3) it is evident that Υ acts continuously from Cq,ϑ into −1 −1 −1 lq,ϑ = lq,ϑ (Z, C♥i ) for all ϑ ∈ R. It is easy to see that Υ maps lq,ϑ (Z, C♥i ) into −1 −1 Cq,ϑ and is continuous. Thus Υ : Cq,ϑ → lq,ϑ is an isomorphism. Since Υ does not −1 −1 depend on q we have that Υ maps Cq ∩ Cq,ϑ onto lq ∩ lq,ϑ . Therefore T ∈ B(Cq−1 ) admits a continuation to Tϑ ∈ B(C −1 ) if and only if Te = ΥT Υ−1 ∈ B(lq ) admits q,ϑ

a continuation to Teϑ ∈ B(lq,ϑ ). By the definition of e(Cq−1 ) and 3.4.1 this means

3.5. EXPONENTIAL STABILITY: CONTINUOUS TIME

185

that T ∈ e if and only if Te ∈ e. It remains to recall that Te ∈ e if and only if Te is induced by a matrix of the class e. (b) Let, e.g., X = Cq . We define the operator Pa = Pa,C : Cq,ϑ → Cq,ϑ as completing the diagram P

−1 −1 Cq,ϑ −−−a−→ Cq,ϑ x x   U U

Cq,ϑ

Cq,ϑ

to a commutative diagram. In other words, Pa,C = U −1 Pa,C −1 U , where Pa,C −1 = −1 −1 Pa : Cq,ϑ → Cq,ϑ , cf. 3.1.1. Since the operators U , U −1 , and Pa,C −1 are defined by the common rule for all q and ϑ > −1 the operator Pa,C is also defined by −1 −1 the common rule for all ϑ > −1. Since the image of Pi − Pi+1 : Cq,ϑ → Cq,ϑ does not depend on q and ϑ the image of Pi − Pi+1 : Cq,ϑ → Cq,ϑ does not depend on q and ϑ, too, and therefore coincides with C♥i . We consider the mapping Υ = Υ0 : Cq,ϑ → lloc (Z, C♥i ) defined by the rule Υ : x 7→ {(Pi − Pi+1 )x}. By the above Υ0 completes the diagram Υ=Υ−1

−1 −1 Cq,ϑ −−−−−→ lq,ϑ (Z, C♥i ) x x   U U Υ=Υ

0 Cq,ϑ −−−−→ lq,ϑ (Z, C♥i )

to a commutative diagram. Here U means the isomorphism {xi } 7→ {U xi }. It is easy to see that Υ = Υ0 acts onto lq,ϑ (Z, C♥i ) and is an isomorphism. The cases of other X are handled similarly. (c) Let, e.g., L : Cq → Cq−1 . Let us consider T = LU −1 : Cq−1 → Cq−1 . Since U, U −1 ∈ e the operators L and T belong to e simultaneously. By what has been proved, T ∈ e if and only if its matrix belongs to e. We consider the diagram U

Υ

U −1

−1 Cq −−−−→ Cq−1 −−−−→ lq (Z, C♥i ) −−−−→ lq (Z, C♥i )       e e yL yT yT yL

Cq−1

Υ

−1 Cq−1 −−−−→ lq (Z, C♥i )

−1 lq (Z, C♥i ),

where Te and Le are defined as completing the diagram to a commutative diagram. According to the definition, by the matrices of T and L we mean the matrices of Te e We recall that U has a main diagonal matrix and its main diagonal entries and L. −1 coincide with U : C♥i → C♥i . Thus U and U −1 belong to e. Consequently the matrices of Te and Le belong to e simultaneously. (d) follows directly from 3.4.2 and (c). ¤

186

III. STABILITY

3.5.3. The operators Qa , Ia , and Ra on weighted spaces. We have seen in 3.5.2 that the operator Pa : X → X admits a continuation Pa = Pa,ϑ : Xϑ → Xϑ for ϑ > −1. Here we discuss the continuation of the operator Qa and the continuations of the operators Ia and Ra , whose definitions (see 3.1.4) are based on the definition of Pa . Proposition. Let X be one of the spaces (1). For all a ∈ R and ϑ > −1 the operators Qa : X → X−∞/a , Ia : X → Xa , and Ra : X−∞/a → X admit natural continuations to Qa : Xϑ → (Xϑ )−∞/a , Ia : Xϑ → (Xϑ )a , and Ra : (Xϑ )−∞/a → Xϑ , respectively. Proof. We observe that the definition of the operator Qa = Qa,ϑ : Xϑ → (Xϑ )−∞/a , where (Xϑ )−∞/a = Xϑ /(Xϑ )a , as the natural projection makes sense for all ϑ ∈ R. We discuss whether Qa,ϑ can be considered as a continuation of Qa : X → X. We recall that we consider Xϑ as a subspace of D0 . Let D0τ = D0τ /+∞ denote the subspace of all distributions f ∈ D0 being equal to zero on (−∞, τ ), and let D0−∞/τ = D0 /D0τ denote the quotient space. The main observation is that (Xϑ )τ /+∞ = Xϑ ∩ D0τ /+∞ . Therefore (Xϑ )−∞/τ = Xϑ /(Xϑ )τ can be embedded naturally in D0−∞/τ , cf. the discussion in 3.3.2. Namely, we identify an element x + (Xϑ )τ ∈ (Xϑ )−∞/τ with x + D0τ ∈ D0−∞/τ . Thus the equality of x ˜ ∈ X−∞/τ and z˜ ∈ (Xϑ )−∞/τ makes sense. Moreover, for all ϑ ∈ R the operator Qa,ϑ : Xϑ → (Xϑ )−∞/a can be interpreted as a restriction of the same natural projection Qa : D0 → D0−∞/a . Consequently Qa,ϑ is really a continuation of Qa : X → X. According to 3.1.4 we define Ia : Xϑ → (Xϑ )a to be the operator of projection that differs from Pa : Xϑ → Xϑ only by the range of values, i.e., Ia x = Pa x for all x ∈ Xϑ . By 3.5.2 Ia : Xϑ → (Xϑ )a is a continuation of Ia : X → Xa for ϑ > −1. Finally, the definition of Ra = Ra,ϑ : (Xϑ )−∞/a → Xϑ as the quotient operator induced by 1 − Pa,ϑ by virtue of the diagram (see 3.1.4) J

(Xϑ )a −−−a−→   y 0

Qa

Xϑ −−−−→ (Xϑ )−∞/a −−−−→ 0   1−P R a,ϑ y y a,ϑ

−−−−→ Xϑ

Xϑ

−−−−→ 0

makes sense for ϑ > −1. Moreover, Ra,ϑ can be considered as a continuation of the operator Ra : X−∞/a → X for all such ϑ. Indeed, let x + Xa/+∞ ∈ X−∞/a , x ∈ X, and z + (Xϑ )a/+∞ ∈ (Xϑ )−∞/a , z ∈ Xϑ , coincide as elements of D0−∞/a = D0 /D0a/+∞ , i.e., x − z ∈ D0a/+∞ . Since Pa x ∈ Xa/+∞ and Pa,ϑ z ∈ (Xϑ )a/+∞ this implies that (x − Pa x) − (z − Pa,ϑ z) ∈ D0a/+∞ , too. Without loss of generality we may assume that a ∈ Z. To be specific we assume also ϑ < 0. Then from 3.5.2(a, b) it is easy to see that Im(1 − Pa ) ⊆ Im(1 − Pa,ϑ ). Hence both x − Pa x and z − Pa,ϑ z belong to Im(1 − Pa,ϑ ) ⊆ Xϑ .

3.5. EXPONENTIAL STABILITY: CONTINUOUS TIME

187

On the other hand, we have D0a/+∞ ∩ Xϑ = (Xϑ )a/+∞ = Ker(1 − Pa,ϑ ). Therefore x − Pa x = z − Pa,ϑ z. 1 We note that if X = Lpq , X = Cq , X = Wpq , or X = Cq1 then Ra,ϑ can be defined by the same explicit formulae as in 3.1.4. ¤ In conclusion we observe that the identities Pa Pb = Pb Pa = Pb , (1 − Pa )(1 − Pb ) = (1 − Pb )(1 − Pa ) = 1 − Pa , Pa LPa = LPa , (1 − Pa )L(1 − Pa ) = (1 − Pa )L, Qa Pa = 0, Ra Qa = 1 − Pa , Qa Ra = 1 (see 3.1.1 and 3.1.4) remain valid on the spaces with exponential weights. Here a < b and L is a causal operator. 3.5.4. The independence of invertibility from q for L ∈ e. Let (X, Y ) be one of the pairs (2). To emphasize the value of q we employ the notation (Xq , Yq ). Proposition. Let 1 ≤ q ≤ ∞ or q = 0, and 1 ≤ r ≤ ∞ or r = 0. (a) Any operator L = Lq ∈ e(Xq , Yq ) admits a continuation to an operator L = Lr ∈ e(Xr , Yr ). (Of course, by saying that Lr is a continuation of Lq we mean that Lq x = Lr x for all x ∈ Xq ∩ Xr .) (b) If the operator Lq is invertible then the operator Lr is also invertible. Proof. (a) By 3.5.2(c) the matrix of L induces the operator Lr . (b) The cases X = Y = Lpq and X = Y = Cq−1 are special cases of 3.4.3. If, for instance, (Xq , Yq ) = (Cq , Cq−1 ) it suffices to observe that the operators Lq , and Tq = Lq U −1 : Cq−1 → Cq−1 , and Tr = Lr U −1 : Cq−1 → Cq−1 , and Lr are invertible simultaneously. The cases of the other pairs are handled similarly. ¤ 3.5.5. The definition of uniform solubility. Let (X, Y ) be one of the pairs (2). We consider the equation Lx = f.

(4)

We assume that equation (4) is locally soluble, see 3.3.1 for the definition. Assume Y = Lpq or Y = Cq−1 . We say that equation (4) is uniformly soluble if there exist δ > 0 and K < ∞ such that for any segment [a, b], b − a < δ, and f ∈ Y and ϕ ∈ X the solution x ∈ X−∞/b of the initial value problem ¡ ¢ Lx (t) = f (t),

a < t < b,

x(t) = ϕ(t),

t 0 and K < ∞ such that for any segment ¡ ¢ [a, b], b−a < δ, and f ∈ Y and ϕ ∈ X satisfying the compatibility condition Lϕ (a) = f (a), the solution x ∈ X−∞/b of (5) satisfies the estimate (6). −1 Assume Y = Wpq . We say that equation (4) is uniformly soluble if there exist δ > 0 and K < ∞ such that for any segment [a, b], b − a < δ, and f ∈ Y and ϕ ∈ X the solution x ∈ X−∞/b of the regularized initial value problem (see 2.4.5) ¡ ¢ ¡ ¢ Pa Lx (t) = Pa f (t), t < b, (7) x(t) = ϕ(t), t 0 is fixed. (b) The definition of uniform solubility does not depend on δ. Proof. (a) Clearly, the uniform solubility implies the uniform invertibility. The converse follows from the formula x = (Ja )−∞/b (La/b )−1 Qb Ia (f − LRa ϕ) + Qb Ra ϕ for the solution of the initial value problem, see 3.1.5. (b) follows from (a) and 2.1.4(b), cf. the proof of 3.2.4(a). Indeed, we assume that for a given δ > 0 we have k(La/b )−1 k ≤ K provided b − a ≤ δ. Now we suppose that b − a ≤ 2δ. We choose t ∈ [a, b] so that b − t, t − a ≤ δ. Then by 2.1.4(b) k(La/b )−1 k ≤ kLk · K 2 + 2K. ¤ 3.5.6. Weighted spaces on semi-axes. Let X be one of the spaces (1). For ϑ, τ ∈ R we set ¡ ¢ Ψϑ,τ x (t) = eϑ(t−τ ) x(t). We denote by Xϑ,τ the space of all functions y such that y = Ψϑ,τ x for some x ∈ X, with the norm kykXϑ,τ = kΨ−ϑ,τ ykX . We note that Xϑ,τ differs from the space Xϑ defined in 3.5.1 only by a factor in the norm. Namely, kxkXϑ,τ = CkxkXϑ , where C = eϑτ . Hence any operator L has the same norm as acting from Xϑ into Yϑ and as acting from Xϑ,τ into Yϑ,τ . The detailed notation for Xϑ,τ will appear as, e.g., 1 Lpq,ϑ,τ or Wpq,ϑ,τ (R, E) and so on. We define the T -direction on Xϑ,τ to be the family of the subspaces (Xϑ,τ )t = { x ∈ Xϑ,τ : x(s) = 0 for s < t }, t ∈ R.

3.5. EXPONENTIAL STABILITY: CONTINUOUS TIME

189

Proposition. Let X be one of the spaces (1). Then for all ϑ > ξ, ϑ, ξ ∈ R, there exists M < ∞ such that (a) (Xξ,τ )τ /+∞ ⊆ (Xϑ,τ )τ /+∞ , and for x ∈ (Xξ,τ )τ /+∞ kxk(Xϑ,τ )τ /+∞ ≤ M kxk(Xξ,τ )τ /+∞ . (We recall that (Xξ,τ )τ /+∞ is a subspace of Xξ,τ . Therefore kxk(Xξ,τ )τ /+∞ is merely kxkXξ,τ . And similarly so for kxk(Xϑ,τ )τ /+∞ .) (b) (Xϑ,τ )−∞/τ ⊆ (Xξ,τ )−∞/τ , and for x ∈ (Xϑ,τ )−∞/τ kxk(Xξ,τ )−∞/τ ≤ M kxk(Xϑ,τ )−∞/τ . We recall that (Xϑ,τ )τ /+∞ ⊆ Xϑ,τ ⊆ D0 and (see 3.5.3) (Xϑ )−∞/τ = Xϑ /(Xϑ )τ can be embedded in D0−∞/τ for all ϑ ∈ R. Thus assertion (b) of the proposition makes sense. Proof. We give a proof only for the case ξ > −1. It will be enough for our applications. To obtain a prove, e.g., for ξ > −c, one should change the operator U x = x˙ + x to Uc x = x˙ + (1 + c)x. Without loss of generality we may assume that ξ = 0. The cases X = Lpq and X = Cq are trivial. −1 We prove, e.g., assertion (b) for the case X = Wpq . We consider the diagram J

(Lpq,ϑ,τ )−∞/τ −−−L−→ (Lpq )−∞/τ     U y yU −1 (Wpq,ϑ,τ )−∞/τ

−1 )−∞/τ . (Wpq

Here U are the isomorphisms from 3.5.1 (note that ϑ > ξ = 0), and JL is the identical embedding (we recall that assertion (b) is assumed to be proved for X = Lpq ; note that the inequality in (b) means kJL k ≤ M ). Clearly, the diagram can be completed to a commutative diagram by the operator JW −1 = U JL U −1 . −1 By 3.5.1 U and U −1 are defined by the common rule on Lpq and Lpq,ϑ,τ , and Wpq −1 and Wpq,ϑ,τ , respectively. Consequently U and U −1 are defined by the common −1 rule on the quotient spaces (Lpq )−∞/τ and (Lpq,ϑ,τ )−∞/τ , and (Wpq )−∞/τ and −1 (Wpq,ϑ,τ )−∞/τ , respectively, as well. Therefore since JL is an embedding the operator JW −1 is an embedding, too. ¤ 3.5.7. Operators on semi-axes. Let (X, Y ) be one of the pairs (2), and let τ ∈ R be a fixed point. We recall that Xτ = Xτ /+∞ (Yτ = Yτ /+∞ ) is the subspace of X (Y ) consisting of all functions which are equal to zero on (−∞, τ ). We take an operator L ∈ e(X, Y ) possessing the property LXτ /+∞ ⊆ Yτ /+∞ or, equivalently, (1 − Pτ )LPτ = 0. (8)

190

III. STABILITY

(We recall that the fulfillment of this condition for all τ ∈ R means the causality of L.) In this case we can consider the restriction Lτ /+∞ : Xτ /+∞ → Yτ /+∞ and the quotient operator L−∞/τ : X−∞/τ → Y−∞/τ . We denote by e(Xτ /+∞ , Yτ /+∞ ) (respectively, by e(X−∞/τ , Y−∞/τ )) the set of all operators Lτ /+∞ (respectively, L−∞/τ ) obtained in this way. We recall that Xτ = Im Pτ . We set X τ = Ker Pτ . And we set similarly for Y . We observe that (8) implies that L has the representation L = (1 − Pτ )L(1 − Pτ ) + Pτ L(1 − Pτ ) + Pτ LPτ , i.e., the matrix of L associated with the decompositions X = X τ ⊕ Xτ and Y = Y τ ⊕ Yτ is triangular. In the preceding definitions of e(Xτ /+∞ , Yτ /+∞ ) and e(X−∞/τ , Y−∞/τ ) without loss of generality we may additionally assume that Pτ L(1 − Pτ ) = 0, too, since the operator (1 − Pτ )L(1 − Pτ ) + Pτ LPτ induces the same operators Lτ /+∞ and L−∞/τ as L, and belongs to e provided L ∈ e. We note that in the last case L can be represented equivalently as L = J τ Lτ Qτ + Jτ Lτ Qτ , where Lτ : Xτ → Yτ and Lτ : X τ → Y τ are restrictions, and Jτ : Yτ → Y and τ τ J τ : Y τ → Y are embeddings, and Qτ : X → X ¡ ¢ τ and Q¡ : X → X are projections. ¢ We define classes e (Xϑ )τ /+∞ , (Yϑ )τ /+∞ and e (Xϑ )−∞/τ , (Yϑ )−∞/τ in a similar way. Proposition. Let (X, Y ) be one of the pairs (2), and let τ ∈ R be a fixed point. ¡ ¢ (a) Any operator L−∞/τ ∈¡ e X−∞/τ , Y−∞/τ admits a continuation to an ¢ operator (L−∞/τ )ϑ ∈ e (Xϑ¡)−∞/τ , (Yϑ )−∞/τ¢ for ϑ close enough to zero. Any operator Lτ /+∞¡ ∈ e Xτ /+∞ , Yτ /+∞ ¢admits a continuation to an operator (Lτ /+∞ )ϑ ∈ e (Xϑ )τ /+∞ , (Yϑ )τ /+∞ for ϑ close enough to zero. (b) If an operator L−∞/τ ∈ e(X−∞/τ , Y−∞/τ ) is invertible then (L−∞/τ )−1 is also an operator of the class e. If an operator Lτ /+∞ ∈ e(Xτ /+∞ , Yτ /+∞ ) is invertible then (Lτ /+∞ )−1 is also an operator of the class e. Let 1 ≤ q ≤ ∞ or q = 0, and 1 ≤ r ≤ ∞ or r = 0. ¡ ¢ (c) Any operator (Lq )−∞/τ ∈ e (Xq¡)−∞/τ , (Yq )−∞/τ admits a continua¢ tion to an operator (Lr )−∞/τ ∈ e (Xr )−∞/τ , (Yr )−∞/τ . If the operator (Lq )−∞/τ is invertible then the ¡operator (Lr )−∞/τ is¢also invertible. Any operator (Lq )τ /+∞ ∈ e (X¡q )τ /+∞ , (Yq )τ /+∞ admits a continua¢ tion to an operator (Lr )τ /+∞ ∈ e (Xr )τ /+∞ , (Yr )τ /+∞ . If the operator (Lq )τ /+∞ is invertible then the operator (Lr )τ /+∞ is also invertible.

3.5. EXPONENTIAL STABILITY: CONTINUOUS TIME

191

Proof. Let us discuss, e.g., the case of −∞/τ . Assume that L−∞/τ is induced by the operator L. Without loss of generality we may assume that L has one of the following special forms. If X = Y we may assume that L = (1 − Pτ )L(1 − Pτ ) + Pτ . And if X 6= Y we may assume that L = (1 − Pτ )L(1 − Pτ ) + Pτ U Pτ . Indeed, since all the operators in the right hand sides of these formulae belong to e, it follows that the improved operator belongs to e as well. It remains to observe that it induces the same operator L−∞/τ . (a) By the definition of e(X, Y ) the operator L admits a continuation to an operator Lϑ : Xϑ → Yϑ . Since L, Pτ ∈ e and the continuations of operators of the ¡ class e are induced by their matrices, we have¢Lϑ = (1 − Pτ )Lϑ (1 − Pτ ) + Pτ respectively, Lϑ = (1 − Pτ )Lϑ (1 − Pτ ) + Pτ U Pτ . We show that (Lϑ )−∞/τ is a continuation of L−∞/τ . To be specific we assume that ϑ < 0 (but ϑ > −1). Let x ∈ X and z ∈ Xϑ , and let x e and ze be their projections into X−∞/τ and e= (Xϑ )−∞/τ , respectively. We must verify that x − z ∈ D0τ /+∞ implies L−∞/τ x (Lϑ )−∞/τ ze. Indeed, replacing x by (1 − Pτ )x and z by (1 − Pτ )z, without loss of generality we may assume that x ∈ Im(1 − Pτ ) ⊆ X and z ∈ Im(1 − Pτ ) ⊆ Xϑ . Since ϑ < 0, for all X under consideration the image of 1 − Pτ : Xϑ → Xϑ is wider than the image of 1 − Pτ : X → X, cf. 3.5.6(b). Therefore x − z belongs to the image of 1 − Pτ : Xϑ → Xϑ . On the other hand, x − z ∈ D0τ /+∞ and D0τ /+∞ ∩ Xϑ = (Xϑ )τ /+∞ = Ker(1 − Pτ ). Therefore x − z = 0, i.e., x = z. Consequently Lx = Lϑ z, and L−∞/τ x e = (Lϑ )−∞/τ ze. (b) To be specific we consider the case X 6= Y . We represent L as L = J τ Lτ Qτ + Jτ Uτ Qτ . We observe that Uτ is invertible since U is causally invertible, and Lτ is invertible since it is similar to L−∞/τ . Clearly, the operator N = J τ (Lτ )−1 Qτ + Jτ (Uτ )−1 Qτ is the inverse of L. Since L ∈ e, by 3.5.2(d) N ∈ e. It is easy to see that (L−∞/τ )−1 = N−∞/τ . Thus (L−∞/τ )−1 is an operator of the class e. (c) To be specific we consider the case X 6= Y . We represent Lq as Lq = (1 − Pτ )Lq (1 − Pτ ) + Pτ U Pτ . We recall from 3.5.2 that all operators of the class e are induced by matrices of the class e. By 1.6.4(a) and 1.6.9(c) the correspondence between matrices of the class e and operators of the class e is an isomorphism. Thus since L, Pτ , U ∈ e we have a similar identity for the matrices, which implies the representation Lr = (1 − Pτ )Lr (1 − Pτ ) + Pτ U Pτ .

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The checking that (Lr )τ /+∞ is a continuation of (Lq )τ /+∞ is similar to the end of the proof of (a). As we have seen in the prove of (b), since Lq has a main diagonal matrix the invertibility of (Lq )−∞/τ implies the invertibility of Lq , which in turn by 3.5.4 implies the invertibility of Lr . But Lr also has a main diagonal matrix. Therefore its main diagonal entries are invertible, which shows that (Lr )−∞/τ is invertible. ¤ 3.5.8. The definition of exponential stability. Let (X, Y ) be one of the pairs (2). Let L ∈ e+ (X, Y ), and let equation (4) be uniformly soluble. Let τ ∈ R be a fixed point, and let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Assume Y = Lpq or Y = Cq−1 . We say that equation (4) is exponentially stable on H if there exist ν > 0 and N < ∞ such that for any [a, b] ⊆ H, −∞ < a < b < +∞, and ϕ ∈ X the solution x ∈ X−∞/b of the homogeneous initial value problem ¡ ¢ Lx (t) = 0, a < t < b, (9) x(t) = ϕ(t), t < a, satisfies the estimate kxk(X−ν,a )−∞/b ≤ N kϕk.

(10)

Assume Y = Cq . We say that equation (4) is exponentially stable on H if there exist ν > 0 and N < ∞ such that for¡ any ¢ segment [a, b] ⊆ H and ϕ ∈ X−∞/a satisfying the compatibility condition Lϕ (a) = 0 the solution x ∈ X−∞/b of (9) satisfies the estimate (10). −1 . We say that equation (4) is exponentially stable on H if Assume Y = Wpq there exist ν > 0 and N < ∞ such that for any segment [a, b] ⊆ H and ϕ ∈ X−∞/a the solution x ∈ X−∞/b of the regularized initial value problem ¡ ¢ Pa Lx (t) = 0, t < b, (11) x(t) = ϕ(t), t < a, satisfies the estimate (10). Taking in (10) the infimum over all ϕ with the same projection into X−∞/b we can rewrite (10) equivalently as kxk(X−ν,a )−∞/b ≤ N kϕkX−∞/a . Sometimes we specify that we are considering stability in the pair (X, Y ) or (Xq , Yq ). We note that exponential stability in the pairs (X0 , Y0 ) and (X∞ , Y∞ ) means the same provided that L acts both in (X0 , Y0 ) and (X∞ , Y∞ ). 3.5.9. Exponential stability and causal invertibility Theorem. Let (X, Y ) be one of the pairs (2). Let L ∈ e+ (X, Y ), and let equation (4) be uniformly soluble. Let τ ∈ R and H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Then the following assumptions are equivalent. (a) Equation (4) is exponentially stable on H. (b) Equation (4) is input–output stable on H. (c) The operator L : X → Y is causally invertible on H.

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193

Remark. In 3.6.4 we shall prove a more general statement. Since that proof is rather more complicated we give here an independent proof of this theorem. Proof. The proof is similar to that of 3.4.7. (It can also be performed as a reduction to 3.4.7, but this transformation is too long.) (b) ⇒ (c) follows from 3.3.6, 3.3.8, and 3.5.4 (or 3.5.7(c)), cf. 3.4.7. (c) ⇒ (a). We consider the initial value problem (9) or (11). We make use of the formula x = −(Ja )−∞/b (La/b )−1 Qb Ia LRa ϕ + Qb Ra ϕ from 3.1.5 for the solution of (9) and (11). According to 3.5.2(d) (or 3.5.7(b)) and the definition of e (or 3.5.7(a)), for ν > 0 small enough the operator L−1 admits a continuation to a bounded operator from Y−ν,a to X−ν,a . We make use of 3.5.3 and, for ν < 1, consider the continuations Ra : (X−ν,a )−∞/a → X−ν,a , Ia : Y−ν,a → (Y−ν,a )a , and Qb : Y−ν,a → (Y−ν,a )−∞/b . Then we note that the embedding Ja : Xa → X trivially admits a continuation to Ja : (X−ν,a )a → X−ν,a for all ν. Clearly, the norms of the operators Ra , Ia , and Ja do not depend on a, and kQb : Y−ν,a → (Y−ν,a )−∞/b k ≤ 1 since Qb : Y−ν,a → (Y−ν,a )−∞/b is the natural projection. Finally, we observe that kϕk(X−ν,a )−∞/a ≤ M kϕkX by 3.5.6(b). Thus from the formula for solution we see that x ∈ (X−ν,a )−∞/b and the estimate (10) holds. −1 . Let us consider the initial (a) ⇒ (b). First we discuss the case Y 6= Wpq value problem (5). We prove that its solution x satisfies the estimate form the definition of input–output stability, see 3.3.4. Without loss of generality we may assume that a and b are integers, say a = n and b = m. Let αk : R → [0, 1], k ∈ Z, be infinitely many times differentiable functions (actually it is sufficient that αk be only continuously differentiable) such that P k k / [k, k + 2], and αk+1 (t) = αk (t − 1). k∈Z α (t) = 1 for all t ∈ R, α (t) = 0 for t ∈ Such a collection {αk } is called a partition of unity, cf. 1.5.2. We redefine αn P+∞ Pn k n to be k=−∞ αk , and αm−1 to be k=m−1 α . So α (t) = 1 for t ≤ n, and αm−1 (t) = 1 for t ≥ m. Pm−1 We represent f as the sum f = k=n f k , where f k = αk f . We shall use only the following properties of f k : f k (t) = 0 for t ∈ / [k, k + 2], k = n + 1, . . . , m − 2; fn (t) = 0 for t > n + 2; fm−1 (t) = 0 for t < m − 1; and kf k kY ≤ M kf kY for some M independent of f , n, m, and m − n. (Note that for concrete Y ’s functions f k with these properties can be constructed in simpler ways.) Let xn be the solution of the initial value problem ¡ n¢ Lx (t) = (f n )(t), n < t < m, (xn )(t) = ϕ(t),

t < n.

And let xk , k = n + 1, . . . , m − 1, be the solutions of the initial value problems ¡ k¢ Lx (t) = (f k )(t), k < t < m, (xk )(t) = 0,

t < k.

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III. STABILITY

Pm−1 We observe that the solution x of (5) is equal to the sum k=n xk . Since the equation is uniformly soluble, for all k we have kxk kX−∞/k+2 ≤ K(kϕk + kf k k) ≤ K(kϕk + M kf k). Next, we observe that because of f k (t) = 0 for t > k + 2, k = n, . . . , m − 2, one can interpret xk as the solution of the homogeneous initial value problem with the initial point k + 2. Therefore by the definition of exponential stability xk satisfies the estimate (for these k) kxk k(X−ν,k+2 )−∞/m ≤ N kxk kX−∞/k+2 .

¢ ¡ (We recall that the last estimate means that the function t 7→ Ψν,k+2 xk (t) = eν(t−k−2) xk (t) is bounded in the norm of X−∞/m by the constant N kxk kX−∞/k+2 .) 1 . We recall that X is assumed to be one of the spaces Cq , Cq1 , Lpq , and Wpq k Summing up the above estimates and taking into account that x (t) = 0 on (−∞, k) for k = n + 1, . . . , m − 1, we obtain the estimate for x from the definition of input–output stability. −1 Now we consider the case Y = Wpq . We prove that the solution x of the initial value problem (7) satisfies the estimate form the definition of input–output stability, see 3.3.4. Without loss of generality we may assume that a and b are integers, say a = n and b = m. −1 −1 Let Pk : Wpq → Wpq be defined as usual (see 2.3.11 and 3.1.1). We set f k = Pk (1 − Pk+1 )f = (Pk − Pk+1 )f , k = n, . . . , m − 1. We shall use the following two properties of f k : Pk+1 f k = 0 and kf k kY ≤ M kf kY for some M . Let xn be the solution of the initial value problem ¡ ¢ Pn Lxn (t) = (Pn f n )(t) = f n (t), t < m, (xn )(t) = ϕ(t),

t < n.

And let xk , k = n + 1, . . . , m − 1, be the solutions of the initial value problems ¡ ¢ Pk Lxk (t) = (Pk f k )(t) = f k (t), t < m, (xk )(t) = 0, t < k. Pm−1 k Pm−1 Since Pn (1 − Pm )f = k=n f the solution x of (7) is equal to the sum k=n xk . Since the equation is uniformly soluble kxk kX−∞/k+1 ≤ K(kϕk + kf k k) ≤ K(kϕk + M kf k). Furthermore, we observe that since Pk+1 f k = 0, by the definition of exponential stability, xk satisfies the estimate kxk k(X−ν,k+1 )−∞/m ≤ N kxk kX−∞/k+1 . The continuation of the proof in now evident. ¤ Remark. An analogue of the remark in 3.4.7 holds.

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195

3.5.10. The independence of stability from q for L ∈ e Corollary. Let 1 ≤ q ≤ ∞ or q = 0, and 1 ≤ r ≤ ∞ or r = 0. Let τ ∈ R, and let H be (−∞, +∞), (−∞, τ ), or [τ, +∞). Let L ∈ e+ (Xq , Yq ), and let equation (4) be uniformly soluble, and exponentially or input–output stable on H in the pair (Xq , Yq ). Then equation (4) is uniformly soluble, and exponentially and input–output stable on H in the pair (Xr , Yr ). Proof. The proof is similar to that of 3.4.8. ¤ 3.5.11. Small perturbations preserve exponential stability Corollary. Let (X, Y ) be one of the pairs (2), and let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Let L, N ∈ e+ (X, Y ). If equation (4) is uniformly soluble and exponentially stable on H, and kL−1 k · kN k < 1 then the perturbed equation (L − N )x = f is also uniformly soluble and exponentially stable on H. Proof. The proof follows from the theorem and 3.3.9, see also 3.5.5 and 2.1.3. ¤ Remark. (a) Clearly, a similar fact holds for equations on Z. (b) See the example in 3.3.9. 3.5.12. Initial value problem on semi-axis. In this subsection we discuss the definition of exponential stability in terms of the initial value problem on [a, +∞). Let (X, Y ) be one of the pairs (2), and let L ∈ e+ (X, Y ). We consider the equation (4). Let ϕ ∈ X. We consider the initial value problem (see 3.3.10) ¡

¢ Lx (t) = 0, x(t) = ϕ(t),

t > a, t < a,

−1 ) (126=Wpq

−1 or, respectively, (Y = Wpq )

¡ ¢ Pa Lx (t) = 0, x(t) = ϕ(t),

t ∈ R, t < a.

−1 ) (12=Wpq

If the operator L acts from X−ν,a into Y−ν,a one can define the solution of (12) for ϕ ∈ (X−ν,a )−∞/a in a way similar to that described in 3.3.10.

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Proposition. Let (X, Y ) be one of the pairs (2), and let L ∈ e+ (X, Y ). Let τ ∈ R, and let H be either [τ, +∞) or (−∞, +∞). Let equation (4) be uniformly soluble. Then the following assumptions are equivalent: (a) Equation (4) is exponentially stable on H. (b) There exist ν > 0 and N < ∞ such that ¡ ¢for all a ∈ H and ϕ ∈ X−∞/a (satisfying the compatibility condition Lϕ (a) = 0 if Y = C) the solution x of the initial value problem (12) lies in X−ν,a and satisfies the estimate kxkX−ν,a ≤ N kϕkX−∞/a . (c) There exist ν > 0 and N < ∞ such that¡for ¢all a ∈ H and ϕ ∈ (X−ν,a )−∞/a (satisfying the compatibility condition Lϕ (a) = 0 if Y = C) the solution x of the initial value problem (12) lies in X−ν,a and satisfies the estimate kxkX−ν,a ≤ N kϕk(X−ν,a )−∞/a . Proof. (a) ⇒ (c). Let equation (4) be exponentially stable on H. Then by 3.5.9 the operator L : X → Y is causally invertible on H. A word for word repetition of the argument of 3.1.5 shows that the function x = −Ja (La/+∞ )−1 Ia LRa ϕ + Ra ϕ is the solution of the initial value problem (12) for ϕ ∈ (X−ν,a )−∞/a with ν > 0 small enough, cf. the part (c) ⇒ (a) of the proof of 3.5.9. Now the estimate is evident. (b) ⇒ (a) is evident. (c) ⇒ (b) follows from 3.5.6(b). ¤

3.6. Exponential dichotomy: continuous time We have seen in 3.3.9 and 3.5.11 that since causal invertibility is equivalent to stability, and causal invertibility is preserved under small perturbations, both input–output stability and exponential stability are preserved under small perturbations. A similar kind of instability is discussed in this section. We call it rough instability. Our main result — theorem 3.6.4 — asserts that for equations of the class e the rough instability is equivalent to exponential dichotomy of solutions of the homogeneous equation. 3.6.1. The definitions of instability. Let (X, Y ) be one of the pairs (Cq , Cq ),

(Lpq , Lpq ),

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ). (1)

In order to stress the value of q we use the notation (Xq , Yq ). Let L ∈ B+ (X, Y ). We consider the equation Lx = f.

(2)

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Let τ ∈ R be a fixed point, and let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). We say that equation (2) is input–output unstable on H if it is not input–output stable on H. We say that equation (2) with L ∈ e is exponentially unstable on H if it is not exponentially stable on H. In particular, we call equation (2) unstable if is not locally (uniformly) soluble on H, see 3.3.4 and 3.5.8. We say that equation (2) is roughly unstable on H if the operator L is invertible on H, but L−1 is not causal. Remark. To be precise, there was no definition of solubility on semi-axes. We leave its formulation to the reader. Proposition. Rough instability implies input–output instability. If L ∈ e then rough instability implies exponential instability. Proof. We begin with the second assertion. If (2) were exponentially stable then by 3.5.9 L−1 would be causal. This contradicts the rough instability. In a similar way from 3.3.6 and 3.3.8 it follows the first assertion except the special cases when q = ∞, and H = (−∞, τ ] and H = (−∞, +∞). First, let us consider the case H = (−∞, τ ]. We assume the contrary: let equation (2) be input–output stable. Then by 3.3.6(b) the operator L0 : (X0 )−∞/τ → (Y0 )−∞/τ is causally invertible, i.e., possesses the property ¡ ¢ ¡ ¢ (L0 )−1 (X0 )−∞/τ a ⊆ (Y0 )−∞/τ a for all a ≤ τ . Clearly, (L0 )−1 is a restriction of ¡ ¢ the¡ inverse of¢ L∞ : (X∞ )τ /+∞ → (Y∞ )τ /+∞ . −1 Consequently (L∞ ) (X0 )−∞/τ a ⊆ (Y0 )−∞/τ a for all a ≤ τ , too. This implies −1 that (L∞ ) is causal. Thus we obtain a contradiction with rough instability. Now we turn to the case H = (−∞, +∞). We assume the contrary: let equation (2) be input–output stable. Then by 3.3.5(a) it is stable on [τ, +∞), and by 3.3.6(a) L is causally invertible on [τ, +∞). But then by 2.1.4 L is invertible on (−∞, τ ]. By what has been proved above, this and stability on (−∞, τ ] imply that L is causally invertible on (−∞, τ ]. To obtain a contradiction, it remains to apply 2.1.6. ¤ 3.6.2. Small perturbations preserve rough instability. The following statement explains the term ‘rough instability’. Proposition. Let (X, Y ) be one of the pairs (1). Let τ ∈ R, and let H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Let L, N ∈ B+ (X, Y ). If equation (2) is roughly unstable, and kL−1 k · kN k < 1 then the perturbed equation (L − N )x = f is also roughly unstable. Proof. We consider the family of operators Kt = L − tN , t ∈ [0, 1]. Clearly, Kt depends on t continuously. By 1.3.6(c) all the operators Kt are invertible. Moreover, by 2.2.12, if (Kt )−1 were causal for at least one value of t then all (Kt )−1 would be causal. Since (K0 )−1 = L−1 is not causal, none of (Kt )−1 is causal. In particular, (K1 )−1 = (L − N )−1 exists, but is not causal. ¤

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3.6.3. The definition of exponential dichotomy. Let (X, Y ) be one of the pairs (1), and let L ∈ e+ (X, Y ). We recall that by the definition of the class e (see 3.5.2) the operator L acts from X−λ into Y−λ for λ close enough to zero. We recall also that the operator L acts from X−λ,τ into Y−λ,τ (see 3.5.6) and kL : X−λ,τ → Y−λ,τ k = kL : X−λ → Y−λ k. Since the operator U −1 (see 3.5.1) and the projectors Pa (see 3.5.2) act from X−λ into Y−λ only for λ < 1, below we assume that the parameter λ satisfies this condition. We consider equation (2). We assume that equation (2) is uniformly soluble in (X, Y ), see 3.5.5 for the definition. Clearly, the initial value problem in the pairs (X−λ , Y−λ ) and (X−λ,τ , Y−λ,τ ) can be considered as a simple modification of the initial value problem in the pair (X, Y ), and, in particular, equation (2) is uniformly soluble in the natural sense in (X−λ , Y−λ ) for all λ. We begin our discussion of the definition of dichotomy with the most diffi−1 cult case: assume Y = Wpq . We say that equation (2) possesses dichotomy in (X−ν , Y−ν ), ν ≥ 0, if the following four conditions are fulfilled. (id ) For any a ∈ R the space (X−ν,a )−∞/a is decomposed by the direct sum + − (X−ν,a )−∞/a = (X−ν,a )−∞/a ⊕ (X−ν,a )−∞/a .

We denote briefly the corresponding projections of ϕ ∈ (X−ν,a )−∞/a into the + − subspaces (X−ν,a )−∞/a and (X−ν,a )−∞/a by ϕ+ and ϕ− , respectively. And we denote by x+ and x− the solutions of the initial value problems (see 3.5.12) ¡

¢ Pa Lx± (t) = 0, ±

±

x (t) = ϕ (t),

t ∈ R,

(3±)

t < a.

(4±)

(iid ) For all b > a the natural projection Qb x+ of x+ into (X−ν,b )−∞/b belongs + to (X−ν,b )−∞/b . (iiid ) The function x− satisfies the equation ¡ −¢ Lx (t) = 0,

t ∈ R,

(5)

and for all b ∈ R the natural projection Qb x− of x− into (X−ν,b )−∞/b lies in − (X−ν,b )−∞/b . (ivd ) There exists N < ∞ such that for all ϕ ∈ (X−ν,a )−∞/a one has x+ ∈ X−ν,a and ϕ− ∈ (Xν,a )−∞/a (note ν, not −ν), and the following estimates hold: kx+ kX−ν,a ≤ N kϕk(X−ν,a )−∞/a , kϕ− k(Xν,a )−∞/a ≤ N kϕk(X−ν,a )−∞/a . Thus x+ exponentially decreases at +∞ and ϕ− exponentially decreases at −∞. We note that by 3.5.6(b) we have that kϕ− k(X−ν,a )−∞/a ≤ M kϕ− k(Xν,a )−∞/a . Therefore the second estimate in (ivd ) implies that the projectors Π− a , a ∈ R, on

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199

− + (X−ν,a )−∞/a onto (X−ν,a )−∞/a parallel to (X−ν,a )−∞/a are uniformly bounded. ± In particular, the subspaces (X−ν,a )−∞/a are closed in (X−ν,a )−∞/a . We say that equation (2) possesses exponential dichotomy in (X−ν , Y−ν ) if, in addition, the following condition is fulfilled. (ve ) There exist ε > 0 and N < ∞ such that for all ϕ ∈ (X−ν,a )−∞/a

kx+ kX−(ν+ε),a ≤ N kϕk(X−ν,a )−∞/a , kϕ− k(Xν+ε,a )−∞/a ≤ N kϕk(X−ν,a )−∞/a . In particular, in the most important case when ν = 0, these inequalities can be rewritten as kx+ kX−ε,a ≤ N kϕkX−∞/a , kϕ− k(Xε,a )−∞/a ≤ N kϕkX−∞/a . If Y = Lpq or Y = Cq−1 the definition of dichotomy is the same. We note only that in these cases equation (3±) can be rewritten in the equivalent and more customary form ¡ ±¢ Lx (t) = 0 t > a. ˘ −ν,a )−∞/a the subspace of all Finally, assume that Y = Cq . We denote by (X ¡ ¢ ϕ ∈ (X−ν,a )−∞/a satisfying the compatibility condition Lϕ (a) = 0. We say that equation (2) possesses dichotomy in (X−ν , Y−ν ), ν ≥ 0, if the following four conditions are fulfilled. ˘ −ν,a )−∞/a is decomposed by the direct sum (id ) For any a ∈ R the space (X ˘ −ν,a )−∞/a = (X + )−∞/a ⊕ (X − )−∞/a . (X −ν,a −ν,a ˘ −ν,a )−∞/a into (X + )−∞/a and We shall denote briefly the projections of ϕ ∈ (X −ν,a − + − (X−ν,a )−∞/a by ϕ and ϕ , respectively. And we shall denote by x+ and x− the solutions of the initial value problems ¡ ±¢ Lx (t) = 0, ±

±

x (t) = ϕ (t),

t > a, t < a.

We stress that since ϕ± satisfies the compatibility condition these initial value problems can be rewritten equivalently as (3±), (4±). (iid ) For all b > a the natural projection Qb x+ of x+ into (X−ν,b )−∞/b belongs + to (X−ν,b )−∞/b . (iiid ) The function x− satisfies the equation ¡ −¢ Lx (t) = 0,

t ∈ R,

− and for all b ∈ R the natural projection of x− into (X−ν,b )−∞/b lies in (X−ν,b )−∞/b .

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III. STABILITY

(ivd ) There exists N < ∞ such that for all ϕ ∈ (X−ν,a )−∞/a one has x+ ∈ X−ν,a and ϕ− ∈ (Xν,a )−∞/a , and the following estimates hold: kx+ kX−ν,a ≤ N kϕk(X−ν,a )−∞/a , kϕ− k(Xν,a )−∞/a ≤ N kϕk(X−ν,a )−∞/a . − + )−∞/a are closed. )−∞/a and (X−ν,a It is easy to see that the subspaces (X−ν,a We say that equation (2) possesses exponential dichotomy in (X−ν , Y−ν ) if, in addition, the following condition is fulfilled. ˘ −ν,a )−∞/a (ve ) There exist ε > 0 and N < ∞ such that for all ϕ ∈ (X

kx+ kX−(ν+ε),a ≤ N kϕk(X−ν,a )−∞/a , kϕ− k(Xν+ε,a )−∞/a ≤ N kϕk(X−ν,a )−∞/a . In particular, if ν = 0 these inequalities can be rewritten as kx+ kX−ε,a ≤ N kϕkX−∞/a , kϕ− k(Xε,a )−∞/a ≤ N kϕkX−∞/a . Proposition. Condition (ve ) implies condition (ivd ). Proof. Assume Y 6= Cq . From 3.5.6(b) and the second estimate in (ve ) we have that ϕ− ∈ (Xν,a )−∞/a , and that the second estimate from (ivd ) holds: kϕ− k(Xν,a )−∞/a ≤ M kϕ− k(Xν+ε,a )−∞/a ≤ M N kϕk(X−ν,a )−∞/a . For the same reasons it follows that ϕ− ∈ (X−ν,a )−∞/a and kϕ− k(X−ν,a )−∞/a ≤ M kϕ− k(Xν+ε,a )−∞/a ≤ M N kϕk(X−ν,a )−∞/a . From the latter estimate we obtain that ϕ+ = ϕ−ϕ− also belongs to (X−ν,a )−∞/a and kϕ+ k(X−ν,a )−∞/a ≤ (1 + M N ) kϕk(X−ν,a )−∞/a . Next, from the definition of x+ and the first estimate in (ve ) we have that ϕ+ = ϕ − ϕ− belongs to (X−(ν+ε),a )−∞/a and kϕ+ k(X−(ν+ε),a )−∞/a ≤ kx+ kX−(ν+ε),a ≤ N kϕk(X−ν,a )−∞/a . We recall from 3.5.3 that Ra admits continuations to Ra : (X−ν,a )−∞/a → X−ν,a and Ra : (X−(ν+ε),a )−∞/a → X−(ν+ε),a provided ν + ε < 1. Let us consider

3.6. EXPONENTIAL DICHOTOMY: CONTINUOUS TIME

201

z + = Ra ϕ+ . Since ϕ+ ∈ (X−(ν+ε),a )−∞/a ⊆ (X−ν,a )−∞/a it follows that z + belongs both to X−(ν+ε),a and X−ν,a , and kz + kX−ν,a ≤ Kν kϕk(X−ν,a )−∞/a , kz + kX−(ν+ε),a ≤ Kν+ε kϕk(X−ν,a )−∞/a , where the constants are Kν = (1 + M N ) · kRa : (X−ν,a )−∞/a → X−ν,a k and Kν+ε = N · kRa : (X−(ν+ε),a )−∞/a → X−(ν+ε),a k. Since z + , x+ ∈ X−(ν+ε),a , and Qa z + = ϕ+ and Qa x+ = ϕ+ we have x+ − z + ∈ (X−(ν+ε),a )a/+∞ . Consequently by 3.5.6(a) x+ − z + ∈ (X−ν,a )a/+∞ ⊆ X−ν,a and kx+ − z + kX−ν,a = kx+ − z + k(X−ν,a )a/+∞ ≤ M kx+ − z + k(X−(ν+ε),a )a/+∞ = M kx+ − z + kX−(ν+ε),a ≤ M kx+ kX−(ν+ε),a + M kz + kX−(ν+ε),a ≤ M N kϕk(X−ν,a )−∞/a + M Kν+ε kϕk(X−ν,a )−∞/a = M (N + Kν+ε M ) kϕk(X−ν,a )−∞/a . Finally, from the representation x+ = z + + (x+ − z + ) we have x+ ∈ X−ν,a and kx+ kX−ν,a ≤ Kν kϕk(X−ν,a )−∞/a + M (N + Kν+ε M ) kϕk(X−ν,a )−∞/a , which yields the first estimate from (ivd ). The case of Y = Cq is handled similarly. ¤ Remark. In a similar way one can prove that (exponential) dichotomy in (X−ν , Y−ν ), ν > 0, implies (exponential) dichotomy in (X−µ , Y−µ ) for µ < ν. 3.6.4. Dichotomy and invertibility Theorem. Let (X, Y ) be one of the pairs (1), let L ∈ e+ (X, Y ), and let equation (2) be uniformly soluble. Then the following assumptions are equivalent: (a) The operator L : X → Y is invertible. (b) Equation (2) possesses dichotomy in (X−ν , Y−ν ) for ν > 0 small enough. (c) Equation (2) possesses exponential dichotomy in (X, Y ). −1 Proof. We give a complete proof for the case Y = Wpq . For Y = Lpq or −1 ˘ Y = Cq , the proof is similar. In the case Y = Cq one should change X to X when necessary and take care on the fulfillment of the compatibility condition. One additional remark will be given at the end of this proof. (b) ⇒ (c). Let us assume that (id )–(ivd ) in X−ν are satisfied for some ν > 0. + − By (id ) we have the decomposition (X−ν,a )−∞/a = (X−ν,a )−∞/a ⊕ (X−ν,a )−∞/a . − By the second estimate in (ivd ), (X−ν,a )−∞/a ⊆ (Xν,a )−∞/a . On the other hand, by 3.5.6(b) (Xν,a )−∞/a ⊆ X−∞/a . Hence − (X−ν,a )−∞/a ⊆ X−∞/a .

202

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Therefore the image of the projector Π− a : (X−ν,a )−∞/a → (X−ν,a )−∞/a onto − + (X−ν,a )−∞/a parallel to (X−ν,a )−∞/a is contained in X−∞,a . Thus Π− a maps the subspace (see 3.5.6(b)) X−∞/a ⊆ (X−ν,a )−∞/a into itself. Consequently the + − − complementary projector Π+ a = 1−Πa onto (X−ν,a )−∞/a parallel to (X−ν,a )−∞/a possesses the same property. Hence the projectors Π± a induce the decomposition + − X−∞/a = X−∞/a ⊕ X−∞/a ,

and (id ) in X holds. + and consider the solution x+ of Let us prove (iid ) in X. We take ϕ+ ∈ X−∞/a (3+), (4+). Since X−∞/b ⊆ (X−ν,b )−∞/b the natural projection Qb x+ of x+ into + X−∞/b belongs to (X−ν,b )−∞/b . As we have seen, Π+ b maps X−∞/b into itself. + + + Hence since Qb x belongs to both X−∞/b and (X−ν,b )−∞/b it belongs to X−∞/b . (iiid ) is proved similarly. Taking ε = ν and using 3.5.6(b) once more, we obtain (ve ) in X: kx+ kX−ε,a ≤ N kϕk(X−ν,a )−∞/a ≤ N kϕkX−∞/a , kϕ− k(Xε,a )−∞/a ≤ N kϕk(X−ν,a )−∞/a ≤ N kϕkX−∞/a . (a) ⇒ (b). We break the reasoning into four steps. (i) Let us take ν ∈ (0, 1) such that L acts from X−ν into Y−ν , and L−1 acts from Y−ν into X−ν . For all a ∈ R we define the operators Π± a : (X−ν,a )−∞/a → (X−ν,a )−∞/a as follows: −1 (1 − Pa )LRa , Π+ a = Qa L −1 Π− Pa LRa , a = Qa L

(6)

− − and for ϕ ∈ (X−ν,a )−∞/a we set ϕ+ = Π+ a ϕ and ϕ = Πa ϕ. − To obtain decomposition (id ), it suffices to show that the operators Π+ a and Πa − are complementary projectors. Since Qa Ra = 1 (see 3.5.3) we have Π+ a + Πa = 1. + 2 + Thus it suffices to show that, e.g., (Πa ) = Πa . Indeed, 2 −1 (Π+ (1 − Pa )LRa · Qa L−1 (1 − Pa )LRa a ) = Qa L

= Qa L−1 (1 − Pa )L(1 − Pa )L−1 (1 − Pa )LRa = Qa L−1 (1 − Pa )LL−1 (1 − Pa )LRa = Qa L−1 (1 − Pa )LRa = Π+ a. Here we use the identities Ra Qa = 1 − Pa , and (1 − Pa )L(1 − Pa ) = (1 − Pa )L, and (1 − Pb )(1 − Pa ) = 1 − Pa , see the end of the subsection 3.5.3, and 3.1.1 and 3.1.4. (ii) To check (iid ) first we observe that the solution of (3+), (4+) is the function x+ = L−1 (1 − Pa )LRa ϕ,

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203

cf. (6). Indeed, we have Pa Lx+ = Pa LL−1 (1 − Pa )LRa ϕ = Pa (1 − Pa )LRa ϕ = 0, Qa x+ = Qa L−1 (1 − Pa )LRa ϕ = Π+ aϕ = ϕ+ . + Next, we show that Qb x+ ∈ (X−ν,b )−∞/b for all b > a. Clearly, it suffices to + + + verify that Πb (Qb x ) = Qb x . Indeed, we have + −1 Π+ (1 − Pb )LRb · Qb L−1 (1 − Pa )LRa ϕ, b Qb x = Qb L

= Qb L−1 (1 − Pb )L(1 − Pb )L−1 (1 − Pa )LRa ϕ, = Qb L−1 (1 − Pb )LL−1 (1 − Pa )LRa ϕ, = Qb L−1 (1 − Pb )(1 − Pa )LRa ϕ, = Qb L−1 (1 − Pa )LRa ϕ = Qb x+ . Here we use the identities Rb Qb = 1 − Pb , and (1 − Pb )L(1 − Pb ) = (1 − Pb )L, and (1 − Pb )(1 − Pa ) = 1 − Pa , see again the end of the subsection 3.5.3, and 3.1.1 and 3.1.4. (iii) Let us verify (iiid ). For any ϕ ∈ (X−ν,a )−∞/a we define x to be the solution of the initial value problem ¡ ¢ Pa Lx (t) = 0, x(t) = ϕ(t), Then we set

t ∈ R, t < a.

x− = x − x+ ,

where x+ is defined on the step (ii). Clearly, x− is the solution of (3−), (4−). Let us prove (5). Indeed, we have Lx− = Lx − Lx+ = Lx − LL−1 (1 − Pa )LRa ϕ = Lx − (1 − Pa )LRa ϕ = Lx − (1 − Pa )LRa Qa x = Lx − (1 − Pa )L(1 − Pa )x = Lx − (1 − Pa )Lx = Pa Lx = 0.

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Here we use the identities Ra Qa = 1 − Pa and (1 − Pa )L(1 − Pa ) = (1 − Pa )L. − )−∞/b for all b ∈ R. It suffices to Finally, we show that Qb x− lies in (X−ν,b − − − verify that Πb (Qb x ) = Qb x . We have − −1 Π− Pb LRb · Qb x− b Qb x = Qb L

= Qb L−1 Pb L(1 − Pb )x− = Qb L−1 L(1 − Pb )x− − Qb L−1 (1 − Pb )L(1 − Pb )x− = Qb (1 − Pb )x− − Qb L−1 (1 − Pb )Lx− = Qb x− . Here we use the identities Rb Qb = 1 − Pb , Pb = 1 − (1 − Pb ), (1 − Pb )L(1 − Pb ) = (1 − Pb )L, Qb Pb = 0, and Lx− = 0. (iv) Let us verify (ivd ). We recall from the step (ii) that the solution x+ of (3+), (4+) can be represented in the form x+ = L−1 (1 − Pa )LRa ϕ. Since all operators in this formula act between spaces with exponential weights, x+ ∈ X−ν,a . Clearly, the first estimate in (ivd ) holds uniformly with respect to ϕ. In a similar way the second estimate follows from the formula −1 ϕ− = Π− Pa LRa ϕ. a ϕ = Qa L

(c) ⇒ (a). We break the reasoning into three steps. (i) Let k ∈ Z. We recall that Pk Pk+1 = Pk+1 Pk = Pk+1 . Consequently we have (Pk − Pk+1 )2 = Pk − Pk+1 , i.e., Pk − Pk+1 is a projector. Assume f ∈ Y belongs to the image of Pk − Pk+1 . On this step, for such an f we construct a solution x ∈ X of the equation Lx = f and establish some estimates for x. We denote by ϕ ∈ X−∞/k+1 the solution of the initial value problem ¡

¢ Pk Lϕ (t) = f (t), ϕ(t) = 0,

t < k + 1,

(7)

t 0 we obtain the estimates kx+ kX−ε,k+1 ≤ N kϕkX−∞/k+1 ≤ N Kkf k, +

kx k(Xε,k+1 )−∞/k+1 = kϕ+ k(Xε,k+1 )−∞/k+1 = kϕ − ϕ− k(Xε,k+1 )−∞/k+1 ≤ kϕk(Xε,k+1 )−∞/k+1 + kϕ− k(Xε,k+1 )−∞/k+1 ≤ N1 kϕkX−∞/k+1 + N kϕkX−∞/k+1 ≤ (N1 + N )Kkf k. Since the norm on X±ε,k+1 differs from the norm on X±ε by the factor e±ε(k+1) , these estimates can be rewritten as kx+ kX−ε = eε(k+1) kx+ kX−ε,k+1 ≤ N2 eε(k+1) kf k, kx+ k(Xε )−∞/k+1 = e−ε(k+1) kx+ k(Xε,k+1 )−∞/k+1 ≤ N2 e−ε(k+1) kf k,

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where N2 = (N1 + N )K. We make use of 3.5.2. We set xi = (Pi − Pi+1 )x+ . We recall from 3.5.2 that X−ε is isomorphic to l−ε and represent x+ as {xi }. From the above estimates we have kxi k ≤ N3 e−εi kx+ kX−ε ≤ M e−ε(i−k−1) kf k,

i ∈ Z,

kxi k ≤ N3 eεi kx+ k(Xε )−∞/k+1 ≤ M eε(i−k−1) kf k,

i ≤ k,

where N3 = kΥ : X±ε → l±ε k and M = N2 N3 . Finally, we rewrite these two estimates as the common one: kxi k ≤ M e−ε|i−k−1| kf k,

i ∈ Z.

(11)

(ii) We show that Im L = Y . Let f ∈ Y . Let us consider f k = (Pk − Pk+1 )f = Pk (1−Pk+1 )f . We construct xk as the solution of the equation Lxk = f k according to the step (i). We set xki = (Pi − Pi+1 )xk . We stress that by (11) kxki k ≤ M e−ε|i−k−1| kf k k,

i ∈ Z.

(12)

According to 3.1.2 we identify Y with lq = lq (Z, Y♥i ) and X with lq = lq (Z, X♥i ). Then f can be interpreted as the family fe = {f k ∈ Y♥k : k ∈ Z} and f k as the family fek = {(fek )j : j ∈ Z}, where (fek )k = f k and (fek )j = 0 for j 6= k. Similarly, xk can be interpreted as the family x ek = {xki ∈ X♥i : i ∈ Z} and xki as the family k k x ei = {(e xi )j : j ∈ Z}, i, k ∈ Z, defined by the rule (e xki )i = xki and (e xki )j = 0 if P eki converges absolutely in the j 6= i. We note that by (12) the series x ek = i∈Z x norm of lq . exk = fek . Let Le : lq → lq be the operator induced by L : X → Y . Clearly, Le Since Le ∈ e, by 3.4.1 we may extend Le to the operator acting from lq,λ,|·| to P P lq,λ,|·| for some λ > 0. It is easy to verify that the series i∈Z k∈Z x eki converges absolutely in the norm of lq,λ,|·| . Therefore ³X X ´ X ³X ´ X X exk = x eki = Le x eki = Le fek = fe, Le i∈Z k∈Z

k∈Z

i∈Z

k∈Z

k∈Z

where the last series converges absolutely in the norm of lq,λ,|·| , too. On the other P P hand, it is evident that i∈Z k∈Z x eki is an element of lq . Thus fe belongs to the image of Le : lq → lq . Consequently f ∈ Im L. (iii) To show that Ker L = {0} we verify that Lx = 0 implies x = 0. So we assume that x ∈ X and Lx = 0. For an arbitrary a ∈ R we denote by − ϕa = Qa x the natural projection of x into X−∞/a . Let ϕa = ϕ+ a + ϕa be the ± ± decomposition of ϕ according to (id ) and x = xa be the solutions of (3±), (4±).

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207

From the uniqueness of the solution of the initial value problem we obtain x + x− = x. − We take an arbitrary b > a and consider ϕb = Qb x, ϕ+ b and ϕb . We show ± that ϕ± b coincide with the natural projections of xa into X−∞/b . Indeed, by (iid ) ± ± the projections Qb x± a of xa lie in X−∞/b . On the other hand, by the definition ± − ± + + − of ϕ± b we have ϕb ∈ X−∞/b . But xa + xa = x and ϕb + ϕb = ϕb . Since the sum in (id ) is direct we obtain that ϕ± b coincide with the natural projections of ± xa into X−∞/b . We consider the functions x± b . They are solutions of the initial value problems +

¡

¢ Pb Lx± b (t) = 0, ± x± b (t) = ϕb ,

t ∈ R, t < b.

We show that x± a are also solutions of these initial value problems. Indeed, by what has been just proved, xa± also satisfy these initial conditions. Since Pb Pa = Pb the ± functions x± a satisfy the equation Pb Lxb = 0, too. From unique solubility it follows ± ± that x± b = xa , i.e., the functions x do not depend on the choice of a. By (ve ) we have the estimates kx+ kX−ε,a ≤ N kϕa kX−∞/a ≤ N kxk, −

kx k(Xε,a )−∞/a = kϕ− a k(Xε,a )−∞/a ≤ N kϕa kX−∞/a ≤ N M kxk, which, by the arbitrariness of a, imply x+ = 0 and x− = 0. Thus x = 0 too. Now we turn to the case Y = Cq . The main changes in the prove are trivial, and we restrict ourselves to the only remark. ˘ −ν,a )−∞/a We observe that the operators Π± (X a given by the formula (6)¡ map ¢ ˘ into itself. Indeed, the condition ¡ ¢ ϕ ∈ (X−ν,a )−∞/a means that ¡ Lϕ (a) = ¢0, which since¡L is causal implies LR ϕ (a) = 0. Therefore we have (1−Pa )LRa ϕ (a) = 0 a ¢ and Pa LRa ϕ (a) = 0. Finally, let us, for instance, verify that the function −1 ϕ satisfies ¢the compatibility condition. ¡z =¢Qa L ¡Pa LRa−1 ¡ −1 ¢ ¡ Clearly, ¢we have that Lz (a) = LQa L Pa LRa ϕ (a) = LL Pa LRa ϕ (a) = Pa LRa ϕ (a) = 0. ¤ 3.6.5. Dichotomy, stability, and instability Corollary. Let (X, Y ) be one of the pairs (1), let L ∈ e+ (X, Y ), and let equation (2) be uniformly soluble. (a) If equation (2) possesses dichotomy in (X−ν , Y−ν ) for some ν ≥ 0 then − dim(X−ν,a )−∞/a does not depend on a. (b) Equation (2) is exponentially stable if and only if it possesses dichotomy − in (X−ν , Y−ν ) for some ν > 0 with (X−ν,a )−∞/a = {0}.

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(c) Equation (2) is exponentially stable if and only if it possesses exponential dichotomy in (X, Y ) with (X − )−∞/a = {0}. (d) Equation (2) is roughly unstable if and only if it possesses dichotomy in − (X−ν , Y−ν ) for some ν > 0 with (X−ν,a )−∞/a 6= {0}. (e) Equation (2) is roughly unstable if and only if it possesses exponential dichotomy in (X, Y ) with (X − )−∞/a 6= {0}. Proof. (a) Assume for definiteness that Y 6= Cq . Let us take arbitrary a < b. − )−∞/a we define U − (b, a)ϕ− to be Qb x− . By (iiid ) the operator For ϕ− ∈ (X−ν,a − − )−∞/b . U − (b, a) acts from (X−ν,a )−∞/a to (X−ν,b − − And for ψ ∈ (X−ν,b )−∞/b we define U − (a, b)ψ − to be the restriction Qa ψ − . − − Again, by (iiid ) the operator U − (a, b) acts from (X−ν,b )−∞/b to (X−ν,a )−∞/a . Clearly, U − (a, b)U − (b, a) = 1. − Conversely, let ψ − ∈ (X−ν,b )−∞/b and ϕ− = U − (a, b)ψ − . According to (5), both ψ − and U − (b, a)ϕ− are solutions on the initial value problem

¡

¢ Pa Lx (t) = 0, x(t) = ϕ− (t),

t < b, t < a.

By the uniqueness of a solution ψ − = U − (b, a)ϕ− , which means that U − (b, a)U − (a, b) = 1. − − Thus we have proved that (X−ν,a )−∞/a and (X−ν,b )−∞/b are isomorphic. The case Y = Cq is handled similarly. (b) and (c) follow from 3.5.12. (d) and (e) follow from (b) and (c), and 3.6.4. ¤

Remark. Assume a < b. For ϕ ∈ (X−ν,a )−∞/a we define U(b, a)ϕ to be x ∈ (X−ν,b )−∞/b , where x is the solution of the initial value problems (with the −1 usual changes for Y = Wpq and Y = Cq ) ¡ ¢ Lx (t) = 0, x(t) = ϕ(t),

a < t < b, t < a.

The operator U(b, a) : (X−ν,a )−∞/a → (X−ν,b )−∞/b , a < b, is called the shift along the trajectories of the equation (2). It is often used in the theory of stability of ordinary differential equations (see, e.g., [DaK]). We observe that U(b, a), a < b, cannot be contractive if ν = 0 because x is the continuation of ϕ and hence kxkXb ≥ kϕkXa . But in the case ν > 0 one may have kxkX−ν,b < kϕkX−ν,a . Thus if one wants to describes stability as the contractive property of U (b, a), one may use the phase spaces X−ν,a with ν > 0 instead of X.

CHAPTER IV

SHIFT INVARIANT OPERATORS AND EQUATIONS The main examples of shift invariant equations are the integral equation Z x(t − s)g(s) dλ(s) = f (t), the difference equation

∞ X

am x(t − tm ) = f (t),

m=1

and various kinds of differential equations with constant coefficients, e.g., the differential difference equation ∞ X m=1

am x(t ˙ − tm ) +

∞ X

bm x(t − tm ) = f (t).

m=1

Shift invariant operators and equations are the most important for two reasons. First, shift invariant equations are widespread in applications. Second, this is probably a unique class in which invertibility and stability can be investigated more or less completely.

4.1. Algebras of bounded measures A rather general example of a shift invariant operator is the convolution with a bounded measure. In particular, the convolution with an absolutely continuous measure induces an integral operator, and the convolution with a discrete measure induces a difference operator. In this section we define the convolution of two measures and the convolution of a measure with functions of the classes C0 and L1 . The main result of the section is theorem 4.1.10. It states that absolutely continuous and discrete measures form full subalgebras of the algebra of all bounded measures. 4.1.1. Convolution operators on C0 . Let G be a locally compact abelian group with the Haar measure λ, see 1.6.2. The most important example of G is the group R; therefore we denote elements of G by the symbols t, s, and so on. Let µ ∈ M(G, C), see 1.8.4 for the definition of M. We consider the operator Z Z ¡ ¢ Tµ x (t) = x(t − s) dµ(s) = x(s) dµ(t − s). (1)

209

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IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

The operator Tµ is called the operator of convolution. The function Tµ x is called the convolution of µ and x and is denoted by the symbol µ ∗ x. The operator Tµ acts on many functional spaces. We begin our consideration with the simplest case — the space C0 . R Remark.R To be precise (see the definitions of x dµ R in 1.5.5 and 1.8.4), if x ∈ C0 , by x(t − s)¡dµ(s) ¢ we mean hISt x, µi, ¡ and ¢ by x(s) dµ(t − s) we mean hx, I 0 (S−t )0 µi, where St x (s) = x(s − t) and Ix (s) = x(−s). It is useful to note that ISt = S−t I and that it is reasonable to interpret (S−t )0 µ as St µ, and I 0 µ as Iµ. Proposition. ¡ ¢ For any µ ∈ M(G, C) the formula (1) defines an operator Tµ ∈ B C0 (G, C) with kTµ k = kµk. Proof. Clearly, for all µ ∈ M, x ∈ C0 , and t ∈ G the integral (1) is well defined. To prove that Tµ x is continuous we recall from 1.6.12 that the function h 7→ Sh is continuous in the strong topology, i.e., Sh x depends on h continuously for all x ∈ C0 , see 1.1.5. In particular, the s 7→ x(t − s) depends on t continuously ¡ integrand ¢ in the norm of C0 . Therefore Tµ x (t) depends on t continuously. From the estimate ¯¡ ¢ ¯ ¯ Tµ x (t)¯ ≤ kµk · kxk. it follows that Tµ acts from C0 to C and kTµ : C0 → Ck ≤ kµk. The opposite inequality kTµ : C0 → Ck ≥ kµk follows from the definition of kµk. Finally, we show that Tµ maps C0 into C0 . If µ is compactly supported then, clearly, Tµ maps C00 into C00 and consequently C0 into C0 . From 1.8.5 and the estimate kTµ : C0 → Ck ≤ kµk it follows that Tµ maps C0 into C0 for all µ. ¤ 4.1.2. Shift invariant operators. Let G be a locally compact abelian group. Let X and Y be functional spaces on G such that the shift operators ¡

¢ Sh x (t) = x(t − h),

h ∈ G,

act continuously both on X and Y . Example. Let E be a Banach space. (a) For arbitrary G one can take for X and Y the spaces Lp (G, E), Lpq (G, E), or Cq (G, E), where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, see 1.6.3. (b) If G = R one can take for X and Y the spaces 1 −1 Cq1 (R, E), or Wpq (R, E), or Cq−1 (R, E), or Wpq (R, E), see 2.3.2 and 2.3.6 for the definitions. An operator T ∈ B(X, Y ) is called shift invariant if it commutes with the shift operators, i.e., T Sh = Sh T for all h ∈ G. We denote the set of all shift invariant operators T ∈ B(X, Y ) by the symbol A = A(X, Y ). If X = Y we employ the brief notation A(X).

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211

Proposition. (a) The set A(X, Y ) is a closed subspace of B(X, Y ). The product of shift invariant operators and the inverse of a shift invariant operator (provided it exists) are shift invariant operators, too. In particular, the set A(X) is a closed full subalgebra ¡ of B(X). ¢ (b) An operator T ∈ B C0 (G, C) is an operator of¢ the convolution with a ¡ bounded measure if and only if T ∈ A C0 (G, C) . Remark. We recall that Tµ , µ ∈ M, acts on many other functional spaces. Assertion (b) remains true in the space L1 (Rn , C), see [Ste2 , ch. 1, theorem 3.19]. Nevertheless, it is not the case in most of the other spaces. For instance, from the example in 5.1.11 below is follows that there are shift invariant operators on l∞ (Z, C) which are not convolution operators. See also the example in 5.6.9 below. Proof. (a) The proof is essentially a repetition of that of 1.4.4(b). (b) Clearly, an operator of convolution is shift invariant. ¡ ¢ We prove the converse. Let T ∈ A. For any fixed t ∈ G the mapping x 7→ T x (t) is a linear functional on C0 . Therefore there exist uniquely determined measures µt , t ∈ G, such that the operator T has the representation Z ¡ ¢ T x (t) = x(t − s) dµt (s). We recall that by the definition of a shift invariant operator, Sh T S−h = T . It is straightforward to verify that Z ¡ ¢ Sh T S−h x (t) = x(t − s) dµt−h (s), which implies that measures µt do not depend on t.

¤

4.1.3. The convolution of measures. Let G be a locally compact abelian group. By 4.1.2(b) and 4.1.1 the Banach ¡ space M ¢ = M(G) is isometrically isomorphic to the Banach algebra A = A C0 (G, C) . This isomorphism induces on M the structure of Banach algebra. The multiplication of measures µ, ν ∈ M induced by this isomorphism is called the convolution of measures andRis denoted by the symbol µ∗ν. The unit of M is the measure δ defined by the rule x dδ = x(0); it corresponds to the identity operator. Clearly, kµ ∗ νk ≤ kµk · kνk. For any µ ∈ M we consider the operator of convolution Tµ ν = µ ∗ ν acting on M (cf. 1.7.9). Obviously, kTµ : M → Mk ≤ kµk. And since µ ∗ δ = µ we have the equality kTµ : M → Mk = kµk. We recall from 1.5.8 that the product of µ and ν is the measure µ ⊗ ν on G × G and kµ ⊗ νk ≤ kµk · kνk.

212

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Proposition. Let µ, ν ∈ M. Then for all x ∈ C (not only for x ∈ C0 ) one has Z ZZ x d(µ ∗ ν) = x(t + s) d(µ ⊗ ν)(t, s) Consequently the Banach algebra M(G, C) is commutative. Proof. We consider the operator B = Tµ Tν acting on C0 . It is shift invariant. By 4.1.2(b) it possesses the representation Bx =¡ξ ∗ x¢ with aR measure ξ ∈ M. We observe that ξ can be restored from the equality Bx (0) = x(−s) dξ(s), x ∈ C0 . Namely, for x ∈ C0 we have Z Z x(−t) d(µ ∗ ν)(t) = x(−t) dξ(t) Z ¡ ¢ = Tν x (0 − t) dµ(t) Z ³Z ´ = x(−t − s) dν(s) dµ(t). To finish the proof it remains to establish the identity Z ³Z ZZ ´ x(t + s) dν(s) dµ(t) = x(t + s) d(µ ⊗ ν)(t, s). We observe that the function (t, s) 7→ x(t + s) belongs to C(G × G, C). Therefore it is measurable and bounded. If µ and ν are positive the identity follows from Fubini’s theorem. The general case follows from 1.8.4. Thus the proof for the case x ∈ C0 is complete. Let now x ∈ C. Then it is easy to see that both left and right integrals exist and they are estimated by kxk · kµk · kνk. We observe that the case when µ and ν are compactly supported is reduced simply to the case x ∈ C0 . Therefore the general case x ∈ C follows from 1.8.5. ¤ 4.1.4. The ideal Mac of absolutely continuous measures. Let G be a locally compact abelian group with the Haar measure λ. We recall from 1.8.10(a) (see also 1.8.7) that a measure µ ∈ M(G) is absolutely continuous (with respect to λ) if there exists a density g ∈ L1 (G, C) such that Z Z x dµ = x(t)g(t) dλ(t) for all x ∈ C00 . In this case kµkM = kgkL1 . We denote the set of all absolutely continuous measures µ ∈ M(G) by the symbol Mac = Mac (G) = Mac (G, C). We stress that the operator of the convolution with an absolutely continuous measure µ = gλ is the shift invariant integral operator (see also 4.4.11) Z ¡ ¢ Tµ x (t) = x(t − s)g(s) dλ(s) for all x ∈ C0 .

4.1. ALGEBRAS OF BOUNDED MEASURES

213

Proposition. The set Mac (G, C) is a closed ideal of the algebra M(G, C). Proof. Clearly, Mac is a closed subspace of M. Let µ ∈ M and ν ∈ Mac be positive measures. We show that µ ∗ ν ∈ Mac . We make use of the Lebesgue–Radon–Nikodym theorem (see 1.8.8). Since ν is bounded the function h(t) ≡ 1, t ∈ G, belongs to L1 (G, C, ν). Assume x ∈ C0 , 0 ≤ x(t) ≤ 1 = h(t). From 1.8.8(b) it follows that for any ε > 0 there exists δ > 0 such that kxkL1 ≤ δ implies (we recall that ν is positive) ¡ ¢ Tν x (t) =

Z x(t − s) dν(s) ≤ ε

for all t ∈ G.

Thus kTν xkC0 ≤ ε. Consequently Z

¡ ¢ x(−t) d(µ ∗ ν)(t) = Tµ Tν x (0) ≤ kµk · ε.

This estimate implies that condition 1.8.8(c) holds for µ∗ν. Hence µ∗ν is absolutely continuous. The case of non-positive µ and ν follows from 1.8.4(c). ¤ 4.1.5. The push transform. Let E be a Banach space. By the push transform we mean the transform x 7→ u which takes a function x : G → E of one variable to the function u(t, s) = x(t − s) of two variables. Lemma. Let E be a Banach space, and let µ ∈ M(G, C). (a) For any function x ∈ L1 (G, E, λ) the function u(t, s) = x(t − s) belongs to L1 (G × G, E, λ ⊗ µ). Moreover, kukL1 = kxkL1 · kµk. (b) For any λ-measurable function x : G → E the function u(t, s) = x(t − s) is λ ⊗ µ-measurable. Proof. (a) We make use of the definition of the Lebesgue integral, see 1.5.5. For functions ¡ − ¢ defined on G × G we consider ¡ + ¢ the two mutually inverse transformations: I x (t, s) = x(t − s, s) and I x (t, s) = x(t + s, s). Clearly, they map C00 (G G, E) onto itselfR and M + (G × G) onto itself, too. Since λ is shift invariR × ant I ± y d(λ ⊗ |µ|) = y d(λ R ∗ ⊗±|µ|) for all y ∈ RC∗00 . Then from the definition it follows immediately that I zR d(λ ⊗ |µ|) = z d(λ ⊗ |µ|) for all z ∈ M + . R∗ ± ∗ Consequently I x d(λ x d(λ ⊗ |µ|) for all x : G × G → [0, +∞], and R∗ ± R ∗ ⊗ |µ|) = I |x| d(λ ⊗ |µ|) = |x| d(λ ⊗ |µ|) for all x ∈ F(T, E). Hence I ± map isometrically F1 onto itself. Since L1 is the closure of C00 in F1 , it follows that I ± map isometrically L1 onto itself as well. ¡ Now ¢let x ∈ L1 (G, E, λ). By Fubini’s theorem (or by 1.7.4(c)) the function x ⊗ 1G (t, s) = x(t)1G (s) = x(t) is λ ⊗ µ-integrable and its L1 -norm is equal to kxkL1 · kµk. By what has been proved, the function u(t, s) = x(t − s) is λ ⊗ µintegrable, too. Moreover, its L1 -norm is equal to the norm of the function x ⊗ 1G . (b) We make use of the definition of a measurable function (see 1.5.6).

214

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

By 1.1.1 the co-ordinate projection of a compact subset of G×G to G is compact. Hence any compact subset of G × G is contained in a compact subset of the form N × M , where N, M ⊆ G are compact. Thus to verify the measurability it suffices to show that the restriction of u to a set of the form N × M is measurable. So let N, M ⊆ G be compact. We define the compact set K to be the image of the set N × M under the mapping (t, s) 7→ t − s; thus K = N − M . For an arbitrary ε > 0 we choose a compact set K1 ⊆ K such that λ(K \ K1 ) < ε and the restriction of x to K1 is continuous. Applying (a) to the characteristic functions of the sets K, K1 , and K \ K1 we obtain that the sets e = { (t, s) : t − s ∈ K }, K e 1 = { (t, s) : t − s ∈ K1 }, K e \K e 1 = { (t, s) : t − s ∈ K \ K1 } K^ \ K1 = K are λ ⊗ |µ|-summable and, moreover, λ ⊗ |µ|(K^ \ K1 ) = λ(K \ K1 ) · kµk < ε · kµk. e \K e 1 . By definition the set K e 1 is the pre-image of K1 Clearly, K^ \ K1 = K under the continuous mapping (t, s) 7→ t − s from G × G to G. Therefore the e 1 is closed, and consequently the set (N × M ) ∩ K e 1 is compact. Evidently, set K e Hence N × M ⊆ K. ¡ ¢ e 1 = (N × M ) \ K e1 ⊆ K e \K e1 (N × M ) \ (N × M ) ∩ K = K^ \ K1 . From this inclusion it follows easily that ³ ¡ ¢´ e λ ⊗ |µ| (N × M ) \ (N × M ) ∩ K1 ≤ λ ⊗ |µ|(K^ \ K1 ) < ε · kµk. e1 ⊆ K e 1 is It remains to observe that the restriction of u to the set (N × M ) ∩ K continuous. ¤ 4.1.6. Convolution operators on L1 . By 4.1.4, for any µ ∈ M the operator of convolution Tµ ν = µ ∗ ν on M maps the subspace Mac into itself and thus induces the operator Teµ on Mac . Since Mac ' L1 we can consider the operator Teµ , µ ∈ M, as acting on L1 with kTeµ : L1 → L1 k ≤ kµk. Thus one can obtain the definition of the convolution of a measure and a function different from (1). In assertion (b) of the theorem below we show that these definitions are in agreement. We also define the convolution f ∗ g of functions f, g ∈ L1 (G, C) to be the density of the convolution of measures f λ and gλ or, equivalently, as the result of the action of Tgλ : L1 → L1 on f .

4.1. ALGEBRAS OF BOUNDED MEASURES

215

Theorem. Let G be a locally compact abelian group. (a) Let µ ∈ M(G, C). For any x ∈ L1 (G, C) the function s 7→ x(t − s) is µ-integrable for ¡ ¢ λ-almost all t ∈ G. Formula (1) defines the operator Tµ ∈ B L1 (G, C) with kTµ k ≤ kµk. Thus the convolution µ ∗ x for x ∈ L1 and µ ∈ M is well defined. (b) Let g ∈ L1 and ν = gλ. Then the definitions of µ ∗ g and µ ∗ ν are in agreement, namely, µ ∗ ν is the measure (µ ∗ g)λ. (c) The convolution of f, g ∈ L1 (G, C) can be calculated by the formulae Z ¡ ¢ f ∗ g (t) = f (t − s)g(s) dλ(s), Z ¡ ¢ f ∗g = Sh f g(h) dλ(h). Here the first integral exists for almost all t ∈ G, and in the second integral the function h 7→ Sh f takes its values in L1 (G, C). Remark. In 4.4.6 we shall show that kTµ : L1 → L1 k = kµk. Proof. (a) Let x ∈ L1 (G, C). By 4.1.5 and Fubini’s theorem the internal integral in Z ³Z ´ x(t − s) dµ(s) dλ(t) exists for almost all t and defines the function Tµ x in L1 (G, C) with norm less than or equal to kµk · kxkL1 . In particular, the estimate kTµ xk ≤ kµk · kxkL1 implies that Tµ x depends on the equivalence class of x only. Thus (1) defines a bounded linear operator on L1 . (b) By 4.1.3, for any x ∈ C00 we have (we employ the shift invariance of the Haar measure λ) Z ZZ x d(µ ∗ ν) = x(t + s) d(µ ⊗ ν)(t, s) Z ³Z ´ = x(t + s) dν(s) dµ(t) Z ³Z ´ = x(t + s)g(s) dλ(s) dµ(t) Z ³Z ´ = x(r)g(r − t) dλ(r) dµ(t) Z ³Z ´ = x(r) g(r − t) dµ(t) dλ(r) Z = x(r)(µ ∗ g)(r) dλ(r), which means that µ ∗ ν is absolutely continuous with the density µ ∗ g. (c) The first formula follows from (a). Indeed, we can interpret f ∗ g as the result of the action of Tgλ : L1 → L1 on f .

216

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

The second formula follows from 1.5.9. Indeed, it suffices to observe that in accordance ¡ ¢ with the isomorphism L1 (G, L1 (G, C)) ' L1 (G × G, C) the function h 7→ Sh f g(h) corresponds to the function (t, s) 7→ f (t − s)g(s) and, furthermore, the integration with respect to s in L1 (G × G, C) corresponds to the integration with respect to h in L1 (G, L1 (G, C)). ¤ 4.1.7. The ideal Mc of continuous measures. A measure µ ∈ M is called continuous if µ({t}) = 0 for any one point set {t}, t ∈ G. We denote the set of all continuous measures by Mc = Mc (G) = Mc (G, C). Proposition. The set Mc (G, C) is a closed ideal of the algebra M(G, C). Proof. Clearly, Mc is a subspace. We show that Mc is closed. Assume µ e ∈ M belongs to the closure of Mc . Then for any ε > 0 there exists µ ∈ Mc such that kµ − µ ek ≤ ε. From the estimate (for the definition and the discussion of |µ|, see 1.8.4) ¯¡ ¯ ¢ ¯ µ−µ e ({t})¯ ≤ |µ − µ e|({t}) ≤ |µ − µ e|(G) = kµ − µ ek 0 there exists U such that |xV − 1{0} | dξ < ε for all V ⊆ U . Indeed, 1{0} dξ = ξ({0}) = inf{ ξ(H) : 0 ∈ H, H is open }. Therefore we may take for U an open set H such that ξ(H) − ξ({0}) < ε. In a similar way the functions s 7→ xU (t − s) converge to 1{t} . Thus ξ is continuous if and only if the functions Z ¡ ¢ t 7→ Tξ xU (t) = xU (t − s) dξ(s) converges to zero at all points t ∈ G. Let us return to the measures µ and ν. First we show that Tν xU converges uniformly to zero. Assume that ν is concentrated on a compact set K. Then it is

4.1. ALGEBRAS OF BOUNDED MEASURES

217

easy to verify that Tν xU is supported in K + U . We fix a compact neighbourhood U0 of 0 ∈ G. Clearly, the set K + U0 is compact. Let us take an arbitrary ε > 0. For any t ∈ G we choose an open neighbourhood Vt ⊆ U0 such that ν(t + Vt + Vt ) < ε. Clearly, the family { t+Vt : t ∈ K +U0 } covers K +U0 . Since K +U0 is compact we can choose a finite family t1 +Vt1 , . . . , tn +Vtn which covers K+U0 . We consider the neighbourhood W = ∩nk=1 Vtk . We take an arbitrary t ∈ K + U0 . By assumption there exists k such that t ∈ tk + Vtk . Consequently t + W ⊆ tk + Vtk + Vtk , which implies ν(t + W ) ≤ ν(tk + Vtk + Vtk ) < ε. Therefore ¡ ¢ Tν xW (t) ≤ ν(t + W ) < ε and since W ⊆ U0 ,

¡ ¢ Tν xW (t) = 0

for t ∈ K + U0

for t ∈ / K + U0 .

The result proved implies that ¡

¢ Tµ Tν xW (t) =

Z

¡ ¢ Tν xW (t − s) dµ(s)

≤ ε · µ(t − K − U0 ), ¡ ¢ which means that the functions t 7→ Tµ Tν xW (t) converge to zero at all points t ∈ G. Thus the measure µ ∗ ν is continuous. ¤ 4.1.8. The subalgebra Md of discrete measures. For t ∈ G we denote R by δt the atom measure concentrated at t, i.e., x dδt = x(t). ¡ We ¢ observe that the operator of the convolution with δh is the shift operator Sh x (t) = x(t − h). P∞ Let tn ∈ G, and let an ∈ C, n ∈ N, be such that n=1 |an | < ∞. The measure of the form ∞ X µ= an δtn (2) n=1

is called a discrete measure. We denote by Md = Md (G) = Md (G, C) the set of all discrete measures. We stress that the operator of the convolution with the discrete measure (2) is the difference operator (see also 4.4.10) ∞ X ¡ ¢ an x(t − tn ). Tµ x (t) = n=1

Proposition. The set Md (G, C) is a closed subalgebra of the algebra M(G, C). If tn ∈ G are distinct points then for the measure (2) one has |µ| =

∞ X n=1

|an |δtn

and

kµk =

∞ X n=1

|an |.

218

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

The algebra Md (G, C) is naturally isometrically isomorphic to M(Gd , C). Proof. From the formula Sh x = δh ∗ x, by the definition of the convolution of measures it follows δt ∗ δs = δt+s . Hence it is evident that Md is a subalgebra. Let µ be a measure of the form (2). From kδn k = 1 it follows that kµk ≤

∞ X

|an |.

(3)

n=1

Pn Therefore the sequence µn = k=1 ak δtk converges to µ in norm. Pn−1 We observe that δtn and ¯P singular measures. Therek=1 ak δtk are ¯ mutually °Pn ° Pn n ° ° = fore by 1.8.9 and induction ¯ k=1 ak δtk ¯ = |a |δ and a δ k tk k tk k=1 k=1 Pn k=1 |ak |. By continuity similar identities P∞ hold for infinite sums. From 1.1.2 and the identity kµk = n=1 |an | it follows that Md is closed. In conclusion, we observe that both Md (G, C) and M(Gd , C) are isometrically isomorphic to l1 (Gd , C). ¤ 4.1.9. The main decomposition. Let G be a locally compact abelian group with the Haar measure λ. We recall from 1.8.10 that a measure µ ∈ M(G) is singular with respect to the Haar measure λ if it is concentrated on a locally null (with respect to λ) set. We denote by Msc = Msc (G) = Msc (G, C) the set of all continuous measures µ ∈ M(G) which are singular with respect to the Haar measure λ. Example. Let G = R2 . The partial integral Z

1

x 7→

x(t1 , t2 ) dt1 ,

x ∈ C00 ,

−1

is a singular continuous measure. Theorem ([Hew, theorem 19.20]). (a) Let G be non-discrete. Then M = Md ⊕ Mc and Mc = Msc ⊕ Mac . Thus M = Md ⊕ Msc ⊕ Mac . For the corresponding representation µ = µd + µsc + µac of µ ∈ M |µ| = |µd | + |µsc | + |µac |

and

kµk = kµd k + kµsc k + kµac k.

(b) Let G be discrete. Then M = Md = Mac and Msc = {0}. Proof. (a) First we show that M = Md ⊕ Mc . We take an arbitrary µ ∈ M and consider the set D = { t ∈ G : µ({t}) 6= 0 }. The set D is countable. Indeed, if for some ε > 0 the set Dε = { t ∈ GP : |µ({t})| > ε } contains m elements then kµk exceeds mε. For the same reasons t∈D |µ({t})| < ∞. Therefore the formula P µd = t∈D µ({t})δt defines a discrete measure µd correctly. Clearly, µc = µ − µd is a continuous measure. Thus M ⊆ Md + Mc .

4.1. ALGEBRAS OF BOUNDED MEASURES

219

On the other hand, any discrete measure is concentrated on a countable set, and a continuous measure of a countable set is equal to zero. Thus a discrete measure and a continuous measure are mutually singular in the sense of 1.8.9. Hence Md ∩ Mc = {0} and, by 1.8.9 |µ| = |µd | + |µc |

and

kµk = kµd k + kµc k.

Since G is non-discrete, by 1.6.2(d) λ is continuous, i.e., λ({t}) = 0, t ∈ G. Then by the Lebesgue–Radon–Nikodym theorem Mac ⊆ Mc . Consequently by a simple reasoning of linear algebra the decomposition M = Ms ⊕ Mac (see 1.8.10) implies Mc = Msc ⊕ Mac (we recall that Msc = Ms ∩ Mc by definition), and for µc = µsc + µac we have |µc | = |µsc | + |µac |

and

kµc k = kµsc k + kµac k.

(b) If G is discrete, λ({t}) = 1 for all t ∈ G. Therefore C0 coincides with l0 , and consequently by 1.8.1 M = C00 coincides with l1 . Thus M = Md . Clearly, Mac = Md . As we have seen in the course of the proof of (a), Md ∩ Mc = {0}. Therefore Mc = {0}. ¤ 4.1.10. Full subalgebras of M. Let G be non-discrete. Since the convolution with δ is the identity operator, from 4.1.9(a) it follows that the algebras Mc and gc and M ] Mac do not contain the unit of M. We denote by M ac the algebras Mc ] and Mac with adjoint units, see 1.4.1. We realize M ac as the subalgebra of M consisting of all measures of the form αδ + µac , where α ∈ C and µac ∈ Mac . By gc similarly. 4.1.9 kαδ + µac k = |α| + kµac k. And we realize M We denote briefly the set Md ⊕ Mac by the symbol Md⊕ac . Theorem. Let G be a non-discrete locally compact abelian group. Then Md , g ] Mc , M ac , and Md⊕ac are full subalgebras of the algebra M. Proof. Let µd ∈ Md be invertible and ν = νd + νc , where νd ∈ Md and νc ∈ Mc , be its inverse. From the equality µd ∗ ν = µd ∗ νd + µd ∗ νc = δ, and 4.1.7, and 4.1.8, and 4.1.9 we have µd ∗ νd = δ and µd ∗ νc = 0, which implies νd = (µd )−1 and (since µd is invertible) νc = 0. Let αδ + µc , µc ∈ Mc , be invertible and ν = νd + νc , where νd ∈ Md and νc ∈ Mc , be its inverse. From the equality (αδ + µc ) ∗ ν = αδ ∗ νd + αδ ∗ νc + µc ∗ νd + µc ∗ νc = δ, and 4.1.7, and 4.1.8, and 4.1.9 we obtain αδ ∗ νd = δ, which implies νd = (1/α)δ.

220

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Let αδ + µac , µac ∈ Mac , be invertible and ν = νd + νsc + νac , where νd ∈ Md , νsc ∈ Msc , and νac ∈ Mac , be its inverse. From the equality (αδ + µac ) ∗ ν = αδ ∗ νd + αδ ∗ νsc + αδ ∗ νac + µac ∗ νd + µac ∗ νsc + µac ∗ νac = δ, and 4.1.4, and 4.1.7, and 4.1.8, and 4.1.9 we obtain αδ ∗ νd = δ (which implies α 6= 0 and νd = (1/α)δ) and αδ ∗ νsc = 0, which implies νsc = 0. Let µd + µac , µd ∈ Md and µac ∈ Mac , be invertible and ν = νd + νsc + νac , where νd ∈ Md , νsc ∈ Msc , and νac ∈ Mac , be its inverse. From the equality (µd + µac ) ∗ ν = µd ∗ νd + µd ∗ νsc + µd ∗ νac + µac ∗ νd + µac ∗ νsc + µac ∗ νac = δ, and 4.1.4, and 4.1.7, and 4.1.8, and 4.1.9 we obtain µd ∗ νd = δ (which means that µd is invertible in the algebra Md ) and µd ∗ νsc = 0, which implies νsc = 0. ¤ 4.1.11. The algebra M+ . Let G = R. We denote by M+ = M+ (R) = M+ (R, C) the set of all µ ∈ M(R) concentrated on [0, +∞). We set M+ d = + + + + + + + Md ∩ M , Mc = Mc ∩ M , Msc = Msc ∩ M and Mac = Mac ∩ M . We note that M+ ac is isometrically isomorphic to L1 [0, +∞); and L1 [0, +∞) can be interpreted as the subspace of L1 (−∞, +∞) consisting of all functions which are supported in [0, +∞). Similarly, M+ d is isometrically isomorphic to l1 [0, +∞). We denote by M+ (Rd , C) the set of all µ ∈ M(Rd ) concentrated on [0, +∞). And we denote by M+ (Z, C) the set of all µ ∈ M(Z) concentrated on [0, +∞). Proposition. (a) A measure µ ∈ M(R, C) belongs to M+ (R, C) if and only if the operator Tµ : C0 → C0 is causal. (See also 4.4.12(a).) (b) The set M+ (R, C) is a closed subalgebra of M(R, C). The set M+ d (R, C) + + is a closed subalgebra of M (R, C). The sets Mac (R, C) and M+ c (R, C) + are closed ideals in M (R, C). (c) Let, according to 4.1.9, µ ∈ M+ be represented as µ = µc + µsc + µac . Then µc , µsc , µac ∈ M+ . Thus + + M+ = M+ d ⊕ Msc ⊕ Mac .

(d) The set M+ (Rd , C) is a closed subalgebra of M(Rd , C). The algebra M+ (Rd , C) is naturally isometrically isomorphic to M+ d (R, C). Proof. (a) Let Tµ be Then for all x ∈ C00 the x(s) = 0, ¡ causal. ¢ ¡ assumption ¢ s < 0, implies hx, µi = Tµ x (0) = 0. Consequently |µ| (−∞, 0) = 0, i.e., µ is concentrated on [0, +∞). The opposite statement is evident. (b) By (a) and 4.1.2 M+ is isometrically isomorphic to the intersection of the Banach algebras B+ (C0 ) and A(C0 ). Hence it is closed. The other assertions follow from the definitions and analogous assertions for M. (c) By the definition of the modulus of measure, |µ| is concentrated on [0, +∞). From the formula |µ| = |µd | + |µsc | + |µac | it follows that |µd |, |µsc |, |µac | are also concentrated on [0, +∞). Hence µd , µsc , µac are concentrated on [0, +∞), too. (d) is evident. ¤

221

4.2. THE FOURIER TRANSFORM

g+ (and M + ] 4.1.12. Full subalgebras of M+ . Let G = R. We denote by M c ac ) + the subalgebra of M consisting of all measures of the form αδ + µ, where α ∈ C + and µ ∈ M+ c (respectively, µ ∈ Mac ), cf. 4.1.10. + + We denote briefly the set M+ d ⊕ Mac by the symbol Md⊕ac . g+ ] + + Corollary. Let G = R. Then M+ d , Mc , Mac , and Md⊕ac are full subalgebras of the algebra M+ . Proof. The proof follows immediately from 4.1.10.

¤

4.2. The Fourier transform According to 4.1.2 the invertibility of the shift invariant operator Tµ on C0 is equivalent to the invertibility of µ in the algebra M(G, C). (What happens in other functional spaces, we shall discuss later.) In this section we discuss whether the invertibility of µ in M is reduced to the pointwise invertibility of its Fourier transform µ ˆ. The main idea is the following. The algebra M is commutative. Therefore the invertibility in it can be investigated by means of the Gel0 fand transform (see 1.4.11). To use the Gel0 fand transform one should describe effectively the space of characters of M. Unfortunately, in general this problem is very difficult. But for the subalgebra Md⊕ac a solution can be given in terms of the Fourier transform. 4.2.1. The dual group. Let G be a locally compact abelian group. We denote by U the multiplicative group { z ∈ C : |z| = 1 }. A character or a continuous character of the group G is defined to be a continuous mapping χ : G → U which preserves the group operation, i.e., χ(t + s) = χ(t)χ(s),

t, s ∈ G.

Clearly, this implies χ(0) = 1. (In other words, a character can be defined to be a morphism of topological groups from G to U, or as a unitary representation of G in B(C).) We denote the set of all (continuous) characters of G by X = X(G). We denote by Gd the group G considered with the discrete topology, and by Xb = Xb (G) the set of all characters of Gd . Thus Xb consists of all mappings χ : G → U preserving the group operation. We call elements of Xb discontinuous characters of G. (As we shall see below, the notation Xb has the following meaning: Xb is the Bohr compactification of X.) Let χ1 , χ2 ∈ X. The product of χ1 and χ2 is defined by the pointwise rule ¡

¢ χ1 χ2 (t) = χ1 (t)χ2 (t),

t ∈ G.

It is easy to see that with respect to this operation, X is an abelian group with the character 1(t) ≡ 1 as a unit and χ−1 (t) = χ(t) = χ(−t) as the inverse of χ; here

222

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

χ(t) is the complex conjugate of χ(t). The group X = X(G) is called the character group of the group G or the dual group of the group G. We endow X(G) with the topology of uniform convergence on compact sets or, briefly, the topology of compact convergence. It is convenient to describe this topology in terms of neighbourhoods of points. Namely, for χ0 ∈ X, a compact set K ⊆ G, and ε > 0 we set WK,ε (χ0 ) = { χ : |χ(t) − χ0 (t)| < ε for all t ∈ K }. The family of all sets WK,ε (χ0 ) is a neighbourhood base (see 1.1.1) at χ0 . It is a fact of general topology (see, e.g., [Bou1 , ch. 10, §1, 3] or [Kel, ch. 7]) that this system of neighbourhoods actually determines a topology. Proposition. (a) Let G be a locally compact abelian group. Then X(G) is a topological group. (See also 4.2.4.) (b) Let G1 and G2 be locally compact abelian groups. Then X(G1 × G2 ) = X(G1 ) × X(G2 ). (c) Let G be a locally compact abelian group, and let H be a closed subgroup of it. Then the group X(G/H) is naturally isomorphic to the subgroup H⊥ = { χ ∈ X(G) : χ(h) = 1 for all h ∈ H } and X(H) is naturally isomorphic to the quotient group X(G)/H⊥ . Proof. (a) First we verify that X is a Hausdorff space. Let χ1 6= χ2 , i.e., χ1 (t0 ) 6= χ2 (t0 ) for some t0 . Assume that ε < |χ1 (t0 ) − χ2 (t0 )|/2. Then the intersection of the neighbourhoods W{t0 },ε (χ1 ) and W{t0 },ε (χ2 ) is empty. Next, we show that the mapping (χ, κ) 7→ χκ from X × X to X is continuous at an arbitrary point (χ0 , κ0 ). We observe that |χ(t)κ(t) − χ0 (t)κ0 (t)| ≤ |χ(t) − χ0 (t)| + |κ(t) − κ0 (t)| because characters take their values in U. Consequently the mapping (χ, κ) 7→ χκ maps WK,ε (χ0 ) × WK,ε (κ0 ) into WK,2ε (χ0 κ0 ). The continuity of the mapping χ 7→ χ−1 is verified in a similar way. (b) Let χ1 ∈ X(G1 ) and χ2 ∈ X(G2 ). Then, obviously, χ(t, s) = χ1 (t)χ2 (s) is a character of X(G1 × G2 ). Conversely, assume χ ∈ X(G1 × G2 ). Then χ1 (t) = χ(t, 0) and χ2 (s) = χ(0, s) are characters of G1 and G2 , respectively, and χ(t, s) = χ1 (t)χ2 (s). (c) Let q : G → G/H be the natural projection. Clearly, for any κ ∈ X(G/H) the mapping χ = κq is a continuous character which is equal to 1 on H. Conversely, it is easy to see (cf. 1.2.4) that any χ ∈ H⊥ induces the mapping κ which completes the diagram q

G −−−−→ G/H     χy κy C

C

to a commutative diagram. Clearly, κ is a (discontinuous) character. To prove that κ is continuous we observe that by the definition of the topology on G/H,

4.2. THE FOURIER TRANSFORM

223

the pre-image κ −1 (O) of an open set O is open if and only if (κq)−1 (O) = χ−1 (O) is open. Finally, we show that the mapping χ 7→ κ is a topological isomorphism. We observe that if K ⊆ G is compact then by 1.1.1 its natural projection q(K) to G/H is compact as well. Let us prove the converse: any compact set N ⊆ G/H is the image of a compact set K ⊆ G. Let U be a compact neighbourhood of the zero of G. We chooseSa finite covering of N by the sets of the form q(tk + U ). Clearly, the set K0 = (tk + U ) is compact and q(K0 ) ⊇ N . On the other hand, q −1 (N ) is closed. Therefore the set K = K0 ∩ q −1 (N ) is compact. It remains to observe that the isomorphism χ 7→ κ maps the neighbourhood WK,ε (χ0 ) = { χ ∈ H ⊥ : |χ(t) − χ0 (t)| < ε for all t ∈ K } onto the neighbourhood WN,ε (κ0 ) = { κ ∈ X(G/H) : |κ(s) − κ0 (s)| < ε for all s ∈ N }. The second statement follows immediately from the first and Pontrjagin’s theorem, see below 4.2.5. (We formulate assertion (c) here for completeness and shall not use it until 4.2.5.) ¤ Example. (a) We show that X(Z) ' T. Clearly, for any u ∈ U the function χu (n) = un ,

n ∈ Z,

is a character of Z. Conversely, assume χ : Z → U is a character. We set u = χ(1). Then, evidently, χ(n) = un for all n ∈ Z. Thus X(Z) is set-theoretically isomorphic to U. It is easy to see that the operations of multiplication on X(Z) and U agree with this isomorphism. Finally, every compact subset of Z is contained in a segment [−n, n], n ∈ N. Therefore the sets W[−n,n],ε (u0 ) = { u : |uk − uk0 | < ε, |k| ≤ n }, n ∈ N and ε > 0, form a neighbourhood base at u0 ∈ U ' X(Z). Obviously, these neighbourhoods induce the natural topology on U. Thus X(Z) is isomorphic to U as a topological group. We recall that U is isomorphic to the additive group T = R/2πZ. It is convenient to think that X(Z) ' T with the isomorphism χt (n) = eitn ,

t ∈ T.

(b) We show that X(R) ' R. Clearly, for any ω ∈ R the function χω (t) = eiωt ,

t ∈ R,

is a character of R. Conversely, let χ : R → U be a√(continuous) character. We take δ > 0 such that t ∈ [−δ, δ] implies |χ(t) − 1| < 2. Let us represent χ(δ) as eiωpwith |ω| < π/2. We observe that χ(δ/2) can be only √ one of two complex values iω/2 iω/2 of χ(δ), i.e., e or −e . Since |χ(δ/2) − 1| < 2 it must be the nearest to 1 p k value of χ(δ). Thus χ(δ/2) = eiω/2 . For the same reason χ(δ/2k ) = eiω/2 for all k k ∈ N. Consequently χ(δm/2k ) = eiωm/2 for all k, m ∈ Z. Hence by continuity χ(δt) = eiωt for all t ∈ R. Thus all characters of R have the form χ(t) = eiωt , ω ∈ R, i.e., X(R) is set-theoretically isomorphic to R. It is easy to see that the multiplication on X(R) and the addition on R agree with this isomorphism.

224

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Finally, every compact subset of R is contained in a segment [−n, n]. Therefore the sets W[−n,n],ε (ω0 ) = { ω : |eiωt − eiω0 t | < ε, |t| ≤ n } form a neighbourhood base at ω0 ∈ R. Clearly, these neighbourhoods induce the natural topology on R. (c) The isomorphism X(T) ' Z follows from (b) and assertion (c) of the proposition. We note only that if one represents T as (−π, π] then χn (t) = eint ,

t ∈ (−π, π].

(d) Next, we describe characters of the group Qd of rational numbers considered with the discrete topology. We follow [Hew, 25.5]. Let χ be a character of Qd . We set µ ¶ 1 , n ∈ N. un = χ n! It is easy to see that

(un+1 )n+1 = un ,

n ∈ N.

(1)

We that the sequence {un } determines χ completely. Indeed, we have ¡ mobserve ¢ χ n! = (un )m for any n ∈ N and m ∈ Z. Conversely, let { un ∈ U : n ∈ N } ¡m¢ be a sequence satisfying (1). Then the rule χ n! = (un )m defines a character of Qd . Note that the property (1) ensures the independence of this definition from m the representation of q ∈ Qd in the form q = n! . Clearly, the group operation on sequences {un } should be defined as the componentwise one. In conclusion we note that sometimes it is convenient to employ the description of the group X(Qd ) in terms of the group of a-adic numbers, see [Hew, 25.4] for details. (e) Now we turn to the description of the set Xb (R). We consider the group Rd as a linear space over the field Q of rational numbers. A family H ⊆ Rd is called linearly independent over the field Q if for any finite subcollection {h1 , h2 , . . . , hn } ⊆ H the equality r1 h1 + r2 h2 + · · · + rn hn = 0 with r1 , r2 , . . . , rn ∈ Q implies r1 = 0, r2 = 0, . . . , rn = 0. If, in addition, any t ∈ Rd can be represented in the form t = r1 h1 + r2 h2 + · · · + rn hn with hk ∈ H and rk ∈ Q, the set H is called a Hamel basis of Rd over Q. A standard application of Zorn’s lemma (see, e.g., [Kel]) shows that any linearly independent family can be extended to a Hamel basis. Let P H = { hα : α ∈ A } be a Hamel basis of Rd over Q. Using the representation t = α∈A rα hα , we identify Rd with the weak Cartesian product QA∗ , i.e., with the group of all families { rα ∈ Q : α ∈ A } with a finite number of non-zero members rα . Let { χα : α ∈ A } be an arbitrary family of characters of Qd , see example (d). It is easy to see that the rule {rα } 7→

Y

χα (rα )

α∈A

defines a character of Rd . And conversely, any character of Rd has this form. Thus X(Rd ) ' X(Qd )A . Cf. assertion (b) of the proposition. Usually (see, e.g., the proof of 4.5.7 below) there is no need to have a complete Hamel basis of Rd . If one is interested in the action of characters of Rd on a subset D of Rd , it is enough to have a Hamel basis for the span of D.

225

4.2. THE FOURIER TRANSFORM

(f) Let us discuss X(Qd /Z). According to assertion (c) of the proposition, X(Qd /Z) can be identified with the subgroup Z⊥ of X(Qd ). By example (d) we represent X(Qd ) as the collection of sequences {un ∈ U} satisfying (1). Clearly, Z⊥ consists of sequences {un } with u1 = 1. (g) Now we turn to the set X(T), where T = R/2πZ. We represent R as QA∗ d , see example (e). Clearly, the subgroup 2πZ is contained entirely in one of the factors Qd . Thus to obtain X(T) from the isomorphism X(R) ' X(Qd )A one must change the corresponding factor to X(2πQd /2πZ), see example (f). (h) From assertion (b) of the proposition we have X(Rn ) ' Rn . Namely, any ω = (ω1 , . . . , ωn ) ∈ Rn induces the character χω (t) = eihω,ti , where hω, ti =

Pn k=1

t ∈ Rn ,

ωk tk .

4.2.2. The character space of Mac . Let G be a locally compact abelian group, let X = X(G), and let µ ∈ M(G). We call the function Z χ(t) dµ(t),

µ ˆ(χ) =

χ ∈ X,

the Fourier transform of µ. Here the bar means the complex conjugation. The function Z µ ˇ(χ) = χ(t) dµ(t), χ ∈ X, is called the Fourier cotransform of µ. We stress that µ ˆ(χ) = µ ˇ(χ). Thus essentially µ ˆ and µ ˇ are equivalent transforms. We note that µ ˆ and µ ˇ are continuous functions on X. Indeed, if µ is compactly supported, this assertion follows from the definition of the topology on X; and if µ is arbitrary, it follows from 1.8.5. For g ∈ L1 (G, C) we define the Fourier transform and the Fourier cotransform by the rule gˆ = (gλ)∧ and gˇ = (gλ)∨ . Thus

Z gˆ(χ) =

Z χ(t)g(t) dλ(t)

and

gˇ(χ) =

χ(t)g(t) dλ(t).

Example. (a) Let G = R. We represent X(R) as R, see example (b) in 4.2.1. Then the Fourier transform of µ ∈ M(R, C) is the function Z µ ˆ(ω) =

e−iωt dµ(t),

ω ∈ R.

226

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

And for g ∈ L1 (R, C) the Fourier transform is the function Z e−iωt g(t) dt,

gˆ(ω) =

ω ∈ R.

(b) Let G = T. We represent T as (−π, π] and X(T) as Z. Then the Fourier transform of µ ∈ M(T) is the sequence Z

π

e−int dµ(t),

µ ˆn =

n ∈ Z.

−π

Usually the numbers µ ˆn are called the Fourier coefficients of µ. In particular, for g ∈ L1 (T, C) its Fourier transform (Fourier coefficients) is the sequence Z

π

gˆn =

e−int g(t) dt,

n ∈ Z.

−π

(c) Let G = Z. We represent T = X(Z) as (−π, π]. Then the Fourier transform of µ ∈ M(Z) is the function µ ˆ(ω) =

X

e−inω µ({n}),

ω ∈ (−π, π]

n∈Z

or, if we represent µ as the sequence c = {cn } ∈ l1 (Z, C), is the Fourier series cˆ(ω) =

X

e−inω cn ,

ω ∈ (−π, π].

n∈Z

We recall from 4.1.3 that M is a commutative algebra. We denote by X(M), X(Md ), and X(Mac ) the corresponding character spaces. Theorem ([Bou4 , ch. 2, §1, 1], [Nai, §31]). Let G be a locally compact abelian group. (a) For any χ ∈ X(G) the mapping ζχ defined by the rule Z ζχ (µ) = µ ˇ(χ) =

χ(t) dµ(t)

(2)

is a character of M and, in particular, is a character of Mac . Conversely, any non-zero character of Mac can be represented in the form (2). (b) The mapping χ 7→ ζχ establishes a topological isomorphism from X(G) onto X(Mac ). Remark. If G is non-discrete then there must exist characters of M which do not possess representation (2), cf. 4.2.12 below.

4.2. THE FOURIER TRANSFORM

227

Proof. (a) First we show that ζχ is a character of M. It is not evident only that ζχ (µ ∗ ν) = ζχ (µ)ζχ (ν) for all µ, ν ∈ M. By 4.1.3 we have Z ζχ (µ ∗ ν) = χ d(µ ∗ ν) ZZ = χ(t + s) d(µ ⊗ ν)(t, s) ZZ = χ(t)χ(s) d(µ ⊗ ν)(t, s) =µ ˇ(χ)ˇ ν (χ) = ζχ (µ)ζχ (ν). Next, we show that any non-zero character of the algebra Mac can be represented in the form (2) for some χ ∈ X(G). Let ζ : Mac → C be a non-zero character of Mac . We fix µ0 ∈ Mac such that ζ(µ0 ) 6= 0. Replacing µ0 to µ0 /ζ(µ0 ), without loss of generality we may assume that ζ(µ0 ) = 1. We define the function χ = χζ : G → C by the formula χ(h) = ζ(Sh µ0 ),

h ∈ G,

where Sh on Mac is defined by virtue of the isomorphism Mac ' L1 , i.e., by the rule Sh (f λ) = (Sh f )λ. Since for any g ∈ L1 the function h 7→ Sh g is continuous in the norm of L1 , the function χ is continuous. Furthermore, for any h1 , h2 ∈ G we have χ(h1 )χ(h2 ) = ζ(Sh1 µ0 )ζ(Sh2 µ0 ) ¡ ¢ = ζ (Sh1 µ0 ) ∗ (Sh2 µ0 ) ¡ ¢ = ζ (Sh1 +h2 µ0 ) ∗ µ0 = ζ(Sh1 +h2 µ0 ) ∗ ζ(µ0 ) = χ(h1 + h2 ). (Here we use the evident identity (Sh µ) ∗ ν = δh ∗ µ ∗ ν = µ ∗ Sh ν.) Thus χ is a character of G. Finally, we prove that ζ is induced by χ according to (2). We identify Mac with L1 , and the measure µ0 = g0 λ with g0 ∈ L1 . Then by 4.1.6(c), for any f ∈ L1 we have ζ(f ) = ζ(g0 )ζ(f ) = ζ(g0 ∗ f ) µZ ¶ ¡ ¢ =ζ Sh g0 f (h) dλ(h) Z ¡ ¢ = ζ Sh g0 f (h) dλ(h) Z = χ(h)f (h) dλ(h).

228

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

(b) First we note that for different characters of G the formula (2) defines different characters of Mac . Indeed, let χ1 and χ2 be different characters, i.e., different elements of C(G, C). Clearly, there exists a function g ∈ L1 such that R R χ1 g dλ 6= χ2 g dλ. So the mapping j : χ 7→ ζχ is a set-theoretic isomorphism from X(G) onto X(Mac ). We show that this isomorphism preserves topology. First we verify that the mapping j : X(G) → X(Mac ) is continuous at an arbitrary point χ0 ∈ X. For any ε > 0 and µ ∈ Mac we consider the neighbourhood V = Vµ,ε (ζχ0 ) = { ζχ ∈ X(Mac ) : |ζχ (µ) − ζχ0 (µ)| < ε } of ζχ0 in X(Mac ). (We recall that finite intersections of the sets Vµ,ε (ζχ0 ) form a neighbourhood base at χ0 , see 1.4.11.) We take η > 0 and according to 1.8.5 choose a measure µK concentrated on a compact set K ⊆ G such that kµ − µK k < η. After that, we consider the neighbourhood W = WK,η (χ0 ) = { χ : |χ(h) − χ0 (h)| < η for all h ∈ K } of χ0 in X(G). For χ ∈ WK,η (χ0 ) we have ¯Z ¯ ¯ ¯ |ζχ (µ) − ζχ0 (µ)| = ¯ (χ − χ0 ) dµ¯ ¯Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ≤ ¯ (χ − χ0 ) dµK ¯ + ¯ (χ − χ0 ) d(µ − µK )¯ Z Z ≤ |χ − χ0 | d|µK | + (|χ| + |χ0 |) d|µ − µK | K

≤ ηkµK k + 2kµ − µK k ≤ η(kµk + η) + 2η, which is less than ε provided η is small enough. Thus jW ⊆ V . A passage to finite intersections of neighbourhoods Vµ,ε is evident. Next, we show that j −1 : X(Mac ) → X(G) is continuous. Let χ0 ∈ X be a fixed point, and let ζχ0 be the corresponding element of X(Mac ). We show that j −1 is continuous at the point ζχ0 . Let a compact set K ⊆ G and ε > 0 be arbitrary. We consider the neighbourhood W = WK,ε (χ0 ) = { χ : |χ(h) − χ0 (h)| < ε for all h ∈ K } of χ0 in X(G). As in the proof of (a), we choose µ0 ∈ Mac so that ζχ0 (µ0 ) = 1.

(3)

We observe that for all χ ∈ X(G) (not only for χ0 ) χ(h) = ζχ (Sh µ0 )/ζχ (µ0 ),

h ∈ G,

(4)

provided ζχ (µ0 ) 6= 0, cf. the proof of (a). We recall that the function h 7→ Sh µ0 is continuous in the norm of Mac . Hence by 1.1.1 the set M = { Sh µ0 : h ∈ K } ∪ {µ0 } is compact in Mac . For any µ ∈ M we denote by Uµ the η/3-neighbourhood of µ. Thus we have kν − µkM < η/3 for all ν ∈ Uµ . Assume that Uµ1 , Uµ2 , . . . , Uµn is a finite

4.2. THE FOURIER TRANSFORM

229

covering of M . Let us consider the neighbourhood V = Vµ1 ,µ2 ,...,µn ,η/3 (χ0 ) = { ζχ ∈ X(Mac ) : |ζχ (µk ) − ζχ0 (µk )| < η/3, k = 1, 2, . . . , n } of ζχ0 . We show that j −1 V ⊆ W provided η is sufficiently small. Indeed, assume ζχ ∈ V and ν ∈ M . Then since ν ∈ Uµk for some k, we have |ζχ (ν) − ζχ0 (ν)| ≤ |ζχ (ν) − ζχ (µk )| + |ζχ (µk ) − ζχ0 (µk )| + |ζχ0 (µk ) − ζχ0 (ν)| < kχkC · kν − µk kM + η/3 + kχ0 kC · kν − µk kM < η. By the definition of M this estimate implies that for all ζχ ∈ V and h ∈ K, |ζχ (Sh µ0 ) − ζχ0 (Sh µ0 )| < η

and

|ζχ (µ0 ) − ζχ0 (µ0 )| < η.

In view of (3) and (4) from these estimates it follows that |χ(h) − χ0 (h)| < ε for all ζχ ∈ V and h ∈ K provided η is small enough.

¤

4.2.3. The character space of Md Corollary. Let G be a locally compact abelian group. (a) For any χ ∈ Xb (G) the mapping ζχ defined by rule (2) is a character of Md . And conversely, any character of Md is defined by P this rule. ∞ We note that if µ is represented in the form µ = n=1 an δhn with P∞ n=1 |an | < ∞ then ζχ (µ) =

∞ X

χ(hn )an .

n=1

(b) The mapping χ 7→ ζχ establishes a topological isomorphism from Xb (G) onto X(Md ). Proof. It suffices to recall that Md (G) = Mac (Gd ) and then refer to 4.2.2.

¤

4.2.4. The topology on X(G) Corollary. Let G be a locally compact abelian group. (a) The group X = X(G) is locally compact, too. (b) If G is discrete then X is compact. (c) If G is compact then X is discrete. Proof. (b) We recall from 1.4.11 that the space of characters of a commutative Banach algebra with a unit is compact. If G is discrete then Mac = M is an algebra with a unit. Therefore X(Mac ) ' X(G) is compact. (a) If a commutative Banach algebra has no unit, its space of characters is locally compact, see 1.4.11 again. Therefore X(Mac ) ' X(G) is locally compact. (c) Consider the neighbourhood√ WG,ε (1) = ¡{ χ ∈¢X : |χ(t)−1| < ε for all t ∈ G } n of the unit character 1 with ε < 3. Since χ(t) = χ(t + t + · · · + t), for all χ ∈ WG,ε (1) and n ∈ N we have |χ(t)n − 1| < ε, which implies that χ(t) = 1. Thus WG,ε (1) consists of one point, i.e., the topology of X is discrete. ¤

230

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

4.2.5. The Pontrjagin theorem. Let G be a locally compact abelian group. For any t ∈ G we consider the function %t (χ) = χ(t) acting from X = X(G) to U. It is trivial to verify that %t is a discontinuous character of the group X. The following theorem states essentially more. Theorem ([Bou4 , ch. 2, §1, 5], [Hew, theorem 24.8], [Nai, ¡ §31,¢ 6]). The mapping t → 7 % is an isomorphism of topological groups from X X(G) onto G. Thus ¡ ¢ t X X(G) ' G. Remark. For the groups R, T, and Z the theorem follows from the examples in 4.2.1 and does not need a proof. Example. If G = Ra × D × K then X ' Ra × K1 × D1 , where K1 = X(D) is a compact group, and D1 = X(K) is a discrete group. Thus this kind of groups is closed under the passage to the dual group. Cf. 1.6.1(b). 4.2.6. The invertibility in Md

P∞ Corollary. Let µ ∈ Md be represented in the form µ = n=1 an δhn with P∞ |a | < ∞. Then the spectrum σ(µ) of µ in the algebra M d is the set n=1 n ∞ nX

o χ(hn )an : χ ∈ Xb (G) .

n=1

In particular, the measure µ is invertible in Md if and only if the function χ 7→

∞ X

χ(hn )an

n=1

does not vanish on Xb (G). Proof. By 1.4.11 σ(µ) coincides with the image of the Gel0 fand transform ϑµ of µ. We recall that by definition the transform ϑµ : X(Md ) → C is the function ϑµ (ζ) =R ζ(µ), ζ ∈ X(M P∞d ). By 4.2.3 X(Md ) ' Xb (G). The representation ζχ (µ) = χ(t) dµ(t) = n=1 an χ(hn ) implies the formula for σ(µ). ¤ g 4.2.7. The invertibility in M ac . Let G be non-discrete. We recall from ] 4.1.10 that we denote by Mac the subalgebra of M consisting of all measures of the form αδ + µac , where α ∈ C and µac ∈ Mac . ] Corollary. Let µ ∈ M ac be represented in the form µ = αδ + gλ, α ∈ C and ] g ∈ L1 . Then the spectrum σ(µ) of µ in the algebra M ac is the set Sn α + gˇ(χ) = α + {α}

Z

o χ(t)g(t) dλ(t) : χ ∈ X(G) .

231

4.2. THE FOURIER TRANSFORM

] In particular, the measure µ is invertible in M ac if and only if α 6= 0 and the function Z χ 7→ α + gˇ(χ) = α + χ(t)g(t) dλ(t) does not vanish on X(G). ] Proof. The proof is similar to that of 4.2.6. We recall only that X(M ac ) is the one point compactification of X(Mac ); and the value α corresponds to the ] character ζ∞ ∈ X(M ac ) \ X(Mac ). ¤ 4.2.8. The character space of Md⊕ac . We recall that we denote the subalgebra Md ⊕ Mac by the symbol Md⊕ac . Let X and Y be sets. We call the union of disjoint copies of X and Y the sum of X and Y and denote it by X t Y . Formally the set X t Y can be defined to be { (x, 0) : x ∈ X } ∪ { (y, 1) : y ∈ Y }. Proposition. Let G be a non-discrete group. Then the space of characters of the algebra Md⊕ac is set-theoretically isomorphic to Xb (G) t X(G). The Gel0 fand transform of µ = µd + µac , µd ∈ Md and µac ∈ Mac , is equivalent to the function ½ ζχ (µd ) for χ ∈ Xb (G), ϑµ (χ) = ζχ (µd + µac ) for χ ∈ X(G). According to this assertion we may endow Xb tX with the topology of the space of characters of the algebra Md⊕ac . We need not discuss the topology on Xb t X in detail. See 4.2.11(b) for its main property. Proof. Let ζ be a character of the algebra Md⊕ac . The two cases are possible. First, we suppose that ζ is equal to zero on Mac . Then substantially ζ is a character of the algebra Md . Hence by 4.2.3 ζ is induced by some χ ∈ Xb (G). Now we suppose that ζ(µ0 ) 6= 0 for some µ0 ∈ Mac . Then by 4.2.2 there exists χ ∈ X(G) such that ζ coincides on Mac with ζχ , where ζχ is given by (2). We show that ζ coincides with ζχ on the whole of Md⊕ac . Let µd ∈ Md . Since both ζ and ζχ are characters of Md⊕ac we have ζ(µd ∗ µ0 ) = ζ(µd ) · ζ(µ0 )

and

ζχ (µd ∗ µ0 ) = ζχ (µd ) · ζχ (µ0 ).

We recall that ζ coincides with ζχ on Mac , and µ0 ∈ Mac . Since Mac is an ideal we have µd ∗ µ0 ∈ Mac , too. Therefore these equalities imply ζ(µd ) = ζχ (µd ). ¤ 4.2.9. The invertibility in Md⊕ac Corollary. Let G be a non-discrete group, and let µ ∈ Md⊕ac be represented in the form µ = µd + µac , where µd ∈ Md and µac ∈ Mac . Then the spectrum σ(µ) of µ in the algebra Md⊕ac is the set { ζχ (µd ) : χ ∈ Xb (G) } ∪ { ζχ (µd + µac ) : χ ∈ X(G) }. In particular, the measure µ is invertible in Md⊕ac if and only if the functions χ 7→ ζχ (µd )

and

χ 7→ ζχ (µd + µac )

do not vanish on Xb (G) and X(G), respectively. Proof. The proof is similar to the preceding ones.

¤

232

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

4.2.10. The Kronecker theorem Theorem. Let G be a non-discrete locally compact abelian group. (a) ([Hew, theorem 26.15]) Let H be a (not obligatorily closed) subgroup of G. Let κ : H → U be a (discontinuous) character of the group Hd , and let h0 , h1 , . . . , hk ∈ H be arbitrary elements. Then for any ε > 0 there exists a (continuous) character χ ∈ X(G) such that |χ(hm ) − κ(hm )| < ε

for all m = 0, . . . , k.

(b) The subset X ⊆ Xb is dense in Xb . Consequently the spectrum of µ ∈ Md is the closure of the set {µ ˇ(χ) : χ ∈ X(G) }. In particular, a measure µ ∈ Md is invertible if and only if the function χ 7→ µ ˇ(χ) is bounded away from zero on X(G), i.e., there exists ε > 0 such that |ˇ µ(χ)| > ε for all χ ∈ X(G). Proof. (a) First we consider the case where H = G. We recall that the topology on Xb is the topology of uniform convergence on compact subsets. Compact subsets of Gd are finite subsets. The assertion (with H = G) states that on a finite subset any character κ ∈ Xb can be approximated by a character χ ∈ X. Therefore this means that X is dense in Xb . We denote by X the closure of X in Xb . By the definition of the topology on Xb , X is a subgroup. We consider the quotient group Xb /X. By Pontrjagin’s theorem and 4.2.1(c) its dual group is the group { g ∈ Gd : χ(g) = 1 for all χ ∈ X }. But even χ(g) = 1 for all χ ∈ X implies g = 0. Thus the dual group of Xb /X is zero. Consequently so is Xb /X. Now let H be arbitrary. From the isomorphism X(Hd ) ' X(Gd )/H⊥ it follows (see 4.2.1) that the character κ can be represented as the restriction to Hd of a character κ e of the group Gd . Applying what has been proved to κ e , we complete the proof. (b) The density of X in Xb has been proved above. The other assertions follow from (in the notation of 4.2.6) P∞ 4.2.6. To prove them it suffices to observe that 0 ˇ(χ) for χ ∈ X, and that the Gel fand transform of µ is continn=1 χ(hn )an = µ uous on Xb . ¤ 4.2.11. A generalization of the Kronecker theorem Theorem. Let G be a non-discrete locally compact abelian group. (a) Let H be an arbitrary subgroup of G. Let κ : H → U be a (discontinuous) character of the group Hd , and let h1 , h2 , . . . , hk ∈ H be arbitrary elements, and let g1 , g2 , . . . , gl ∈ L1 (G, C) be arbitrary functions. Then for any ε > 0 there exists a (continuous) character χ ∈ X(G) such that |χ(hm ) − κ(hm )| < ε

for all m = 1, . . . , k

4.2. THE FOURIER TRANSFORM

233

and |ˇ gm (χ)| < ε

for all m = 1, . . . , l.

(b) The subset X ' ∅ t X ⊆ Xb t X is dense in Xb t X. Consequently the spectrum of µ ∈ Md⊕ac is the closure of the set {µ ˇ(χ) : χ ∈ X(G) }. In particular, the measure µ is invertible if and only if the function χ 7→ µ ˇ(χ) is bounded away from zero on X(G), i.e., there exists ε > 0 such that |ˇ µ(χ)| > ε for all χ ∈ X(G). Remark. Let us compare the embedding X ⊆ Xb t X in this theorem with the embedding X ⊆ Xb in the previous theorem. In 4.2.10(b) we interpreted χ ∈ X as a special element of Xb . On the contrary in this theorem we identify X with the second summand in Xb t X. Proof. (a) First of all we observe that without loss of generality we may assume that in the formulations of 4.2.10(a) the group H is the group H0 generated by h0 , h1 , . . . , hk (where h0 will be defined below) and in the assertion being proved the group H is the group H1 generated by h1 , h2 , . . . , hk . To simplify notation we set G = {g1 , g2 , . . . , gl }. We fix δ > 0 and pick a neighbourhood W of the zero of G such that Z |g(s − h0 ) − g(s)| dλ(s) < δ

for all h0 ∈ W and all g ∈ G.

(5)

We consider two cases. First, suppose that for any n ∈ N there exists a point h0 ∈ W such that h0 , 2h0 , . . . , nh0 ∈ / H1 . (We note that such a situation arise, e.g., if G = R.) We continue the character κ from the assumptions of the theorem from H1 to H0 (here h0 is defined as in the last paragraph). If there exists N ∈ N such that N h0 ∈ H1 , we denote by N the smallest of such N (clearly, N > n and therefore we may assume that N p is sufficiently large; here we use pthe assumption of the first case) and set κ(h0 ) = N κ(N h0 ), with the value of N κ(N h0 ) chosen so that |κ(h0 ) + 1| < δ.

(6)

If there is no N ∈ N such that N h0 ∈ H1 , we set κ(h0 ) = −1. Thus |κ(h0 ) + 1| = 0.

(60 )

Clearly, κ possesses a continuation to the whole of H0 . We choose a character χ in accordance with 4.2.10(a). From (6) or (60 ) we obtain the estimate |χ(h0 ) + 1| ≤ |χ(h0 ) − κ(h0 )| + |κ(h0 ) + 1| ≤ ε + δ. Then by

234

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

the shift invariance of λ, for all g ∈ G we have Z 2

Z χg dλ =

Z χg dλ +

Z = =

Z χg dλ +

Z

χ(s − h0 )g(s − h0 ) dλ(s) χ(s)χ(−h0 )g(s − h0 ) dλ(s)

¡ ¢ χ(s) χ(−h0 ) + 1 g(s) dλ(s) Z ¡ ¢ + χ(s)χ(−h0 ) g(s − h0 ) − g(s) dλ(s),

which, by |χ(h0 ) + 1| ≤ ε + δ and (5), implies ¯Z ¯ ¯ ¯ |ˇ g (χ)| = ¯¯ χg dλ¯¯ ≤

¢ 1¡ (ε + δ)kgkL1 + δ . 2

Now we turn to the second case: assume there exists n ∈ N such that for any point h0 ∈ V there exists N ≤ n such that N h0 ∈ H1 . (We note that such a situation arise, e.g., if G = (Z2 )N .) We take a neighbourhood V of the zero of G such that nV ⊆ W (here nV = V + V + · · · + V ). Clearly, from (5) we have Z |g(s − jh0 ) − g(s)| dλ(s) < δ

for h0 ∈ V , g ∈ G, and j = 1, 2, . . . , n.

(7)

We show that there must exist h0 ∈ V \ H1 . Indeed, the set H1 consists of the Pk elements of the form m=1 αk hk , αk ∈ Z. Therefore H1 is countable or finite. Since G is not discrete λ(H1 ) = 0. But the Haar measure of an open set is not zero. Hence V \ H1 has non-zero measure and consequently is not empty. p So let h0 ∈ V \ H1 . Again, we set κ(h0 ) = N κ(N h0 ), where N ∈ N is the smallest number for which N h0 ∈ H1 (evidently, N > 1) and the complex value of the root is chosen so that ¡ ¢2 ¡ ¢N κ(h0 ) + κ(h0 ) + · · · + κ(h0 ) = 0. Clearly, κ possesses a continuation to the whole of H0 . We choose a character χ by 4.2.10(a). We note that from |χ(h0 ) − κ(h0 )| < ε it ¯¡ ¢j ¯ ¢j ¡ follows that ¯ χ(h0 ) − κ(h0 ) ¯ < jε for all j ∈ N, which implies the estimate ¯ ¡ ¢ ¯ ¡ ¢ ¯χ(h0 ) + χ(h0 ) 2 + · · · + χ(h0 ) N ¯ < N (N +1) ε. By the shift invariance of λ, for 2

4.2. THE FOURIER TRANSFORM

235

all g ∈ G we have Z N Z X N χg dλ = χ(s − jh0 )g(s − jh0 ) dλ(s) j=1

=

N Z X

χ(s)χ(−jh0 )g(s − jh0 ) dλ(s)

j=1

=

N X

Z χ(−jh0 )

χ(s)g(s − jh0 ) dλ(s)

j=1

=

N ³X

´Z χ(−jh0 ) χg dλ

j=1

+

N X

Z χ(−jh0 )

¡ ¢ χ(s) g(s − jh0 ) − g(s) dλ(s),

j=1

¯ ¡ ¢2 ¡ ¢N ¯ which, by ¯χ(h0 ) + χ(h0 ) + · · · + χ(h0 ) ¯ < N (N2+1) ε and (7), implies ¯ ¯Z ¯ ¯ |ˇ g (χ)| = ¯¯ χg dλ¯¯ µ ¶ 1 N (N + 1) ≤ εkgkL1 + N δ . N 2 It remains to recall that ε is independent from N and N < n. (b) It is necessary to show that the closure of X in Xb t X contains Xb . We take κ ∈ Xb . We show that every neighbourhood of κ contains a character in X. From the definition of the topology on the character space of a commutative Banach algebra (see 1.4.11) it follows that any neighbourhood of κ contains a finite intersection of the sets of the form Vµ,η = { χ ∈ Xb t X : |ϑµ (χ) − ϑµ (κ)| < η }, where µ ∈ Md⊕ac and η > 0. P∞ We take arbitrary µ , µ , . . . , µ ∈ M . Assume µ = 1 2 l d⊕ac m k=1 akm δhkm + gm λ P∞ with akm ∈ C, k=1 |akm | < ∞, and gm ∈ L1 . It follows immediately from (a) Tl that m=1 Vµm ,ηm contains a continuous character. ¤ g 4.2.12. A remark on the character space of M sc . In this subsection we g show that an analogue of 4.2.10(b) and 4.2.11(b) does not hold for M sc . Namely, g the spectrum of a measure ξ ∈ Msc can be essentially wider than the image of its ˇ Fourier cotransform ξ. R For any µ ∈ M we define the conjugate measure µ∗ by the rule x dµ∗ = R x(−t) dµ(t). A measure µ ∈ M is called Hermitian if µ∗ = µ. Clearly, the Fourier cotransform µ ˇ of a Hermitian measure µ is a real-valued function. Theorem ([Will1 , theorem 2]). Let G be a non-discrete group. Then there ˇ exists a measure ξ ∈ Msc such that ξ(χ) ∈ [−1, 1] for all χ ∈ X(G), but i ∈ σM (ξ). f Thus X(G) is not dense in X(M). The measure ξ is positive and Hermitian, and is supported in [−1, 1].

236

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Proof. We give a proof only for the case G = R. A proof for an arbitrary G can be found in [Will1 ]. Our proof is a variant of that of [Will1 ]. We break the argument into six steps. (i) The group R can be considered as a linear space over the field Q of rational numbers. Clearly, the dimension of PnR over Q is infinite. Therefore there exists a sequence { hk : k ∈ N } such that k=1 qk hk = 0, n ∈ N and qk ∈ Q, implies that all qk are equal to zero. In particular, it follows that the subgroup in R generated by h1 , h2 , . . . , hn is naturally isomorphic to Zn . Obviously, without loss of generality we may assume that hk > 0 and hk decreases as fast as desired. Let Hnr denote the set of all numbers of the form Pn r k=−n lk hk with lk ∈ Z, |lk | ≤ r. Let dn be the smallest distance between elements of Hnr . Clearly, drn ≥ dnn for r < n. We require hn+1 < dnn /(4n). Then, in particular, hn+1 < hn /(4n) and ∞ X

hk < hn+1

k=n+1

< = (ii) We set νk =

dnn n dnn 3n

∞ X

1

k=n+1 ∞ X k=n+1

4k−(n+1) 1

4k−n

.

1 (δ−hk + 2δ0 + δhk ), 4

k ∈ N.

For any µ ∈ M we designate by µm the m-th power µ ∗ µ ∗ · · · ∗ µ of µ. Clearly, (νk )m =

l=m 1 X (2m)! δlh . 4m (m + l)!(m − l)! k l=−m

For all n ∈ N we consider the measure ξn = ν1 ∗ ν2 ∗ · · · ∗ νn and its powers (ξn )m = (ν1 )m ∗ (ν2 )m ∗ · · · ∗ (νn )m , m = 1, 2, . . . . To simplify the notation, without loss of generality we assume that hk < 2−k for all k ∈ Z. We show that the sequence ξn = ν1 ∗ν2 ∗· · ·∗νn has a limit in the ∗-weak topology (see 1.1.7) of M = C00 . Indeed, we observe that νn+1 ∗ νn+2 ∗ · · · ∗ νn+l is concentrated on the segment [−2−n , 2−n ]. Employing this it is easy to see that for any x ∈ C0 the convolution of νn+1 ∗ νn+2 ∗ · · · ∗ νn+l with x is Rclose enough to x in the norm of C0 provided n is sufficiently large. Therefore x dξn has a limit as n → ∞ for all x ∈ C0 . Let ξ denote the ∗-weak limit of ξn . Note that ξ is supported in [−1, 1] because hk < 2−k . In a similar way one proves that (ξn )m converges to ξ m in the ∗-weak sense.

4.2. THE FOURIER TRANSFORM

237

(iii) We show that if n is much bigger than r then the powers (ξn )1 , (ξn )2 , . . . , (ξn )r are almost mutually singular. Let us fix r and ε > 0, and take n > r. Clearly, (ξn )m is concentrated on Hnm , see (i) for the definition of Hnm . We observe that (ξn )m (Hnm−1 ) = (1 − 2/4m )n ≤ (1 − 2/4r )n

for m ≤ r

¡ ¢ here (ξn )m (Hnm−1 ) means the (ξn )m -measure of the set Hnm−1 . We choose N > r so that (1 − 2/4r )N < ε and consider the measures m−1 (πN )(m) (E) = (ξN )m (E \ HN ).

It is easy to see that k(πN )(m) − (ξN )m k < ε.

(8)

m−1 m \ HN . Hence the meaClearly, the measure (πN )(m) is concentrated on HN (m) sures (πN ) , m = 1, 2, . . . , r, are mutually singular. We note that the distances between the supports of (πN )(m) , m = 1, 2, . . . , r, are less than or equal to dN N. (iv) We show that for all r ∈ N and α1 , α2 , . . . , αr ∈ C r r °X ° X ° m° αm ξ ° = |αm |. ° m=1

(9)

m=1

We recall that µ∗1 = kµk·1 for a positive measure µ; here 1 is the unity function. Clearly, νk , ξn , (ξn )m , ξ, and ξ m are positive measures. Since kνk k = 1 it follows m m that Prk(ξn ) k = 1, and kξk = 1, and kξ k = 1. Consequently Pr kξn k = m1, and k m=1 αm ξ k ≤ m=1 |αm |. Let us prove the opposite inequality. We take an arbitrary ε > 0 and choose N as in (iii). We consider the measures (m)

ηN,l = (πN )(m) ∗ (νN +1 ∗ νN +2 ∗ · · · ∗ νN +l )m . (m)

Clearly (cf. (ii)), for fixed m (and N ) the sequence ηN,l converges to some measure (m)

η (m) as l → ∞. By (8) kηN,l − (ξN +l )m k < ε. We recall (see 1.1.9) that the unit ball of M is ∗-weak closed. Therefore the last inequality implies kη (m) − ξ m k ≤ ε. By (i)

∞ X k=N +1

hk
0 such that |χ(t) − 1| < 1 for all t ∈ [0, δ]. ¡ character. ¢ We set τ = − ln χ(δ) /δ, where the value of ln χ(δ) is chosen with the imaginary part situated in [−π/2, π/2]. Clearly, χ(δ) = e−τ δ and τ ∈ C+ . We observe that χ(δ/2) is either e−τ δ/2 or −e−τ δ/2 . Since |χ(δ/2) − 1| < 1 we k conclude that χ(δ/2) = e−τ δ/2 . For the same reason χ(δ/2k ) = e−τ δ/2 for all k k ∈ N. Consequently χ(δm/2k ) = e−τ δm/2 for all k, m ∈ N. Hence by continuity χ(δt) = e−τ δt for all t ∈ R+ . Thus all characters of R+ have the form (1), i.e., X+ is set-theoretically isomorphic to C+ . (b) Clearly, formula (2) defines a non-trivial character of R+ d . We prove the + converse. Assume χ is a non-trivial character of Rd . We consider the real-valued function κ(t) = |χ(t)|, t ∈ R+ . It is easy to verify that κ is a character of R+ d, too. We take t, s > 0. For any natural n and m such that t/s ≤ m/n we have + sm − tn ∈ R+ . Since κ takes its values κ(ms − nt) ≤ 1, which ¡ ¢inn U ,¡this ¢implies m can be rewritten equivalently as κ(t) ≤ κ(s) or ln κ(t)/ln κ(s) ≤ m/n. Taking in the last estimate the infimum over all natural n and m such that t/s ≤ m/n we obtain ln κ(t)/ln κ(s) ≤ t/s. Substituting of t for s and vice versa gives ln κ(s)/ln κ(t) ≤ s/t, which implies ln κ(t)/ln κ(s) = t/s. Consequently ln κ(t)/t is independent of t. Thus there exists γ such that κ(t) = e−γt for all t ∈ R+ . Since κ is bounded we may conclude that γ ≥ 0. Now, it suffices to observe that the function χ0 (t) = χ(t)/κ(t), t ∈ R+ , takes its values in U and therefore possesses a continuation to a character of Rd . (c) is evident, cf. the example 4.2.1(a). ¤

4.3. THE LAPLACE TRANSFORM

241

+ Remark. We shall use X+ , X+ b , and X(Z ) only for the calculation of spectra of measures in M+ and causal spectra of corresponding convolution operators. So we need not discuss semi-group operations and natural (see also 4.3.10) topologies on X+ and X+ b . In the previous section the situation was different because the theory of dual groups is richer than the theory of dual semi-groups. For example, the formulation of Pontrjagin’s theorem employs the group properties and topological properties of X(G). + 4.3.2. The character space of M+ ac . Let µ ∈ M . The function

Z

∞

µ e(τ ) =

e−τ t dµ(t),

τ ∈ C+ ,

0

is called the Laplace transform of µ. Because of 4.3.1 µ e can be defined equivalently as the function Z χ(t) dµ(t), χ ∈ X+ . µ e(χ) = R+

f or, in For g ∈ L1 (R+ , C) we define the Laplace transform of g by the rule ge = gλ detailed form, Z ∞ ge(τ ) = e−τ t g(t) dt, τ ∈ C+ . 0

We recall from 4.1.11 that M+ is a commutative algebra. We denote by + X(M+ ), X(M+ d ), and X(Mac ) the corresponding character spaces. Theorem ([Gel, §17]). For any τ ∈ C+ the mapping ζτ defined by the rule Z

∞

ζτ (µ) = µ e(τ ) =

e−τ t dµ(t)

(3)

0

is a character of the algebra M+ and, in particular, is a character of M+ ac . And conversely, any non-zero character of the algebra M+ is given by the rule (3). ac + + Thus X(Mac ) can be identified with C . Proof. (Cf. 4.2.2). First we show that µ] ∗ ν(χ) = µ e(χ)e ν (χ) for all µ, ν ∈ M+ and χ ∈ X+ . By 4.3.1(a) the two cases are possible. First: assume γ = 0. Then χ is the restriction of a character of X(R) to R and the formula follows from 4.2.2. The second case, when γ > 0, can be considered similarly to that in 4.2.2. But we would like to demonstrate a different idea. ¡ ¢ R∞ Let µ ∈ M+ . Since the operator Tµ x (t) = 0 x(t − s) dµ(s) is causal it induces naturally (see 2.1.1) the operator (Tµ )−∞/0 acting on (C0 )−∞/0 ' C0 (−∞, 0]. Assume that χ ∈ X+ possesses the representation (2) with γ > 0. We denote by xχ the function xχ (t) = χ(−t). Since γ > 0 we have xχ ∈ (C0 )−∞/0 . Clearly, the (one-dimensional) span of xχ is an invariant subspace for (Tµ )−∞/0 , and (Tµ )−∞/0 xχ = µ e(χ) · xχ , which trivially implies that µ] ∗ ν(χ) = µ e(χ)e ν (χ) for all µ, ν ∈ M+ .

242

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Thus formula (3) defines a character of M+ . Next, we show that any character of the algebra M+ ac can be represented in the + form (3) with some τ ∈ C . + + Let ζ : M+ ac → C be a non-zero character of Mac . We choose µ0 ∈ Mac such that ζ(µ0 ) 6= 0. Changing µ0 to µ0 /ζ(µ0 ), without loss of generality we may assume that ζ(µ0 ) = 1. We denote by L1 [0, +∞) = (L1 )0 the subspace of L1 (R, C) consisting of all functions vanishing on (−∞, 0) (see 2.1.1). We define the function χ = χζ by the rule χ(h) = ζ(Sh µ0 ), h ∈ R+ , + where Sh on M+ ac is defined by virtue of the isomorphism Mac ' L1 [0, +∞), i.e., by the rule Sh (f λ) = (Sh f )λ. Since for any g ∈ L1 the mapping h 7→ Sh g is continuous in the norm of L1 , the function χ is continuous. For any h1 , h2 ∈ R+ we have

χ(h1 )χ(h2 ) = ζ(Sh1 µ0 )ζ(Sh2 µ0 ) ¡ ¢ = ζ (Sh1 µ0 ) ∗ (Sh2 µ0 ) ¡ ¢ = ζ (Sh1 +h2 µ0 ) ∗ µ0 = ζ(Sh1 +h2 µ0 ) ∗ ζ(µ0 ) = χ(h1 + h2 ). Since kζk ≤ 1 the function χ takes its values in U+ . Thus χ ∈ X+ . Finally, we prove that ζ is induced by χ. We identify M+ ac with the subspace L1 [0, +∞) of L1 , and we identify the measure µ0 = g0 λ with g0 ∈ L1 [0, +∞). Then for any f ∈ L1 [0, +∞), by 4.1.6(c) we have ζ(f ) = ζ(g0 )ζ(f ) = ζ(g0 ∗ f ) ³Z ∞ ¡ ´ ¢ =ζ Sh g0 f (h) dh Z ∞0 ¡ ¢ = ζ Sh g0 f (h) dh Z0 ∞ = χ(h)f (h) dh, 0

which completes the proof. ¤ g + 4.3.3. The invertibility in M ac + ] Corollary. Let µ ∈ M ac be represented in the form µ = αδ + gλ, α ∈ C and + ] g ∈ L1 [0, +∞). Then the spectrum of µ in the algebra M ac is the set

{α}

Sn

Z α + ge(τ ) = α + 0

∞

o e−τ t g(t) dt : τ ∈ C+ .

243

4.3. THE LAPLACE TRANSFORM

+ ] In particular, the measure µ is invertible in M ac if and only if α 6= 0 and the function Z ∞ τ 7→ α + ge(τ ) = α + e−τ t g(t) dt 0

does not vanish on C+ . Proof. The proof is based on 1.4.11 and similar to that of 4.2.7. ¤ 4.3.4. The character space of M+ d Theorem. For any χ ∈ X+ b the mapping ζχ defined by the rule Z

∞

ζχ (µ) =

χ(t) dµ(t)

(4)

0

is a character of M+ d . We note that if µ is represented in the form µ = P∞ with n=1 |an | < ∞ and hn ≥ 0 then ζχ (µ) =

∞ X

P∞ n=1

an δhn

χ(hn )an .

n=1

Conversely, any character of M+ d is defined by this rule. Thus X(M+ ) can be identified with X+ d b . Proof. The proof is similar¡to that of ¢4.3.2. Let µ ∈ M+ d . We consider the operator of convolution Tµ ∈ B l∞ (Rd , C) induced by µ, see 1.6.8. Since Tµ is causal it induces naturally (see 2.1.1) the operator (Tµ )−∞/0 acting on (l∞ )−∞/0 . Let χ ∈ X+ b . We denote the natural projection of χ into (l∞ )−∞/0 by the symbol χ−∞/0 . Clearly, the span of χ−∞/0 is an invariant subspace for (Tµ )−∞/0 and (Tµ )−∞/0 χ−∞/0 = ζχ (µ) · χ−∞/0 , which trivially implies ζχ (µ ∗ ν) = ζχ (µ)ζχ (ν) for all µ, ν ∈ M+ d. + Thus formula (4) defines a character of Md . Next, we show that any character of the algebra M+ d can be represented in the form (4) with some τ ∈ C+ . + + Let ζ : M+ d → C be a character of Md . We define the function χ = χζ : Rd → C by the rule χ(h) = ζ(δh ), h ∈ R+ d. For any h1 , h2 ∈ R+ d we have χ(h1 )χ(h2 ) = ζ(δh1 )ζ(δh2 ) = ζ(δh1 ∗ δh2 ) = ζ(δh1 +h2 ) = χ(h1 + h2 ).

244

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Since kζk ≤ 1 the function χ takes its values in U+ . Thus χ ∈ X(R+ d ). Finally, we prove that ζ is induced by χ. We denote by ζχ the character induced + by χ. By definition ζ(δh ) = ζχ (δh ) for all h ∈ R+ d . Since Md is the closure of the + span of the family { δh : h ∈ Rd }, and both ζ and ζχ are continuous they coincide on the whole of R+ d. It remains to note that for different characters of R+ d the formula (4) defines + different characters of Md . Indeed, let χ1 and χ2 be different characters, i.e., + different R ∞C(Rd , C). Clearly, there exists a function g ∈ l1 [0, +∞) such R ∞ elements of that 0 χ1 g dλ 6= 0 χ2 g dλ. ¤ 4.3.5. The invertibility in M+ d

P∞ Corollary. Let µ ∈ M+ n=1 an δhn with d be represented in the form µ = P∞ + n=1 |an | < ∞ and hn ≥ 0. Then the spectrum of µ in the algebra Md is the set ∞ nX o χ(hn )an : χ ∈ X+ . b n=1

In particular, the measure µ is invertible in M+ d if and only if the function χ 7→

∞ X

χ(hn )an

n=1

does not vanish on X+ b . Proof. The proof is evident. ¤ 4.3.6. The character space of M+ (Z). We denote by M+ = M+ (Z) = M+ (Z, C) the subset of M(Z) consisting of all measures concentrated on Z+ . One can identify M+ (Z) with l1 (Z+ , C). Clearly, M+ (Z) is a subalgebra of M(Z). Let µ ∈ M+ (Z). The function ∞ X

µ e(u) =

un µ({n}),

u ∈ U+ ,

n=0

is an analogue of the Laplace transform. It is usually called the z-transform of µ (because the letter ‘z’ is usually used instead of our ‘u’). If one represents µ as the sequence {an = µ({n}) : n ∈ Z+ } ∈ l1 then µ e takes the form µ e(u) =

∞ X

un an ,

u ∈ U+ .

n=0 +

Theorem. For any u ∈ U the mapping ζu defined by the rule ∞ X ζu (µ) = un µ({n}) n=0 n

(we recall that we interpret u as 1 if u = 0 and n = 0) is a character of the algebra M+ (Z). And conversely, any character of the algebra M+ (Z) is defined by this rule. Thus X(M+ (Z)) can be identified with Z+ . Proof. The proof is a simplified version of that of 4.3.4.

¤

4.3. THE LAPLACE TRANSFORM

245

4.3.7. The invertibility in M+ (Z) Corollary. Let us identify µ ∈ M+ (Z) with the sequence {an : n ∈ Z+ } ∈ l1 . Then the spectrum of µ in the algebra M+ (Z) is the set ∞ nX o un an : u ∈ U+ . n=0

In particular, the measure µ is invertible in M+ d if and only if the function ∞ X un an u 7→ n=0 +

does not vanish on U . Proof. The proof is evident. 4.3.8. The character space of M+ d⊕ac . We recall from 4.1.12 that the sub+ + algebra Md ⊕ Mac is denoted by the symbol M+ d⊕ac . Proposition. The space of characters of the algebra M+ d⊕ac is set-theoretically + + 0 isomorphic to Xb t X . The Gel fand transform of µ = µd + µac , where µd ∈ M+ d and µac ∈ M+ ac , is equivalent to the function ½ ζχ (µd ) for χ ∈ X+ b , ϑµ (χ) = ζχ (µd + µac ) for χ ∈ X+ b . + Proof. Let ζ be a character of the algebra M+ d ⊕ Mac . The two cases are possible. First, assume ζ is equal to zero on M+ ac . Then + essentially ζ is a character of the algebra Md and, by 4.3.4, ζ is induced by some χ ∈ X+ b . Second, assume that ζ(µ0 ) 6= 0 for some µ0 ∈ M+ ac . Then by 4.3.2 there exists + + χ ∈ X such that ζ coincides with ζχ on Mac , where ζχ is given by (3). We show that (3) holds for all µ ∈ M+ d⊕ac . + Let µd ∈ Md . Since both ζ and ζχ are characters of M+ d⊕ac we have

ζ(µd ∗ µ0 ) = ζ(µd ) · ζ(µ0 )

and

ζχ (µd ∗ µ0 ) = ζχ (µd ) · ζχ (µ0 ).

+ + We recall that ζ coincides with ζχ on M+ ac . Since Mac is an ideal µd ∗ µ0 ∈ Mac . Therefore these equalities imply ζ(µd ) = ζχ (µd ). ¤

4.3.9. The invertibility in M+ d⊕ac Corollary. Let µ ∈ M+ d⊕ac be represented in the form µ = µd + µac , where µd ∈ Md and µac ∈ Mac . Then the spectrum of µ in the algebra M+ d⊕ac is the set + { ζχ (µd ) : χ ∈ X+ b } ∪ { ζχ (µd + µac ) : χ ∈ X }. In particular, the measure µ is invertible in M+ d⊕ac if and only if the functions χ 7→ ζχ (µd ) do not vanish on

X+ b

and

and X+ , respectively.

Proof. The proof is evident. ¤

χ 7→ ζχ (µd + µac )

246

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

+ + 4.3.10. X+ is dense in X+ b and in Xb t X

Proposition. + + (a) The subset X+ ⊆ X+ b is dense in Xb endowed with the topology of X(Md ), see 4.3.4. Consequently the spectrum of µ ∈ M+ d is the closure of the set {µ e(χ) : χ ∈ X+ }. In particular, the measure µ is invertible if and only if the function χ 7→ µ e(χ) is bounded away from zero on X+ , i.e., there exists ε > 0 such that |e µ(χ)| > ε for all χ ∈ X+ . + + (b) The subset X+ ' ∅ t X+ ⊆ X+ is dense in X+ endowed b t X b t X with the topology of X(M+ ), see 4.3.8. Consequently the spectrum of d⊕ac + µ ∈ Md⊕ac is the closure of the set {µ e(χ) : χ ∈ X+ }. In particular, the measure µ is invertible if and only if the function χ 7→ µ e(χ) is bounded away from zero on X+ . Proof. Let us, for example, prove (b). The proof of (a) is similar. + + It is necessary to show that the closure of X+ in X+ b tX contains Xb . Let us fix a non-trivial character κ ∈ X+ b . We show that every neighbourhood of κ contains a character χ ∈ X+ . From the definition of the topology of the character space of a commutative Banach algebra (see 1.4.11) it follows that any neighbourhood of + κ contains a neighbourhood of the form V + = Vµ+1 ,µ2 ,...,µn ,η (κ) = { χ ∈ X+ b tX : |ϑµk (χ) − ϑµk (κ)| < η, k = 1, 2, . . . , n }, where µk ∈ M+ , k = 1, 2, . . . , n, and R ∞ d⊕ac η > 0. We recall from 4.3.4 that ϑµ (κ) = ζκ (µd ) = 0 κ dµd since κ ∈ X+ b , and R∞ + ϑµ (χ) = ζχ (µ) = 0 χ dµ if χ ∈ X . (We are looking just for such a χ.) By 4.3.1 we represent κ as κ = eκ0 , where κ0 ∈ Xb (R) and e(t) = e−γt , t ∈ R+ , for some γ ≥ 0. Next, for a measure µ on R+ or on R+ d we define the measure eµ R R by the usual rule x d(eµ) = ex dµ. We consider the neighbourhood V = Vµ1 ,...,µn ,η (κ0 ) = { χ ∈ Xb t X : |ϑeµk (χ) − ϑeµk (κ0 )| < η, k = 1, . . . , n } of κ0 in X(Md ⊕ Mac ) ' Xb (R) t X(R) with the same µk and e as above. By the Kronecker theorem V contains a continuous character χ0 ∈ X(R). We show that the character χ+ = eχ0 belongs to V + . Indeed, we have |ϑµk (χ+ ) − ϑµk (κ)| = |ϑµk (eχ0 ) − ϑµk (eκ0 )| = |ϑeµk (χ0 ) − ϑeµk (κ0 )| < η, which implies that χ+ ∈ V + . Now we consider the trivial character χ∞ ∈ X+ , see 4.3.1 for the definition. It is approximated by the continuous characters χγ (t) = e−γt , t ∈ R, because ϑµ (χγ ) = R +∞ −γt e dµ converges to ϑµ (χ∞ ) = µ({0}) as γ → +∞ for all µ ∈ M+ d⊕ac . ¤ 0

4.4. CONVOLUTION OPERATORS

247

g + 4.3.11. A remark on the character space of M sc . This subsection is g + similar to 4.2.12. Here we show that an analogue of 4.3.10 does not hold for M sc . g + Namely, the spectrum of a measure ξ ∈ M sc can be wider than the image of its e Laplace transform ξ. Theorem ([Will2 ]). There exists a measure ν ∈ M+ sc concentrated on [1, 5] such that |e ν (τ )| ≤ 1/3 for τ ∈ C+ , but 1 ∈ σM (ν). (We recall from 1.4.5 that σM (ν) ⊆ σM+ (ν).) Thus X+ is not dense in X(M+ ). Proof. Let ξ be the measure from 4.2.12. We set ξ 2 to be ξ ∗ ξ. Since a character of a commutative Banach algebra is a morphism into the algebra C, and the mapping µ 7→ µ ˇ(χ) is a character for all χ ∈ X(R), we have ξb2 (χ) ∈ [0, 1] for all χ ∈ X(R). We recall that the spectrum of a measure µ coincides with the image of its Gel0 fand transform ϑµ in the algebra M. Therefore there exists a character ζ of M such that ζ(ξ) = i. Clearly, ζ(ξ 2 ) = −1. Let δ3 be the measure defined by the rule hx, δ3 i = x(3). Since kδ3 k = 1 and kδ−3 k = 1, by 1.4.2 and 1.4.3 σ(δ3 ) ⊆ {z : |z| = 1}. Consequently |ζ(δ3 )| = 1. We observe that ξ 2 is concentrated on [−2, 2]. Therefore δ3 ∗ ξ 2 is concentrated on [1, 5] and consequently δ3 ∗ ξ 2 ∈ M+ . We set µ ¶ 1 δ3 ∗ ξ 2 ν= 1−2 . 3 ζ(δ3 ) For the same reasons as above, we have |ˇ ν (χ)| ≤ 1/3 for all χ ∈ X(R). On the other hand, it is easy to see that ζ(ν) = 1. Hence 1 ∈ σ(ν). To complete the proof we show that |e ν (τ )| ≤ 1/3 for all τ ∈ C+ , too. We observe that νˇ can be interpreted as the restriction of νe to the imaginary axis iR = {τ : Re τ = 0}. Therefore the statement follows from the maximum principle for holomorphic functions. ¤

4.4. Convolution operators Previously we have considered operators of convolution Tµ x = µ ∗ x on the spaces C0 and L1 . They were necessary as an auxiliary tool for the discussion of convolution in the algebra M. Now we proceed to the detailed consideration of convolution operators Tµ on other functional spaces. The vector-valued case is essentially based on the theory of tensor products, see §1.7. 4.4.1. Convolution operators on Lp , Lpq , and Cq (scalar case). We recall that in the definition of the spaces Lpq and Cq the representation of the locally compact abelian group G in the form Rc × V and the subgroup K are assumed to be fixed, see 1.6.1. Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let G be a locally compact abelian group, and let µ ∈ M(G, C). (a) For any x ∈ Lp (G, C) the function s 7→ x(t − s) is µ-integrable for λ-almost all t ∈ G if p 6= 0, ∞, and for λ-locally almost all t ∈ G if p = 0, ∞. In all

248

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

cases the formula ¡

¢ Tµ x (t) =

Z x(t − s) dµ(s) Z

=

(1) x(s) dµ(t − s)

¡ ¢ defines an operator Tµ ∈ B Lp (G, C) with kTµ k ≤ kµk. (b) For any x ∈ Lpq (G, C) the function s 7→ x(t−s) is µ-integrable for λ-locally ¡ ¢ almost all t ∈ G. Formula (1) defines an operator Tµ ∈ s Lpq (G, C) with kTµ k ≤ 2c kµk. See 1.6.8 for the definition of s. (c) For any x ∈ Cq (G, C) the function s 7→ x(t−s) is µ-integrable for all t ∈ G. ¡ ¢ Formula (1) defines an operator Tµ ∈ s Cq (G, C) with kTµ k ≤ 2c kµk. Proof. (a) The case p = 1 has been considered in 4.1.6. Suppose p 6= 1. Let p0 be the conjugate index, andRxR∈ Lp and y ∈ Lp0 . We show that Fubini’s theorem is applicable to the integral x(t − s)y(t) d (λ ⊗ µ)(t, s). 0 Indeed, since p 6= 0, ∞, by 1.6.2(e) the function y is equal to zero outside a σ-compact set. On the other hand, |µ|(G) < ∞ since µ is bounded. Consequently the function (t, s) 7→ x(t−s)y(t) is equal to zero outside a σ-finite set. Furthermore, by 4.1.5(b) the function (t, s) 7→ x(t − s)y(t) is λ ⊗ µ-measurable. And, finally, Z µZ

¶ Z |x(t − s)y(t)| dλ(t) d|µ|(s) ≤ kxkLp · kykLp0 d|µ| = kxkLp · kykLp0 · kµk.

Thus the assumptions of Fubini’s theorem are satisfied. By Fubini’s theorem the function s 7→ x(t − s)y(t) is µ-integrable for λ-almost all t and ¯Z µZ ¯ ¶ ¯ ¯ ¯ x(t − s)y(t) dµ(s) dλ(t)¯¯ ≤ kxkLp · kykLp0 · kµk. ¯ Taking ¡ ¢ for Ry characteristic functions of compact sets we see that the function Tµ x (t) = x(t−s) dµ(s) is defined locally almost everywhere and λ-measurable. Let x ∈ Lp , p 6= 0, be fixed. For all y ∈ Lp0 we have the estimate Z

¯ Z ¯Z ¯ ¯ |Tµ x| · |y| dλ = ¯¯ x(t − s) dµ(s)¯¯ · |y(t)| dλ(t) ≤ kxkLp · kykLp0 · kµk,

which, by 1.8.1(b), implies that Tµ x belongs to Lp , i.e., Tµ x coincides locally almost everywhere with a function lying in Lp . In particular, we have that Tµ x ∈ L∞ provided x ∈ L∞ . Thus formula (1) defines an operator Tµ : L∞ → L∞ with kTµ k ≤ kµk.

4.4. CONVOLUTION OPERATORS

249

Next, we show that Tµ takes L0 ⊆ L∞ into itself. If µ and x have compact supports then, evidently, Tµ x has a compact support, too. It remains to recall that compactly supported measures form a dense subset in M (see 1.8.5), and compactly supported functions form a dense subset in L0 . Suppose p 6= 0, ∞. By 1.6.2(e) the function x ∈ Lp is equal to zero outside a σ-compact set. By 1.8.5 µ is concentrated on a σ-compact set, too. Finally, by 1.1.1(a) the sum of two compact sets is compact. Hence the sum of two σ-compact sets is σ-compact. Consequently Tµ x is also equal to zero outside a σ-compact set. Therefore Tµ x is a function lying in Lp provided x ∈ Lp . Thus Tµ acts on Lp . By the Riesz–Thorin theorem we have kTµ : Lp → Lp k ≤ kµk. (b) Let us consider the group I = Zc × D (see 1.6.3) and for i = (k, d) ∈ I the e i = (k + (0, 1]c ) × d. set Q e i , i ∈ I. By 1.8.5 and 1.6.3 µ is concentrated on a countable union of the sets Q Let { Tij : i, j ∈ I } be the matrix of the operator Tµ considered as acting on Lp , e j , E) to Lp (Q e i , E). Assume i = (k, d) see 1.6.4. We recall that Tij acts from Lp (Q c and j = (l, b), where k, l ∈ Z and d, b ∈ D. It is easy to verify that the matrix element Tij is determined by the restriction of µ to the set (k −l+(−1, 1)c )×(d−b) and thus ¡ ¢ e j , C) → Lp (Q e i , C)k ≤ |µ| (k − l + (−1, 1)c ) × (d − b) . kTij : Lp (Q e i , i ∈ I, from this Since µ is concentrated on a countable union of the sets Q estimate it follows that the matrix {Tij } has only a countable number of non-zero diagonals and, moreover, {Tij } ≤ 2c kµk, see 1.6.8 for the definition of · . Thus {Tij } is a matrix of the class s. By 1.6.8 the matrix {Tij } induces a linear bounded operator, say R, acting on Lpq for all q. We show that this operator coincides with the operator (1). We fix x ∈ Lpq and i ∈ I. By the definition of the operator induced by a matrix, e i we have for t ∈ Q X¡ ¡ ¢ ¢ (Rx)i (t) = Tij xj (t) j∈I

=

XZ

1Qe j (s)x(s) dµ(t − s),

j∈I

where

R

1Qe j (s)x(s) dµ(t − s) makes sense by (a); in particular, this integral exe i . Next, we observe that since {Tij } ≤ ∞ the series ists for almost all t ∈ Q P e ij xj converges absolutely in the norm of Lp (Qi , C). Therefore by 1.5.5 the j∈I T R P series j∈I 1Qe j (s)x(s) dµ(t − s) converges almost everywhere. e i the integral (1) can be represented as On the other hand, for t ∈ Q Z x(s) dµ(t − s) =

XZ j∈I

1Qe j (s)x(s) dµ(t − s)

250

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

e j , j ∈ I. By what has since µ is concentrated on a countable union of the sets Q been proved, it exists for locally almost all t ∈ G. Hence formula (1) defines the operator R. It remains to recall that kTµ : Lpq → Lpq k ≤ {Tij } . (c) First assume that x ∈ C. Since µ is bounded the integral in (1) makes sense for all t. We recall that C is embedded in L∞ isometrically. Therefore, by what has been proved, Tµ x ∈ L∞ and kTµ xkL∞ ≤ kµk · kxkC . It suffices to show that Tµ x is continuous. If µ is compactly supported then this assertion is evident. The case of an arbitrary µ follows from 1.8.5. Thus Tµ acts on C. The operator Tµ acts on Cq since Cq = C ∩ L∞q and Tµ acts both on C and on L∞q . To show that Tµ ∈ s we make use of the explicit formula for Πi from 1.6.10. It is straightforward to verify that ¡ ¢ kTij : C♦j → C♦i k ≤ 2|µ| (k − l + (−2, 1)c ) × (d − b) . The estimate kTµ : Cq → Cq k ≤ 2c kµk holds since Cq is embedded isometrically in L∞q . ¤ Remark. The fact the operators of the convolution with µ ∈ M act on Lp1 (respectively, C1 ), implies that the set of all measures of the form gλ, g ∈ Lp1 (respectively, C1 ), is an ideal in M, cf. 4.1.6, 5.3.2, and 5.4.6. 4.4.2. The ordering in the space of measures. Let T be a locally compact topological space. We denote by Mloc (T, R) the (real) space of all real measures µ ∈ Mloc (T, C), and by M(T, R) the (real) Banach space of all real µ ∈ M(T, C), see 1.8.4. Clearly, M(T, R) is the intersection of Mloc (T, R) and M(T, C). For ξ, µ ∈ Mloc (T, R) we say that ξ dominates µ and write µ ≤ ξ if Z Z + x dµ ≤ x dξ for all x ∈ C00 . In other words, µ ≤ ξ if ξ − µ is a positive measure. By the definition of the upper integral (see 1.5.5), if 0 ≤ µ ≤ ξ then Z ∗ Z ∗ |x| dµ ≤ |x| dξ for all x : T → E (2) (here E is a Banach space). Let N ⊆ Mloc (T, R) be a family. We say that N is lower bounded by µ ∈ Mloc (T, R) if µ ≤ ν for all ν ∈ N . We say that µ ∈ Mloc (T, R) is the infimum of N and write µ = inf N if µ ≤ ν for all ν ∈ N and, for any µ0 such that µ0 ≤ ν for all ν ∈ N , we have µ0 ≤ µ. Clearly, the infimum is unique provided it exists. Proposition. Let T be a locally compact set. (a) ([Bou3 , ch. 3, §1, 5, theorem 3]) For any pair ν1 , ν2 ∈ Mloc (T, R) there exists µ = inf{ν1 , ν2 }. A non-empty lower bounded family N ⊆ Mloc (T, R) has the infimum. If N ⊆ M and N is lower bounded by µ ∈ M, then, evidently, inf N ∈ M.

4.4. CONVOLUTION OPERATORS

251

(b) Let ξ ∈ Mloc (T, R) be a positive measure, let µ ∈ M(T, C), and let E be a Banach space. Assume that ξ dominates |µ|, i.e., |µ| ≤ ξ. Then for all p 0 is arbitrary, implies Z ¯ ¯ ¯Φ(y)¯ ≤ |x| · |y| dξ for y = ϕe0 , ϕ ∈ L1 (T, C, η). Let E1 , E2 , . . . , En ⊆ T be η-summable and pairwise disjoint sets, and let 1E1 , 1E2 , . . . , 1En be characteristic P functions of them; and let e01 , e02 , . . . , e0n ∈ E0 be n arbitrary vectors. Then for z = m=1 1Em e0m we have n n ¯ X ¯ ¯ ¯ X ¯ ¯ 0 ¯ ¯Φ(z)¯ = ¯¯Φ( ¯Φ(1Em e0m )¯ 1Em em )¯ ≤ m=1

≤

n Z X

|x| · 1Em · |e0m | dξ

m=1

Z =

m=1

|x| · |z| dξ.

254

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Pn We recall that sums z = m=1 1Em e0m are dense in L1 (T, E0 , η). Assume an P n arbitrary y ∈ L1 (T, E0 , η) is approximated by z = m=1 1Em e0m . Then we have ¯ ¯ ¯Φ(y)¯ ≤ |Φ(y − z)| + |Φ(z)| Z Z ≤ |x| · |y − z| dη + |x| · |z| dξ Z Z Z ≤ |x| · |y − z| dη + |x| · |y| dξ + |x| · |y − z| dη, which implies the estimate Z ¯ ¯ ¯Φ(y)¯ ≤ |x| · |y| dξ

for all y ∈ L1 (T, E0 , η)

and, in particular, for all y ∈ C00 (T, E0 ). Thus (5) holds for all y ∈ C00 (T, E0 ). Conversely, assume that (5) is true for all x and y. We fix x ∈ C00 (T, E) and consider the function y = ϕe0 , where e0 ∈ E0 and ϕ ∈ C00 (T, C) such that ϕ(t) = 1 for all t ∈ supp x. Then Z Z hdµ x, yi = hdµ x, ϕe0 i Z = hdµ (xϕ), e0 i Z = hdµ x, e0 i DZ E = dµ x, e0 . Therefore

¯DZ ¯ E¯ ¯Z ¯ ¯ ¯ 0 ¯ dµ x, e ¯ = ¯ hdµ x, yi¯ ¯ Z ≤ |x| · |y| dξ Z = |x| dξ.

Taking here the supremum over all e0 with ke0 k ≤ 1 we obtain (4). (b) Clearly, |µ| ≤ ξ and kak k · |µk | ≤ ξ. By 4.4.2 it follows that x is µ-integrable + and µk -integrable. By the definition of ξ, for x ∈ C00 we have Z ∞ Z X x d (kak k · |µk |) = x dξ. k=1

Next, we observe that by the definition of the Lebesgue integral (see 1.5.5), for all x ∈ M + and also for all x : T → [0, +∞] we have Z ∗ ³X n Z ∗ n ´ X x d (kak k · |µk |) = xd kak k · |µk | . k=1

k=1

255

4.4. CONVOLUTION OPERATORS

Consequently for all x : T → [0, +∞] we obtain ∞ Z X

Z

∗

∗

x d (kak k · |µk |) = lim

xd

n→∞

k=1

Z

n ³X

´ kak k · |µk |

k=1

∗

≤

x dξ.

Hence for all x ∈ L1 (T, E, ξ) we have Z Z ∞ ∞ ¯ ¯ X ¯X ¯ ¯ ¯ ¯ ¯ ak dµk x¯ ≤ ¯ ¯ak dµk x¯ k=1

≤

k=1 ∞ Z X

|x| d (kak k · |µk |)

k=1

Z ≤

|x| dξ.

R P∞ In particular, this estimate shows that the series k=1 ak dµk x converges absolutely. As we know, for all x ∈ L1 (T, E, ξ) also holds the estimate ¯Z ¯ Z ¯ ¯ ¯ dµ x¯ ≤ |x| dξ. For x ∈ C00 (T, E) the identity (6) is valid by definition. The two last estimates show that by continuity it is true for x ∈ L1 (T, E, ξ) as well. (c) Let us fix a representation P∞ (3) of µ ∈ M(T, B) with ak 6= 0 and consider the positive measure ξ = k=1 kak k · |µk |. By the Lebesgue–Radon–Nikodym theorem µk are absolutely continuous with respect to ξ. Therefore according to 1.8.10(a) there P∞ exist gk ∈ L1 (T, C, ξ) such that µk = gk ξ with kµk kM = kgk kL1 . Thus µ = k=1 ak ⊗ (gk ξ). We set G(t) =

∞ X

gk (t)ak .

k=1

P∞

P∞

Clearly, since k=1 kak k·kgk kL1 = k=1 kak k·|µk | < ∞, it follows that G belongs to L1 (T, B, ξ), and for all x ∈ C00 (T, E) and y ∈ C00 (T, E0 ), Z Z dµ x = Gx dξ, Z Z ­ ® hdµ x, yi = G(t) x(t), y(t) dξ(t). We consider the measure ζ defined by the rule dζ(t) = kG(t)k dξ(t).

256

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

We note that ζ is a positive scalar measure. Clearly, ζ satisfies (4) and (5). We claim that estimate (5) with ζ is explicit, i.e., ζ is the smallest positive measure which satisfies this estimate for all x ∈ C00 (T, E) and y ∈ C00 (T, E0 ). To prove this ¯R assertion ¯ we R assume the contrary: we suppose that |µ| < ζ. By definition ¯ hdµ x, yi¯ ≤ |x| · |y| d|µ| for all x ∈ C00 and y ∈ C00 , and consequently for 0 all x ∈ ¯R Lp (T, E, µ) ¯ and y ∈ Lp0 (T, E , µ). In particular, if |x| ≤ 1 and |y| ≤ 1 we have ¯ hdµ x, yi¯ ≤ |µ|(T ). Since |µ| ≤ ξ we obtain L∞ (T, E, ξ) ⊆ Lp (T, E, ξ) ⊆ Lp (T, E, µ) and similarly so for p0 and E0 . Therefore we can state that ¯ ¯Z ¯ ¯Z ­ ® ¯ ¯ ¯ ¯ ¯ G(t) x(t), y(t) dξ(t)¯ = ¯ hdµ x, yi¯ ≤ |µ|(T ) for all x ∈ L∞ (T, E, ξ) and y ∈ L∞ (T, E0 , ξ) such that |x| ≤ 1 and |y| ≤ 1. PnNext, we take an arbitrary ε > 0 and choose a function B of the form B = such k=1 1Ek bk , where bk ∈ B and Ek ⊆ T are ξ-summable and pairwise disjoint, Pn that kG − BkL1 (T,B,ξ) ≤ ε. Then we take x and y of the form x = k=1 1Ek ek Pn and y = k=1 1Ek e0k , where ek ∈ E, kek k = 1, and e0k ∈ E0 , ke0k k = 1. Clearly, Z

­

n X ® B(t) x(t), y(t) dξ(t) = hbk ek , e0k i

Z 1Ek dξ.

k=1

We observe that theP supremumRof the modulus of the right side over all considered n x and y is equal to k=1 kbk k 1Ek dξ = kBkL1 (T,B,ξ) . Hence we can fix x and y such that ¯Z ­ ¯ ® ¯ ¯ ¯ B(t) x(t), y(t) dξ(t)¯ ≥ (1 − ε)kBkL1 (T,B,ξ) . For these x and y we have ¯Z ­ ¯ ¯Z ¯ ® ¯ ¯ ¯ ¯ ¯ G(t) x(t), y(t) dξ(t)¯ ≥ ¯ hB(t) x(t), y(t)i dξ(t)¯ ¯Z ­¡ ¯ ¢ ® ¯ ¯ −¯ G − B (t) x(t), y(t) dξ(t)¯ ≥ (1 − ε)kBkL1 (T,B,ξ) − kB − GkL1 (T,B,ξ) ≥ (1 − ε)kBk − ε ≥ (1 − ε)kGk − (1 − ε)kB − Gk − ε ≥ (1 − ε)kGkL1 (T,B,ξ) − 2ε = (1 − ε)ζ(T ) − 2ε, ¯R ­ ¯ ® which contradicts the inequality ¯ G(t) x(t), y(t) dξ(t)¯ ≤ |µ|(T ) < ζ(T ). Thus we have proved that ζ = |µ|. Consequently d|µ|(t) = kG(t)k dξ(t). We set ± ½ G(t) kG(t)k if G(t) 6= 0, H(t) = 0 if G(t) = 0.

257

4.4. CONVOLUTION OPERATORS

We recall that G ∈ L1 (T, B, ξ). Therefore for ξ-almost all t we have the equality G(t) = H(t) · kG(t)k (it may be violated only when G(t) is undefined). Hence the function t 7→ H(t) · kG(t)k belongs to L1 (T, B, ξ), too. Since G ∈ L1 (T, B, ξ) and d|µ|(t) = kG(t)k dξ(t), by 1.8.7(c) the equality G(t) = H(t) · kG(t)k implies that H ∈ L1 (T, B, µ). By definition |H(t)| ≤ 1 for almost all t ∈ T . Hence H ∈ L∞ (T, B, |µ|). The proof of the equalities (7) is trivial. For example, Z

Z dµ x =

Gx dξ Z

= Z =

¡

¢ H(t) · kG(t)k x(t) dξ(t)

¡ ¢ H(t) x(t) kG(t)k dξ(t)

Z =

Hx d|µ|.

P∞ We turn to the proof of (8). Let H = k=1 bk hk bePa representation of H with ∞ bk ∈ B and hk ∈ L1 (T, C, µ). We stress that H(t) = k=1 bk hk (t) for |µ|-almost all t. As we have seen above, G(t) = H(t) · kG(t)k for ξ-almost all t. Note that in the two last equalities H’s may differ on a |µ|-null set. But by 1.8.8(e) any |µ|null set is ξ-null. Hence the identity G(t) = H(t) · kG(t)k remains true ξ-almost everywhere if one changes H on a |µ|-null set. Thus ξ-almost everywhere we have P∞ G(t) = H(t) · kG(t)k = k=1 bk hk (t) · kG(t)k. We set fk (t) = hk (t) · kG(t)k. By 1.8.7(c), since hk ∈ L1 (T, C, µ) and d|µ(t)| = kG(t)k P∞ dξ(t) we have fk ∈ L1 (T, C, ξ) with khP k = kf k. Therefore, in particular, k k k=1 kbk k · kfk k < ∞. Clearly, ∞ G(t) = k=1 bk fk (t) for ξ-almost all t. P∞ Representing L1 (T, B, ξ) as B ⊗π L1 (T, C, ξ) we obtain G = k=1 bk ⊗ fk . On P∞ g (t)a the other hand, by the definition of G we have G(t) = k , which means k=1 k P P∞ P∞ ∞ that G = k=1 ak ⊗ gk . Consequently k=1 bk ⊗ fk = k=1 ak ⊗ gk as elements of B ⊗π L1 (T, C, ξ). We consider the natural mapping JPfrom B ⊗π L1 (T, ξ) ' L1 (T, B, ξ) into PC, ∞ ∞ B ⊗π M(T, C) ' M(T,P B) that maps k=1 ck ⊗ uk to k=1 ck ⊗ (uk ξ). Clearly, P ∞ ∞ kc k ⊗ ku k = . Thus J is well definedP and kJk ≤ 1. k k L1 k=1 k=1 kck k ⊗ kuk ξkMP ∞ ∞ Consequently the two representations G = k=1 bk ⊗ fk and G = k=1 ak ⊗ gk induce the same measure. Thus µ=

∞ X k=1

ak ⊗ µk =

∞ X k=1

ak ⊗ (gk ξ) =

∞ X k=1

bk ⊗ (fk ξ) =

∞ X

bk ⊗ (hk |µ|).

k=1

R (d) If dµ x = 0 for all x ∈ C00 then by (a) |µ| = 0. But then by (c) µ possesses the zero representation (8). ¤ Remark. Assertion ¡ ¢ (d) states that the natural embedding of M(T, B) in the space B C0 (T, E), E is injective. We stress that this embedding is not isometric.

258

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Indeed, let T be R and E be C2 with l∞ -norm. We consider the operator-valued measure µ ¶ µ ¶ 1 0 0 0 µ= ⊗ δ1 + ⊗ δ2 . 0 0 0 1 It induces the vector-valued functional µ ¶ µ Z 1 0 0 x 7→ x dµ = x(1) + 0 0 0

0 1

¶ x(2).

¯R ¯ Evidently, ¯ dµ x¯ ≤ 1 for x ∈ C0 (R, E) with kxk ≤ 1. Thus kµk ¡

B C0 (R,E),E

¢ ≤ 1.

On the other hand, since µ ∈ Md ' l1 (R, B) (see 4.4.9 below for the exact definition of Md (R, B)), it is obvious that kµkM(R,B) = 2. 4.4.4. Convolution operators on Lp , Lpq , and Cq (vector case). Let G be a locally compact abelian group, and let µ ∈ M(G, B). We consider the convolution operator Z ¡ ¢ Tµ x (t) = dµ(s) x(t − s) Z (9) = dµ(t − s) x(s). Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let a measure µ∈P M(G, B) be represented in the form (3). As usual we denote by ξ the measure ∞ ξ = k=1 kak k · |µk |. (a) For any x ∈ Lp (G, E) the function s 7→ x(t − s) is ξ-integrable for λ-almost all t ∈ G if p 6= 0, ∞, and for λ-locally almost¡ all t ∈ G¢ if p = 0, ∞. In all cases formula (9) defines an operator Tµ ∈ B Lp (G, E) with kTµ k ≤ kµk. (b) For any x ∈ Lpq (G, E) the function s 7→ x(t−s) is ξ-integrable for λ-locally ¡ ¢ almost all t ∈ G. Formula (9) defines an operator Tµ ∈ s Lpq (G, E) with kTµ k ≤ 2c kµk. (c) For any x ∈ Cq (G, E) the function s 7→ x(t t ∈ G. ¡ − s) is ¢ξ-integrable for all c Formula (9) defines an operator Tµ ∈ s Cq (G, E) with kTµ k ≤ 2 kµk. We recall from 4.4.3(b) that any ξ-integrable function is a fortiori µ-integrable. For the definition of kµk, see 4.4.3 also. Proof. (a, b) First we prove the theorem for the case where µ ∈ M(G, C) is a scalar measure, but E is an arbitrary Banach space (in this case we can interpret ξ as |µ|). For 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, we denote briefly Lp or Lpq by L, and Lp or Lpq by L. We consider the scalar operator T|µ| : L(G, C) → L(G, C), where |µ| is the modulus of µ. For any x ∈ L(G, E) we define the scalar function |x| ∈ L(G, C) by the rule |x|(t) = |x(t)|. Clearly, k |x| k = kxk. Assume x ∈ L(G, E). Let K ⊆ G be a compact set and 1K be its characteristic function. We apply Fubini’s theorem to the function (t, s) 7→ 1K (t)x(t − s). By

4.4. CONVOLUTION OPERATORS

259

4.1.5(b) the function (t, s) 7→ 1K (t)x(t − s) is λ ⊗ µ-measurable. It is equal to zero outside the λ ⊗ µ-summable set K × G. From the equality Z ∗ ¡ ¢ |1K (t)x(t − s)| d|µ|(s) = 1K (t) T|µ| |xn | (t), and the belonging 1K T|µ| |x| ∈ L1 , and the part (b) of Fubini’s theorem, we see that the function (t, s) 7→ 1K (t)x(t − s) is λ ⊗ µ-integrable. Hence by the part (a) of Fubini’s theorem the function s 7→ x(t − s) is µ-integrable for almost all t ∈ K and consequently for locally almost all t ∈ G. Moreover, the function Tµ x is locally integrable and therefore measurable. Suppose temporary that L = Lp , p 6= 0, ∞. Then since T|µ| |x| ∈ Lp , by R∗ 1.6.2(e) the integral |x(tR− s)| d|µ|(s) is equal to zero outside a σ-compact set. Consequently the integral x(t − s) dµ(s) is equal to zero outside a σ-compact R set, too. Hence by what has been proved, the integral x(t − s) dµ(s) exists for almost all t ∈ G. Suppose again that L is arbitrary. We observe that Z ¯ ¯¡ ¢ ¯ ¯¯ ¯ Tµ x (t)¯ = ¯ x(t − s) dµ(s)¯¯ Z ≤ |x(t − s)| d|µ|(s) ¡ ¢ = T|µ| |x| (t) provided that both integrals exist. By this estimate Tµ x ∈ L (we recall that Tµ x is measurable). Moreover, kTµ k ≤ kT|µ| k for all L. Thus the consideration of the case of a scalar measure µ ∈ M(G, C) is completed. Now let µ ∈ M(G, B) be an arbitrary operator-valued measure. By what has been proved, for any x ∈ L(G, E) we have Tµk x ∈PL. Clearly, we have P ∞ ∞ k=1 ak Tµk x converges k=1 kak k · kTµk xk ≤ kµk · kxk < ∞. Thus the series absolutely in the norm P of L(G, E). We show that integral (9) exists for (locally) ∞ almost all t and Tµ x = k=1 ak Tµk x. We consider the scalar operator Tξ : L(G, C) → L(G, C) induced by the measure ξ. Let x ∈ L(G, E). We denote by R the set of all t ∈ G for which the function s 7→ x(t − s) is ξ-integrable. By 4.4.1 the set G \ R is (locally) null. By 4.4.3, for all t ∈ R the function s 7→ x(t − s) is µ-integrable and µk -integrable, and Z ¡ ¢ Tµ x (t) = dµ(s) x(t − s) = =

∞ X k=1 ∞ X

Z ak

x(t − s) dµk (s)

¡ ¢ ak Tµk x (t).

k=1

(c) Let x ∈ C. Clearly, the function s 7→ x(t−s) is ξ-integrable and consequently µ-integrable for all t ∈ G.

260

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Furthermore, for any µ ∈ M(G, B) we have ¯¡ ¢ ¯ ¡ ¢ ¯ Tµ x (t)¯ ≤ T|µ| |x| (t). Clearly, without loss of generality we may assume that in the representation (3) the measures Pn µk are compactly supported. We approximate µ by the partial (n) sums µ = k=1 ak ⊗ µk . It is evident that Tµ(n) x is continuous. The preceding estimate then shows that Tµ x is continuous for an arbitrary µ, too. The preceding estimate together with 4.4.1 also imply that Tµ maps Cq into itself and that the desired estimate for kTµ k holds. (b, c) To prove that Tµ ∈ s one should repeat the corresponding reasoning from the proof of 4.4.1. Namely, the estimates ¡ ¢ e j , E) → Lp (Q e i , E)k ≤ |µ| (k − l + (−1, 1)c ) × (d − b) , kTij : Lp (Q ¡ ¢ kTij : C♦j → C♦i k ≤ 2|µ| (k − l + (−2, 1)c ) × (d − b) show that the matrix of Tµ belongs to s. It is easy to see that Tµ is restored by its matrix. ¤ 4.4.5. The conjugate of a convolution operator. Let 1 ≤ p ≤ ∞ and p0 be the conjugate index, see 1.8.1. Let E be a Banach space and E0 be its conjugate. We recall from 1.8.1 that for x ∈ Lp (G, E) and y ∈ Lp0 (G, E0 ) the function t 7→ hx(t), y(t)i belongs to L1 (G, C) and therefore we may set Z hx, yi =

hx(t), y(t)i dλ(t).

¡ ¢0 This formula defines the isometric embedding of Lp0 (G, E0 ) in Lp (G, E) . Thus we can identify Lp0 with the subspace of (Lp )0 . In a similar way one can identify Lp0 q0 with the subspace of (Lpq )0 . R T For µ ∈ M(G, C) we call the measure µ defined by the equality x dµT = R x(−s) dµ(s) the measure transposed to µ. Let µ P ∈ M(G, B) be represented in ∞ T the form (3). In this case we call the measure µ = k=1 a0k ⊗ µTk ∈ M(G, B(E0 )), the measure transposed to µ. Proposition. Let E be an arbitrary Banach space, and let µ ∈ M(G, B). ¡ ¢0 (a) Let 1 ≤ p ≤ ∞ or p = 0. Then the subspace Lp0 (G, E0 ) ⊆ Lp (G, E) is invariant under the conjugate of the operator Tµ : Lp (G, E) → Lp (G, E). And the restriction of Tµ0 to Lp0 is the operator TµT . ¡ ¢0 (b) Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Then Lp0 q0 (G, E0 ) ⊆ Lpq (G, E) is invariant under the conjugate of Tµ : Lpq (G, E) → Lpq (G, E). And the restriction of Tµ0 to Lp0 q0 is the operator TµT .

4.4. CONVOLUTION OPERATORS

261

Proof. We begin with the following observation. For µ ∈ M(G, C), and x ∈ C00 (G, E) and y ∈ C00 (G, E0 ), we have (we employ the shift invariance of λ) Z ­¡ ¢ ® hTµ x, yi = Tµ x (t), y(t) dλ(t) Z D³Z ´ E = dµ(s) x(t − s) , y(t) dλ(t) Z ³ Z ´ = dµ(s) hx(t − s), y(t)i dλ(t) Z ³ Z ´ = dµ(s) hx(τ ), y(τ + s)i dλ(τ ) Z D ³Z ´E = x(τ ), dµ(s) y(τ + s) dλ(τ ) = hx, TµT yi. By the definition of M(G, B) it follows that the same identity hTµ x, yi = hx, TµT yi holds for µ ∈ M(G, B) as well. (a) Let p 6= 0, 1, ∞. Then C00 is dense both in Lp and Lp0 . Thus by continuity hTµ x, yi = hx, TµT yi

for all x ∈ Lp and y ∈ Lp0

(10)

and the assertion is proved. If p = 1 or p = 0, ∞, (10) is valid by ∗-weak continuity, since C00 is ∗-weak dense in L∞ , see 1.8.2 (the operators Tµ and TµT are ∗-weak continuous by 1.1.10). (b) Let p, q 6= 0, 1, ∞. Then C00 is dense both in Lpq and Lp0 q0 . Therefore hTµ x, yi = hx, TµT yi

for all x ∈ Lpq and y ∈ Lp0 q0 ,

and the proof is complete. By 1.8.2 C00 is dense in L∞q in the hL∞q , L1q0 i-topology. This note allows one to consider the case of arbitrary p and q in a similar way. ¤ 4.4.6. The norm of Tµ on L1 (G, C) and L∞ (G, C) Corollary. For any µ ∈ M(G, C) one has the equalities kTµ : L1 → L1 k = kµk and kTµ : L∞ → L∞ k = kµk. Proof. By 4.1.1 kTµ : C0 → C0 k = kµk. Clearly, kTµ : L∞ → L∞ k ≤ kµk. Since C0 is embedded isometrically in L∞ we have kTµ : L∞ → L∞ k = kµk. The case of L1 follows from 4.4.5: kTµ : L1 → L1 k = k(TµT )0 : L∞ → L∞ k = kµk. ¤ 4.4.7. The preservation of multiplication (scalar case) Theorem. Let L be the space Cq (G, C) or Lpq (G, C), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. Then the mapping µ 7→ Tµ from M(G, C) into B(L) is a morphism of Banach algebras. Proof. If L = C0 the assertion is true by the definition of the convolution in M, see 4.1.3. In all other cases it is only not evident that the mapping µ 7→ Tµ preserves multiplication, i.e., Tµ∗ν = Tµ Tν for all µ, ν ∈ M.

262

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Assume p 6= 0, ∞. Then C00 is dense in Lp and one can extend the proved equality Tµ∗ν x = Tµ Tµ x, x ∈ C00 , from C00 to Lp by continuity. The case L = L∞ follows from 4.4.5. The case L = L0 follows from the isometric embedding L0 ⊆ L∞ . Assume L = Lpq . We recall from 1.6.8 that the correspondence {Tij } 7→ T is an isomorphism from the algebra s(I) of matrices with summable memory onto the algebra s(Lpq ) of operators with summable memory. We represent the mapping µ 7→ Tµ from M into B(Lpq ) as M − → s(Lp ) ' s(Lpp ) − → s(I) − → s(Lpq ). Clearly, all intermediate mappings preserve multiplication. The case L = Cq follows from the isometric embedding Cq ⊆ L∞q . ¤ 4.4.8. The preservation of multiplication (operator-valued case). Let E be an arbitrary Banach space and B = B(E). We recall that M(G, B) is B ⊗π M(G, C). Clearly, the cross-norm π is compatible with multiplication, see 1.7.10 for the definition. Thus M(G, B) is a Banach algebra. We call the multiplication in M(G, B) the convolution and denote it by µ ∗ ν. By definition ∞ ³X k=1

∞ ∞ X ∞ ´ ³X ´ ³X ´ ak bm ⊗ µk ∗ νm . ak ⊗ µk ∗ bm ⊗ νm = m=1

k=1 m=1

Theorem. Let L be the space Cq (G, E) or Lpq (G, E), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. Then the mapping µ 7→ Tµ from M(G, B) into B(L) is a morphism of Banach algebras. Proof. It is not evident only that the mapping µ 7→ Tµ takes the convolution of measures to the composition of operators. We begin with the case of the space C0 (G, E). Let µ, ν ∈ M(G, C) be scalar measures, and let a, b ∈ B(E). We denote by Tµ , Tν : C0 (G, C) → C0 (G, C) the operators induced by µ and ν on the space of scalar functions. Next, we denote by a ⊗ Tµ , b ⊗ Tν : C0 (G, E) → C0 (G, E) the operators induced by the operator-valued measures a ⊗ µ and b ⊗ ν. Let x ∈ C0 (G, E) be a finite-dimensional function, see 1.5.3 for the definition. It is straightforward to verify that (a ⊗ Tµ ) · (b ⊗ Tν )x = (ab ⊗ Tµ∗ν )x. By continuity, the same identity holds for all x ∈ C0 (G, E). (Cf. this argument with the proof of 1.7.4.) Now assume that µ, ν ∈ M(G, measures. According to P∞B) are operator-valued P∞ (3) we represent them as µ = k=1 ak ⊗ µk and ν = m=1 bm ⊗ νm . Then the identity Tµ Tν = Tµ∗ν follows from the chain of equalities Tµ Tν =

∞ ³X

ak ⊗ Tµk

k=1

= =

∞ X ∞ X k=1 m=1 ∞ X ∞ X k=1 m=1

= Tµ∗ν .

∞ ´³ X

´ bm ⊗ Tνm

m=1

ak bm ⊗ Tµk Tνm ak bm ⊗ Tµk ∗νm

4.4. CONVOLUTION OPERATORS

263

The subspace C00 ⊆ C0 is dense in the spaces Lp , p 6= ∞. Therefore on the spaces Lp , p 6= ∞, we obtain the identity Tµ Tν = Tµ∗ν by continuity. We turn to the case of L∞ (G, E). Let µ, ν ∈ M(G, B). By what has been proved, we have the identity Tν T TµT = Tν T ∗µT on L1 (G, E0 ). By 4.4.5 it implies the identity TµT T Tν T T = TµT T ∗ν T T on L∞ (G, E00 ). It remains to observe that L∞ (G, E) is a subspace of L∞ (G, E00 ) invariant under Tµ and Tν . For the case of Lpq we represent the mapping µ 7→ Tµ from M into B(Lpq ) as M− → s(Lp ) ' s(Lpp ) − → s(I) − → s(Lpq ). Cf. the proof of 4.4.7. The case of Cq follows from the case of L∞q . ¤ 4.4.9. The main decomposition (operator-valued case). We denote by Md = Md (G, B) the subalgebra of all µ ∈ M(G, B) possessing at least one representation (3) with µk ∈ Md (G, C). In a similar way we define Mc = Mc (G, B), Mac = Mac (G, B), Msc = Msc (G, B), and Md⊕ac = Md⊕ac (G, B). Theorem. (a) The following natural isometric isomorphisms hold: Md (G, B) ' B ⊗π Md (G, C), Msc (G, B) ' B ⊗π Msc (G, C),

Mc (G, B) ' B ⊗π Mc (G, C), Mac (G, B) ' B ⊗π Mac (G, C),

Md⊕ac (G, B) ' B ⊗π Md⊕ac (G, C). (b) The following natural isometric isomorphisms hold: Mac (G, B) ' L1 (G, B, λ),

Md (G, B) ' l1 (G, B).

(c) Let G be non-discrete. Then M(G, B) = Md (G, B) ⊕ Msc (G, B) ⊕ Mac (G, B). For the corresponding representation µ = µd + µsc + µac of µ ∈ M(G, B) kµk = kµd k + kµsc k + kµac k. In a similar way Md⊕ac (G, B) = Md (G, B) ⊕ Mac (G, B). (d) Let G be discrete. Then M = Md = Mac and Msc = {0}. (e) The sets Mac (G, B) and Mc (G, B) are closed ideals in M(G, B). Proof. (a) Cf. the proof of 1.7.11. We prove, e.g., the first formula. Let J : Md (G, C) → M(G, C) be the natural embedding. We recall from 4.1.9 that M = Md ⊕Msc ⊕Mac with kµk = kµd k+kµsc k+kµac k. Therefore there exists the natural projection Q : M(G, C) → Md (G, C) with kQk ≤ 1. Evidently, QJ = 1. We recall (see the example in 1.7.7) that the cross-norm π is uniform. Hence the operators 1 ⊗ Q : B ⊗π Md (G, C) → B ⊗π M(G, C) and 1 ⊗ J : B ⊗π M(G, C) → B ⊗π Md (G, C) are well defined, and (1 ⊗ Q)(1 ⊗ J) = 1, and k1 ⊗ Qk ≤ 1, and k1⊗Jk ≤ 1. Finally, we observe that the image of 1⊗J is, by definition, Md (G, B).

264

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Thus 1 ⊗ Q and 1 ⊗ J establish isometric isomorphism between Md (G, B) and B ⊗π Md (G, C). The other isomorphisms are proved in a similar way. (b) follows from (a), 1.7.4(c), and the isomorphisms Mac (G, C) ' L1 (G, C, λ) and Md (G, C) ' l1 (G, C, λ). (c) follows from 4.1.9 and 1.7.11. (d) follows immediately from 4.1.9(b). (e) is evident. ¤ 4.4.10. Difference operators. We recall from 4.4.9(b) that Md (G, B) ' l1 (G, B). Let a = P{ ah : h ∈ G } be an element of l1 (G, B) = l1 (Gd , B); thus ah ∈ B(E), kak = h∈G kah k < ∞, and, in particular, only a countable or finite number of ah is not equal to zero. We consider the operator X ¡ ¢ Dx (t) = ah x(t − h). (11) h∈G

We call the operator D a difference operator with constant coefficients or shift invariant difference operator. Corollary. Let E be an arbitrary Banach space, and let a ∈ l1 (G, B), and let µ be the measure induces by a. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. ¡ ¢ (a) Formula (11) defines an operator D ∈ B¡ Lp (G, E) ¢. (b) Formula (11) defines an operator D ∈ s¡Lpq (G, E)¢ . (c) Formula (11) defines an operator D ∈ s Cq (G, E) . In all cases the operator (11) coincides with the operator (9) induced by the measure µ. In the case (a) kDk ≤ kakl1 . In the cases (b) and (c) kDk ≤ 2c kakl1 . Proof. By 1.6.12 formula (11) defines an operator acting on Cq , Lp , and Lpq ; moreover, D ∈ s(Lpq ). Clearly, operators (9) and (11) coincide on C00 . By continuity, they also coincide on Lp , p < ∞. Next, we observe that L1 ∩L0 is dense in L0 (more correctly, L1 ∩L0 is dense in L0 ); therefore they coincide on L0 . Hence they have equal matrices as well. Consequently they define the same operator on Lpq . Since Cq ⊆ L∞q they coincide on Cq , too. ¤ 4.4.11. Integral operators. We recall from 4.4.9(b) that Mac (G, B) ' L1 (G, B). Let g be an element of L1 (G, B). We consider the operator Z ¡ ¢ Gx (t) = g(s)x(t − s) dλ(s) G Z = g(t − s)x(s) dλ(s).

(12)

G

Here we assume that g(s)x(t−s) = 0 if g(s) = 0 and x(t−s) is undefined, cf. 1.8.7. We call the operator G a shift invariant integral operator or integral operator with constant coefficients.

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265

Corollary. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let E be a Banach space. Let g ∈ L1 (G, B) and µ ∈ Mac (G, B) be the measure λg induces by g. (a) For any x ∈ Lp (G, E) the function s 7→ g(s)x(t − s) is λ-integrable for λ-almost all t ∈ G if p 6= 0, ∞, and for λ-locally almost all¡ t ∈ G if¢ p = 0, ∞. In all cases formula (12) defines an operator G ∈ B Lp (G, E) with kGk ≤ kgk. (b) For any x ∈ Lpq (G, E) the function s 7→ g(s)x(t − s) is λ-integrable for G. Formula (12) defines an operator G ∈ ¡ λ-locally¢ almost all t ∈ c s Lpq (G, E) with kGk ≤ 2 kgk. (c) For any x ∈ Cq (G, E) the function s 7→ g(s)x(t − ¡s) is λ-integrable for all ¢ t ∈ G. Formula (12) defines an operator G ∈ s Cq (G, E) with kGk ≤ 2c kgk. (d) For any x ∈ L∞q (G, E) the function s 7→ g(s)x(t −¡ s) is λ-integrable for¢ all t ∈ G. Formula (12) defines an operator G ∈ B L∞q (G, E), Cq (G, E) with kGk ≤ 2c kgk. Operator (12) coincides with the operator (9) induced by the measure µ. P∞ Proof. We represent P∞g ∈ L1 (G, B) in the form g = k=1 ak gk , where gk ∈ L1 (G, C), ak ∈ B, and k=1 kak k · kgk k < ∞. Then according to the isomorphism P∞ Mac (G, B) ' L1 (G, B), µPcan be represented P as µ = k=1 ak ⊗ gk λ. We consider ∞ ∞ the scalar measure ξ = k=1 kak k · |gk λ| = k=1 kak k · |gk |λ, see 1.8.7(a). We show that integral (11) exists and coincides with the integral (9) whenever the function xt (s) = x(t − s) is ξ-integrable. Indeed, if xt is ξ-integrable then by 4.4.3(b) it is |gk |λ-integrable. Next, we have (we use 1.5.5) Z Z ∞ X ¡ ¢ dµ(s) x(t − s) = ak x(t − s) d gk (s)λ(s) =

k=1 ∞ X

Z ak

gk (s)x(t − s) dλ(s)

k=1

=

Z ³X ∞ Z

=

´ ak gk (s) x(t − s) dλ(s)

k=1

g(s)x(t − s) dλ(s).

Thus operator (12) coincides with the operator (9). (a), (b), and (c) follow from 4.4.4. (d) In light of (b) it suffices to show that G maps L∞ into C. Clearly, the function s 7→ g(t − s)x(s) is µ-integrable for all x ∈ L∞ and t ∈ G. The continuity of Gx follows ¯R from the continuityR of the function h 7→¯ Sh g in the norm of L1 and the estimate ¯ g(t1 −s)x(s) dλ(s)− g(t2 −s)x(s) dλ(s)¯ ≤ kSt1 g−St2 gkL1 ·kxkL∞ . ¤ 4.4.12. The algebra M+ (R, B). Let G = R. We denote by M+ = M+ (R, B) the subset of all µ ∈ M(R, B) possessing at least one representation (3) with µk ∈ M+ (R, C), see 4.1.11 for the definition of M+ (R, C). In a similar way we + + + + define M+ d (R, B), Mc (R, B), Mac (R, B), Msc (R, B), and Md⊕ac (R, B).

266

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Proposition. (a) A measure µ ∈ M(R, B) belongs to M+ (R, B) if and only if the operator Tµ is causal in one (or, equivalently, in all) of the spaces Cq (R, E) or Lpq (R, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (b) The following natural isometric isomorphisms hold: M+ (R, B) ' B ⊗π M+ (R, C), + M+ d (R, B) ' B ⊗π Md (R, C),

+ M+ c (R, B) ' B ⊗π Mc (R, C),

+ M+ sc (R, B) ' B ⊗π Msc (R, C),

+ M+ ac (R, B) ' B ⊗π Mac (R, C),

+ M+ d⊕ac (R, B) ' B ⊗π Md⊕ac (R, C).

(c) The set M+ (R, B) is a closed subalgebra of M(R, B). The sets M+ ac (R, B) + + and Mc (R, B) are closed ideals in M (R, B). (d) Let according to 4.4.9 µ ∈ M+ (R, B) be represented as µ = µd + µsc + µac . Then µd , µsc , µac ∈ M+ (R, B). Thus + + M+ = M+ d ⊕ Msc ⊕ Mac

and

+ + M+ d⊕ac = Md ⊕ Mac .

Proof. (a) First of all ¡we note¢ the auxiliary statement: ν ∈ M(R,¢C); ¡assume P∞ −(n−1) , −2−n ] = then one has limε→0+0 |ν| (−ε, 0) = 0. Indeed, n=1 |ν| (−2 ¡ ¢ ¡ ¢ |ν| (−1, 0) < ∞ implies limn→∞ |ν| (−2−(n−1) , 0) = 0. Assume Tµ is causal on one ¡ of ¢the spaces enumerated in the assertion (a). Then x(s) = 0, s < 0, implies ¡ ¢Tµ x (s) = 0, s < 0, for allR x ∈ C00 (R, E), which by continuity implies Tµ x (0) = 0. In other words, dµ(s) x(s) = 0 for all x ∈ C00 (R, E) such that supp x ⊆ (−∞, 0]. We fix a representation of µ in the form (3). Let Hε : R → [0, 1], ε > 0, be a continuous function such that HP ε (t) = 1 for t ≥ −ε, and Hε (t) = 0 for t ≤ −2ε. ∞ We consider the measure µε = k=1 ak ⊗ (Hε µk ). (Evidently, kHε µk k ≤ kµk k. Hence the definition of µε is correct.) For any ε > 0 and (1−Hε )x Ris supported in (−∞, 0]. R x ∈ C00R(R, E) the function R R Therefore we have dµ x = dµ Hε x + dµ (1 − Hε )x = dµ Hε x + 0 = dµε x. Thus µε coincides with µ on C00 . We set H(t) = 1 if t ≥ 0, and H(t) = 0P otherwise. Clearly, H is universally ∞ measurable. We consider the measure µ+ = k=1 ak ⊗(Hµk ). Clearly, µ+ ∈ M+ . By our auxiliary statement, Hε µk converges to Hµk in the norm of M(R, C). Furthermore, kHε µk k ≤ kµk k and kHµk k ≤ kµk k. Therefore we arrive at the P∞ conclusion k=1 kak k · kHε µk − Hµk k → 0 as ε → 0, which implies µε → µ+ as ε → 0 in the norm of M(R, B). Consequently µ coincides with µ+ on C00 . Thus by 4.4.3(d) it follows that µ = µ+ and µ ∈ M+ . Conversely, assume µ ∈ M+ (R, B). By definition this means that µ possesses a representation (3) with µk ∈ M+ (R, C). We show that then Tµ is causal.

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267

Arguing as above isP it easy to show that µk = Hµk . It remains to observe that ∞ for the ¡measure ¢ µ = k=1 ak ⊗ Hµk in (9), it is evident that x(s) = 0, s < t, implies Tµ x (s) = 0, s < t, i.e., that Tµ is causal. (b) Cf. the proof of 4.4.9. Let Q : M(R, C) → M+ (R, C) be the operator Qµ = Hµ, where H(t) = 1 for t ≥ 0 and H(t) = 0 for t < 0. Clearly, kQk ≤ 1. We observe that QJ = 1, where J : M+ (R, C) → M(R, C) is the natural embedding. We recall from the example in 1.7.7 that the cross-norm π is uniform. Hence the operators 1 ⊗ Q : B ⊗π M(R, C) → B ⊗π M+ (R, C) and 1 ⊗ J : B ⊗π M+ (R, C) → B ⊗π M(R, C) are well defined, and (1 ⊗ Q)(1 ⊗ J) = 1, and k1 ⊗ Qk ≤ 1, and k1⊗Jk ≤ 1. Finally, we observe that the image of 1⊗J is, by definition, M+ (R, B). Thus 1 ⊗ Q and 1 ⊗ J establish isometric isomorphism between M+ d (R, B) and B ⊗π M+ (R, C). d The other isomorphisms are proved in a similar way. We only recall from 4.1.11 + + that M+ = M+ d ⊕ Msc ⊕ Mac and from 4.1.9 that kµk = kµd k + kµsc k + kµac k. + These imply, e.g., that the natural embedding J : M+ c (R, C) → M (R, C) has a right inverse Qc with kQc k ≤ 1. (c) follows from (b) and 4.1.11(b). (d) follows from (b), and 4.1.11(c), and 1.7.11. ¤

4.5. Invertibility and causal invertibility We recall from chapter 2 that the causal invertibility of an operator T can be usually interpreted as the input–output or exponential stability of the equation T x = f ; and the ordinary invertibility of T can be interpreted as the rough instability or exponential dichotomy. The investigation of stability and instability are the main applications the results of this section (and the whole chapter) are intended for. Usually these applications are evident. Therefore to save space and the patience of the reader we do not formulate them in detail, but only note some of them as examples. Nevertheless, the formulations of some assertions of this section are very detailed. We take the liberty of writing in such a way because they are the main results of this chapter and are assumed to be used independently of the previous parts of the chapter. 4.5.1. The invertibility in a subalgebra of operator-valued measures. Let G be an arbitrary locally compact abelian group, and let M = M(G, C) be the Banach algebra of all bounded complex measures on G. Let N = N (G, C) be a closed subalgebra with a unit of M(G, C), let X = X(N ) be its character space, and let ϑ : N (G, C) → C(X, C) be the Gel0 fand transform on N , see 1.4.11. Let E be a Banach space and B = B(E). We denote by N (G, B) the Banach algebra B ⊗π N (G, C), cf. 4.4.3 and 4.4.9. As in 1.7.10, we consider the mapping Θ = 1 ⊗ ϑ : B ⊗ N (G, C) → B ⊗ C(X, C) and its extension by continuity Θ = 1 ⊗ ϑ : N (G, B) = B ⊗π N (G, C) → B ⊗ε C(X, C) ' C(X, B). We call Θ the operator-valued Gel0 fand transform. We denote by Θµ the transform Θ applied to the measure µ ∈ N (G, B).

268

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Theorem. Let µ ∈ N (G, B). Then σN (µ) = σC(X,B) (Θµ ). Thus σ(µ) =

¢ ª S© ¡ σ Θµ (ζ) : ζ ∈ X .

In particular, the measure µ ∈ N (G, B) is invertible in N (G, B) if and only if the function Θµ is invertible in C(X, B), i.e., the operator Θµ (ζ) is invertible for all ζ ∈ X. Proof. This is a special case of the Bochner–Phillips theorem. ¤ 4.5.2. The invertibility of measures. Let G be a locally compact abelian group, let B = B(E), and let µ ∈ M(G, B). For any χ ∈ X = X(G) we set Z ¡ ¢ 1 ⊗ ζχ (µ) = χ(t) dµ(t). We define this integral in the following natural way. If µ is represented in the form µ=

∞ X

ak ⊗ µk ,

ak ∈ B, µk ∈ M(G, C),

k=1

with

P∞ k=1

kak k · kµk k < ∞, we set Z χ(t) dµ(t) =

∞ X

Z ak

χ(t) dµk (t).

k=1

Clearly, P∞this definition does not depend on the choice of the representation µ = k=1 ak ⊗ µk . Thus 1 ⊗ ζχ is really the tensor product of 1 : B ¡ → B ¢and the mapping ζχ : M(G, C) → C considered in 4.2.2. We stress that 1 ⊗ ζχ (µ) belongs to B. We call the function ¡ ¢ µ ˇ(χ) = 1 ⊗ ζχ (µ), χ ∈ X, the operator-valued Fourier cotransform of µ. In particular, if µ ∈ Mac (G, B) and µ is represented in the form µ = gλ with g ∈ L1 (G, B, λ) (see 4.4.9(b)) then the Fourier cotransform is µ ˇ(χ) = gˇ(χ) ¡ ¢ = 1 ⊗ ζχ (µ) Z = χ(t)g(t) dλ(t). Let µ ∈ Md (G, B). For χ ∈ Xb = X(Gd ) we set Z ¡ ¢ 1 ⊗ ζχ (µ) = χ(t) dµ(t),

4.5. INVERTIBILITY AND CAUSAL INVERTIBILITY

269

where the integral is defined by the formula Z ∞ X χ(hk )ak χ(t) dµ(t) = k=1

P∞

provided P∞µ is represented in the form µ = k=1 ak ⊗δhk , where hk ∈ G and ak ∈ B with k=1 kak k < ∞, see 4.4.9(b). Since Md (G, B) = M(Gd , B) this definition can be considered as a special case of the previous definition. ] We recall that Md⊕ac = Md ⊕ Mac . We denote by M ac (G, B) the algebra ] Mac (G, B) with an adjoint unit. We embed M ac (G, B) in the algebra M(G, B) ] and, in particular, we identify the unit of Mac (G, B) with the measure δ (or, more correctly, with 1 ⊗ δ). Corollary. Let G be a non-discrete locally compact abelian group. ] (a) Let µ ∈ Mac (G, B). Then the spectrum of µ in M ac (G, B) is the set [ S© ¡ ¢ ª {0} σ µ ˇ(χ) : χ ∈ X(G) . (1) ] In particular, the measure αδ + µ, α ∈ C, is invertible in M ac (G, B) if and only if α 6= 0, and the operator α + µ ˇ(χ) is invertible for all χ ∈ X(G). (b) Let µ ∈ Md (G, B). Then the spectrum of µ in Md (G, B) is the set ¢ ¢ ª S© ¡¡ σ 1 ⊗ ζχ (µ) : χ ∈ Xb (G) . (2) In particular, the¢ measure µ is invertible in Md (G, B) if and only if the ¡ operator 1 ⊗ ζχ (µ) is invertible for all χ ∈ Xb (G). Let µ = µd + µac , where µd ∈ Md (G, B) and µac ∈ Mac (G, B). (c) The spectrum of µ in Md⊕ac (G, B) is the set ¢ ¢ ª [ S© ¡¡ ¢ ¢ ª S© ¡¡ σ 1 ⊗ ζχ (µd ) : χ ∈ Xb (G) σ 1 ⊗ ζχ (µd + µac ) : χ ∈ X(G) . (3) In particular,¡ the measure µ is invertible in Md⊕ac (G, B) if and only if ¢ the operator 1 ⊗ ζ (µ ) is invertible for all χ ∈ Xb (G), and the operator χ d ¡ ¢ 1 ⊗ ζχ (µd + µac ) is invertible for all χ ∈ X(G). (d) The spectrum of µ in Md⊕ac (G, B) is the set ª S© σ(a) : a ∈ ∪{ µ ˇ(χ) : χ ∈ X(G) } , (4) where the bar means the closure in the norm of B. The measure µ is invertible in Md⊕ac (G, B) if and only if the function χ 7→ µ ˇ(χ) is uniformly invertible on X(G), i.e., there exists ε > 0 such that for all χ ∈ X(G) the ¡ ¢−1 ¡ ¢−1 inverse µ ˇ(χ) exists and k µ ˇ(χ) k < 1/ε. (e) The spectrum of µ in Md⊕ac (G, B) is the union of the spectrum of µd in Md (G, B) and the set ¢ ª S© ¡ σ µ ˇ(χ) : χ ∈ X(G) . (5)

270

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

The measure µ is invertible in Md⊕ac (G, B) if and only if the measure µd is invertible in Md (G, B), and the operator µ ˇ(χ) is invertible for all χ ∈ X(G). ] (f) M ac (G, B), Md (G, B), and Md⊕ac (G, B) are full subalgebras of the algebra M(G, B). Let G be a discrete abelian group and µ ∈ M(G, B). (g) The spectrum of µ in M(G, E) is the set (2). In particular, the measure µ is invertible in M(G, B) if and only if the operator µ ˇ(χ) is invertible for all χ ∈ Xb (G). Proof. (c) We recall from 4.2.8 that the space of all characters of the algebra Md⊕ac (R, C) can be identified with Xb (G) t X(G) and the Gel0 fand transform of a measure µ = µd + µac ∈ Md⊕ac (G, C) can be identified with the function ½ ϑµ (χ) =

ζχ (µd )

for χ ∈ Xb (G),

ζχ (µd + µac )

for χ ∈ X(G).

As in 4.5.1, we consider the mapping Θ = 1 ⊗ ϑ : Md⊕ac (G, B) → C(Xb t X, B). It is easily seen that ¢ ½¡ 1 ⊗ ζχ (µd ) for χ ∈ Xb (G), ¢ Θµ (χ) = ¡ 1 ⊗ ζχ (µd + µac ) for χ ∈ X(G). Thus set (3) is the spectrum of Θµ in the algebra C(Xb t X, B). By 4.5.1 the spectrum of µ in the algebra Md⊕ac (G, B) coincides with the set (3). (a) and (b) follow from 4.2.2 and 4.2.3, respectively, in a similar way. (d) We show that (4) coincides with (3). We endow Xb t X with the topology of the character space of Md⊕ac (G, C). We recall that for µ ∈ Md⊕ac (G, C) the Gel0 fand transform ϑµ : Xb t X → C is a continuous function. Hence for µ ∈ Md⊕ac (G, B) the operator-valued Gel0 fand transform Θµ : Xb t X → B is a continuous function, too. We observe that ¡ ¢ the restriction of Θµ to X coincides with the Fourier cotransform µ ˇ(χ) = 1 ⊗ ζχ (µ). Since X is dense in Xb t X (see 4.2.11) and Θµ is continuous the image of Θµ coincides with the set ∪{ µ ˇ(χ) : χ ∈ X(G) }. Therefore (3) and (4) are the same set. If the function Θµ is invertible at all points, it is uniformly invertible because Θµ is a continuous function which domain is a compact set. In particular, since µ ˇ is a restriction of Θµ , µ ˇ is also uniformly invertible. Conversely, if µ ˇ is uniformly invertible then by 1.4.2 Θµ is pointwise invertible. (e) follows from (c) and (b). (f) By 4.2.2, for any χ ∈ X(G) the mapping µ 7→ ζχ (µ) is a character ¡ ¢ of the algebra M(G, C). Therefore for any χ ∈ X(G) the mapping µ 7→ 1 ⊗ ζχ (µ) is a morphism from the algebra M(G, B) = B ⊗π M(G, C) to the algebra ¡ B. On ¢ the other hand, for any µ ∈ M(G, B), its Fourier cotransform µ ˇ(χ) = 1 ⊗ ζχ (µ) is a bounded continuous function. Hence the mapping µ 7→ µ ˇ acts from M(G, B) to C(X, B) and is a morphism of algebras.

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271

Assume µ ∈ Md⊕ac (G, B) is invertible in M(G, B). Then by 1.4.6 µ ˇ is invertible in the algebra C(X, B), i.e., uniformly pointwise invertible, which by (d) implies the invertibility of µ in Md⊕ac . ] Assume µ ∈ M ac or µ ∈ Md is invertible in Md⊕ac . Then by (c), (b), and (a) ] it is invertible in Mac or Md , respectively. (g) follows from 4.2.2 in a way similar to that of the proof of (c). ¤ Remark. It is not true that the spectrum of µ ∈ Md⊕ac (G, B) is the closure of the set ¢ ª S© ¡ σ µ ˇ(χ) : χ ∈ X(G) . We describe briefly a counter-example. Let E be the space lq (Z, C), and let G = R. We define a0 , a1 : E → E as follows. Let . . . , e−1 , e0 , e1 , e2 , . . . , be the standard basis in E = lq . We set a0 e1 = 0 and a1 e1 = e2 , and a0 en = en+1 and a1 en = 0 for all other n. Let g ∈ L1 (G, C) be a function whose Fourier cotransform gˇ does not vanish at all point χ ∈ X(G). We consider the measure µ = a0 ⊗ δ0 + a1 ⊗ g. Clearly,

½ Θµ (χ) =

a0 a0 + gˇ(χ)a1

for χ ∈ Xb (G), for χ ∈ X(G).

It is easy to see that the operator a0 is not invertible. On the other hand, the spectrum of a0 + αa1 , α 6= 0, is the unit circumference, see [Hals2 , problem 85] for © ¡ ¢ ª the proof. Thus ∪ σ µ ˇ(χ) : χ ∈ X(G) = { u : |u| = 1 }, whereas 0 ∈ σ(µ). 4.5.3. The causal invertibility of measures. Let µ ∈ M+ (R, B). For any χ ∈ X+ = X(R+ ) we set Z ∞ ¡ ¢ 1 ⊗ ζχ (µ) = χ(t) dµ(t). 0

We define this integral in the natural way, ¡ ¢ see 4.5.2. We call the function µ e(χ) = 1 ⊗ ζχ (µ) the operator-valued Laplace transform of µ. We recall from 4.3.1 that X+ can be identified with C+ = { τ ∈ C : Re τ ≥ 0 }. Under such an interpretation Z ∞ µ e(τ ) = e−τ t dµ(t) τ ∈ C+ . 0 + + Let µ ∈ M+ d (R, B). For χ ∈ Xb = X(Rd ) we set Z ∞ ¡ ¢ 1 ⊗ ζχ (µ) = χ(t) dµ(t), 0

or, equivalently,

∞ X ¡ ¢ 1 ⊗ ζχ (µ) = χ(hk )ak k=1

272

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

provided P∞µ is represented in the form µ = with k=1 kak k < ∞, see 4.4.12(b).

P∞ k=1

ak ⊗δhk , where hk ≥ 0 and ak ∈ B

+ + ] + We recall that M+ d⊕ac = Md ⊕ Mac . We denote by Mac (R, B) the algebra + ] M+ (R, B) with an adjoint unit. We identify M (R, B) with the subalgebra of ac

ac +

M (R, B) taking 1 ⊗ δ for a unit. Corollary. + ] (a) Let µ ∈ M+ ac (R, B). Then the spectrum of µ in Mac (R, B) is the set [ S© ¡ ª ¢ {0} σ µ e(τ ) : τ ∈ C+ . (6) + ] In particular, the measure αδ + µ, α ∈ C, is invertible in M ac (R, B) if and only if α 6= 0, and the operator α + µ e(τ ) is invertible for all τ ∈ C+ . + (b) Let µ ∈ Md (R, B). Then the spectrum of µ in M+ d (R, B) is the set ¢ ¢ ª S© ¡¡ σ 1 ⊗ ζχ (µ) : χ ∈ X+ . (7) b

In particular, the¢ measure µ is invertible in M+ d (R, B) if and only if the ¡ + operator 1 ⊗ ζχ (µd ) is invertible for all χ ∈ Xb (R). + Let µ = µd + µac , where µd ∈ M+ d (R, B) and µac ∈ Mac (R, B).

(c) The spectrum of µ in M+ d⊕ac (R, B) is the set ¢ ¢ ª [ S ¡¡ ¢ ¢ S© ¡¡ σ 1 ⊗ ζχ (µd ) : χ ∈ X+ { σ 1 ⊗ ζχ (µd + µac ) : χ ∈ X+ }. (8) b (R) In particular, the measure µ is invertible in M+ d⊕ac (R, B) if and only if the ¡ ¢ + ¡operator¢ 1 ⊗ ζχ (µd ) is invertible for all χ+∈ Xb (R), and the operator 1 ⊗ ζχ (µd + µac ) is invertible for all χ ∈ X (R). (d) The spectrum of µ in M+ d⊕ac (R, B) is the set ª S© σ(a) : a ∈ ∪{ µ e(τ ) : τ ∈ C+ } , (9) where the bar means the closure in the norm of B. The measure µ is invertible in M+ e(τ ) is uniformly d⊕ac (R, B) if and only if the function τ 7→ µ + invertible on C , i.e., there exists ε > 0 such that for all τ ∈ C+ the inverse ¡ ¢−1 ¡ ¢−1 µ e(τ ) exists and k µ e(τ ) k < 1/ε. + (e) The spectrum of µ in Md⊕ac (R, B) is the union of the spectrum of µd in M+ d (R, B) and the set ¢ ª S© ¡ σ µ e(τ ) : τ ∈ C+ . (10) The measure µ is invertible in M+ d⊕ac (R, B) if and only if the measure µd is + invertible in Md (R, B), and the operator µ e(τ ) is invertible for all τ ∈ C+ . + + + ] (f) M ac (R, B), Md (R, B), and Md⊕ac (R, B) are full subalgebras of the algebra M+ (R, B).

4.5. INVERTIBILITY AND CAUSAL INVERTIBILITY

273

+ Consider the semi-group R+ d and the algebra M (Rd , B).

(g) Let µ ∈ M+ (Rd , B). Then the spectrum of µ in M+ (Rd , E) is the set (7). In particular, the measure µ is invertible in M+ (Rd , B) if and only if the operator µ e(χ) is invertible for all χ ∈ X+ b . + (h) Let µ ∈ M (Z, B). Then the spectrum of µ in M+ (Z, E) is the set ª ¢ S© ¡ σ µ e(u) : u ∈ U+ ,

(11)

P∞ where µ e(u) = n=0 un µ({n}). In particular, the measure µ is invertible in M+ (Z, B) if and only if the operator µ e(u) is invertible for all u ∈ U+ . Proof. The proof is substantially analogous to that of 4.5.2 and based on the corresponding statements of §4.3. We give only a simplified proof of (f). If, e.g., µ ∈ M+ d (R, B) is invertible in M+ (R, B) then it is a fortiori invertible in M(R, B). By 4.5.2(f) this implies + µ−1 ∈ Md (R, B), which it turn implies µ−1 ∈ M+ d = M ∩ Md . ¤ 4.5.4. A general sufficient criterion. We recall that by B+ = B+ (X, Y ) we denote the space of all bounded causal operators T acting from a Banach space X into a Banach space Y , and by σ + (T ) (see 2.2.2 for details) the causal spectrum of an operator T ∈ B+ (X). Proposition. Let E be a Banach space and B = B(E). (a) Let L be the space Cq (G, E) or Lpq (G, E), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. Suppose µ ∈ M(G, B) is invertible in ¡the algebra ¢ M(G, B). Then the operator Tµ is invertible in the algebra B L(G, E) . Consequently σ(Tµ ) ⊆ σM (µ). (b) Let G = R and L be the space Cq (R, E) or Lpq (R, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Suppose µ ∈ M+ (R, B) is invertible in the algebra M+ (R, B). Then the operator Tµ is causally invertible. Consequently σ + (Tµ ) ⊆ σM+ (µ). We recall from 4.2.12 and 4.3.11 that the inclusions can be proper. Proof. This is an immediate consequence of 1.4.6, 4.4.8, 4.4.9, and 4.4.12.

¤

4.5.5. The invertibility of operators. Let G be a locally compact abelian group and L be the space Cq (G, E) or Lpq (G, E), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. We recall from 4.1.2 that we denote by A = A(L) the algebra of all shift invariant operators T ∈ B(L). We denote by Ad = Ad (L) the set of all operators Tµ , µ ∈ Md (G, B). If G is non-discrete, in a similar way we define Ad⊕ac = Ad⊕ac (L) g g and A ac = Aac (L). Clearly, these sets are subalgebras with a unit of B(L).

274

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

We recall from 4.4.4 that any operator-valued measure µ ∈ M(G, B) induces the convolution operator ¡

¢ Tµ x (t) =

Z dµ(s) x(t − s) Z

=

dµ(t − s) x(s).

In particular, (see 4.4.10 and 4.4.11) measures belonging to Md and Mac induce difference and integral operators, respectively. Hence a measure µ ∈ Md⊕ac induces the difference integral operator Z X ¡ ¢ g(s)x(t − s) dλ(s). Tµ x (t) = ah x(t − h) + G

h∈G

We call the operator

X ¡ ¢ Dx (t) = ah x(t − h) h∈G

the difference part of Tµ . By virtue of 4.4.3(d) and 4.4.9 the difference part D of Tµ is determined uniquely (provided G is non-discrete). In the following ¡ theorem ¢ by the spectrum of an operator we mean the spectrum in the algebra B L(G, E) . Theorem. Let G be a non-discrete locally compact abelian group, and let L be the space Cq (G, E) or Lpq (G, E), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. (a) Let µ ∈ Mac (G, B). Then the spectrum of Tµ is the set (1). In particular, the operator α + Tµ , α ∈ C, is invertible if and only if α 6= 0, and the operator α + µ ˇ(χ) is invertible for all of χ ∈ X(G). (b) Let µ ∈ Md (G, B). Then the spectrum of Tµ is the set (2).¡ In particular, ¢ the operator Tµ is invertible if and only if the operator 1 ⊗ ζχ (µ) is invertible for all of Xb (G). Let µ = µd + µac , where µd ∈ Md (G, B) and µac ∈ Mac (G, B). (c) The spectrum of Tµ is the set (3). the operator Tµ is invert¡ In particular, ¢ ible if and only if ¡the operator 1 ⊗ ζ (µ ) is invertible for all χ ∈ Xb (G), χ d ¢ and the operator 1 ⊗ ζχ (µd + µac ) is invertible for all χ ∈ X(G). (d) The spectrum of Tµ is the set (4). The operator Tµ is invertible if and only if the function χ 7→ µ ˇ(χ) is uniformly invertible on X(G), i.e., there exists ¡ ¢−1 ¡ ¢−1 ε > 0 such that for all χ ∈ X(G), µ ˇ(χ) exists and k µ ˇ(χ) k < 1/ε. (e) The spectrum of Tµ is the union of the spectrum of D and the set (5). The operator Tµ is invertible if and only if the operator D is invertible, and the operator µ ˇ(χ) is invertible for all χ ∈ X(G). ¡ ¢ ¡ ¢ ¡ ¢ g (f) Aac L(G, E) ,¡ Ad L(G, E) , and A L(G, E) are full subalgebras of d⊕ac ¢ the algebra B L(G, E) .

4.5. INVERTIBILITY AND CAUSAL INVERTIBILITY

275

Let G be a discrete abelian group, and let 1 ≤ q ≤ ∞ or q = 0. (g) Let µ ∈ M(G, B). Then the spectrum of Tµ is the set (2). In particular, the operator Tµ is invertible if ¡and only ¢if the operator µ ˇ(χ) is invertible for ¡ ¢ all χ ∈ X(G). The algebra Ad lq (G, E) is full in the algebra B lq (G, E) . Proof. The proof employes corollary 5.2.7 which we shall prove later. (d) If µ ˇ is uniformly invertible then by 4.5.2(d) the measure µ is invertible. Hence by 4.5.4 the operator Tµ is invertible. Conversely, we assume that Tµ is invertible and show that µ ˇ is uniformly in¡ ¢−1 ¡ ¢−1 vertible, i.e., there exists ε > 0 such that µ ˇ(χ) ˇ(χ) exists and k µ k < 1/ε for all¯ χ ∈ ¯X(G). We make use ¯ ¯ of 1.3.2 and show that there exists ε > 0 such that ¯ ¯ ¯ both µ ˇ(χ) + > ε and µ ˇ(χ)¯− > ε for all χ ∈ X(G). ¯ ¯ First we discuss the inequality ¯µ ˇ(χ)¯+ > ε. Assume L is C or Lp∞ . In these cases the functions of the form x(t) = e · χ(t), e ∈ E and χ ∈ X, lie in L and ke · χk = |e|. We observe that ¡ ¢ Tµ (e · χ) = µ ˇ(χ)e · χ. ¯ ¯ Thus kˇ µ(χ)ek ≥ |Tµ |+ · |e| and consequently ¯µ ˇ(χ)¯+ ≥ |Tµ |+ > 0. Now assume that L is the space Cq (G, E) or Lpq (G, E), 1 ≤ p ≤ ∞, 1 ≤ q < ∞ or q = 0. By 4.4.4 Tµ ∈ s. Consequently by 5.2.7 it follows that the invertibility of Tµ on Cq or Lpq implies the invertibility of¯ Tµ on ¯ C∞ or Lp∞ , respectively. For the above reasoning, we obtain the estimate ¯µ ˇ(χ)¯+ ≥ |Tµ : Lp∞ → Lp∞ |+ > 0 or ¯ ¯ ¯µ ˇ(χ)¯+ ≥ |Tµ : C∞ → C∞ |+ > 0, respectively. ¯ ¯ Next, we discuss the inequality ¯µ ˇ(χ)¯− > ε. Let L = Lpq . We consider the operator Tµ as acting on Lp1 . By 5.2.7 it is invertible. Next, we consider the operator TµT of the convolution with the measure µT transposed to the measure µ. As we have seen in 4.4.5, the restriction of the conjugate Tµ0 to Lp0 ∞ (G, E0 ) coincides with TµT . By 1.3.4 |TµT : Lp0 ∞ → Lp0 ∞ |+ ≥ 0 |T ¯ µ |+ ¯ = |T ¯ µ : Lp10 ¯ → Lp1 |− > 0. Repeating the above reasoning we obtain ¯µ ˇ(χ)¯− = ¯(ˇ µ(χ)) ¯+ ≥ |TµT : Lp0 ∞ → Lp0 ∞ |+ > 0. Let L = Cq . We consider the operator Tµ as acting on C1 (G, E) (with the index q = 1). By 5.2.7 it is invertible. We observe that any function y ∈ L1∞ (G, E0 ) induces naturally a functional on C1 . We recall that C1 is embedded isometrically in L∞1 . From 1.8.2 it follows that the unit ball of C1 is dense in the unit ball of L∞1 with respect to the hL∞1 , L1∞ i-topology on L∞1 . This implies that the embedding of L1∞ (G, E0 ) in C10 is isometric. We consider the operator TµT as acting on L1∞ (G, E0 ). Since by 4.4.5 hTµ x, yi = hx, TµT yi for all x ∈ L∞1 and y ∈ L1∞ , the same identity a fortiori holds for all x ∈ C1 . This means¯ that¯ the ˇ(χ)¯− = restriction of Tµ0 to L1∞ coincides with TµT . Now as above we obtain ¯µ ¯ ¯ ¯(ˇ µ(χ))0 ¯+ ≥ |TµT : L1∞ → L1∞ |+ ≥ |Tµ0 : C10 → C10 |+ = |Tµ : C1 → C1 |− > 0. Thus the invertibility of Tµ implies the uniform invertibility of µ ˇ on X(G). Finally, we prove that σ(Tµ ) coincides with set (4). By 4.5.2(d) set (4) is σ(µ). Hence by 4.5.4 σ(Tµ ) is contained in (4).

276

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

Conversely, we assume that λ ∈ / σ(Tµ ) and show that λ is not in (4). Indeed, by what has been proved above, if λ − Tµ is invertible then the function χ → λ − µ ˇ(χ) ¡ ¢−1 is uniformly invertible, i.e., there exists ε > 0 such that λ − µ ˇ(χ) exists and °¡ ¢−1 ° ° λ−µ ° < 1/ε for all χ ∈ X(G). By virtue of 1.4.2 it follows that the ˇ(χ) ¡ ¢ ε-neighbourhood of λ does not intersect σ µ ˇ(χ) . (a, b, c) We take into account 4.5.2(f). In the course of the proof of (d) we have seen that σ(Tµ ) = σ(µ) for all µ ∈ Md⊕ac , where σ(µ) may mean the spectrum in any of the corresponding algebras enumerated in 4.5.2(f). It remains to recall that (1), (2), and (3) coincide with σ(µ). (e) is evident. (f) By what has been proved, if Tµ is invertible then µ is invertible. Hence (Tµ )−1 = Tµ−1 . (g) The proof needs not new ideas. ¤ 4.5.6. The causal invertibility of operators. Let G = R and L = L(R, E) be the space Cq (R, E) or Lpq (R, E), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. We denote by A+ = A+ (L) the algebra of all shift invariant causal operators T ∈ B(L), i.e., + the intersection of A(L) and B+ (L). We denote by A+ d⊕ac = Ad⊕ac (L) the set of + + all operators Tµ , µ ∈ M+ d⊕ac (R, B). In a similar way we define Ad = Ad (L) and g g + + A =A (L). Clearly, these sets are subalgebras with a unit of B+ (L). ac

ac

We recall (see 4.5.5, 4.4.10, and 4.4.11) that an operator-valued measure + µ ∈ M+ d ⊕ Mac induces the difference integral operator Z ∞ X ¡ ¢ Tµ x (t) = ah x(t − h) + g(s)x(t − s) dλ(s) 0

h∈R+

and the operator

X ¡ ¢ Dx (t) = ah x(t − h) h∈R+

is called the difference part of Tµ . We recall (see 4.5.5) that the difference part D is determined uniquely. In the following theorem¡by the causal spectrum of an operator we mean its ¢ + spectrum in the algebra B L(R, E) . Theorem. Let G = R and L be the space Cq (R, E) or Lpq (R, E), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. (a) Let µ ∈ M+ ac (R, B). Then the causal spectrum of Tµ is the set (6). In particular, the operator α + Tµ , α ∈ C, is causally invertible if and only if α 6= 0, and the operator α + µ e(τ ) is invertible for all τ ∈ C+ . + (b) Let µ ∈ Md (R, B). Then the causal spectrum of µ is the set (7). In particular, if and only if the operator ¡ ¢ the operator Tµ is causally invertible + 1 ⊗ ζχ (µd ) is invertible for all χ ∈ Xb (R). + Let µ = µd + µac , where µd ∈ M+ d (R, B) and µac ∈ Mac (R, B). (c) The causal spectrum of Tµ is the set (8). In particular, ¡ ¢ the operator Tµ is causally invertible if and only if the operator 1 ⊗ ζχ (µd ) is invertible

4.5. INVERTIBILITY AND CAUSAL INVERTIBILITY

277

¡ ¢ for all χ ∈ X+ b (R), and the operator 1 ⊗ ζχ (µd + µac ) is invertible for all χ ∈ X+ (R). (d) The causal spectrum of Tµ is the set (9). The operator Tµ is causally invertible if and only if the function τ 7→ µ e(τ ) is uniformly invertible on ¡ ¢−1 + C , i.e., there exists ε > 0 such that for all τ ∈ C+ the inverse µ e(τ ) ¡ ¢−1 exists and k µ e(τ ) k < 1/ε. (e) The causal spectrum of Tµ is the union of the causal spectrum of D and the set (10). The operator Tµ is causally invertible if and only if the operator D is causally invertible, and the operator µ e(τ ) is invertible for all τ ∈ C+ . ¡ ¢ ¡ ¢ ¡ ¢ g + + + (f) A ac L(R, E) , Ad L(R, E) , and Ad⊕ac L(R, E) are full subalgebras of ¡ ¢ the algebra A+ L(R, E) . + Consider the semi-group R+ d and the algebra M (Rd , B). (g) Let µ ∈ M+ (Rd , B). Then the causal spectrum of Tµ on lq (Rd , E), 1 ≤ q ≤ ∞ or q = 0, is the set (7). In particular, the operator Tµ is causally invertible if¡and only ¢if the operator µ e(χ) is invertible for all ¡χ ∈ X+ b . The ¢ + algebra A+ l (R , E) is a full subalgebra of the algebra B l (R d q d , E) . d q Consider the semi-group Z+ and the algebra M+ (Z, B). (h) Let µ ∈ M+ (Z, B). Then the causal spectrum of Tµ on lq (Z, E), 1 ≤ q ≤ ∞ or q = 0, is the set (11). In particular, the operator Tµ is causally invertible if and only if the operator µ e(u) is invertible for all¡u ∈ U+ ¢. The algebra ¡ ¢ + A+ lq (Z, E) . d lq (Z, E) is a full subalgebra of the algebra B

Remark. We recall once again that causal invertibility is equivalent to the stability of the corresponding equation, see ch. 3. ¡ ¢ + Proof. (f) We consider, for example, the case of A+ d L(R, E) . Let µ ∈ Md . Assume that Tµ is causally invertible. Then Tµ is a fortiori ordinary invertible. By 4.5.5(f) its inverse must be an operator Tν , ν ∈ Md . On the other hand, the operator Tν must be causal. By 4.4.12(a) this implies that ν ∈ M+ d. (c, d) By 4.4.3(d) the morphism µ 7→ Tµ is injective. Therefore the algebras + + M+ d⊕ac and Ad⊕ac are isomorphic (not topologically). Hence µ ∈ Md⊕ac is invert+ ible in M+ d⊕ac if and only if Tµ is invertible in Ad⊕ac . But by (f) the invertibility of Tµ in A+ d⊕ac is equivalent to the causal invertibility of Tµ . It remains to recall from 4.5.3 that (8) and (9) coincide with the spectrum of µ in M+ . (a, b, e, g, h) are proved in a similar way. ¤ 4.5.7. The spectrum of a difference operator. The applications of the above results to the calculation of a whole spectrum are usually more convenient than to the verification of invertibility. Below we give some examples. P Theorem. Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. Let am ∈ B, m ∈ Z, and m∈Z kam k < ∞. (a) The spectrum of the operator X ¡ ¢ am x(t − m) Dx (t) = m∈Z

278

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

on lq (Z, E), lq (Rd , E), Cq (R, E), and Lpq (R, E) is the set ´ o [n ³ X σ um am : u ∈ U . m∈Z

(b) Let the family { gi ∈ R : i ∈ N } be linearly independent over the field Q of rational numbers (see example (e) in 4.2.1 for definition). Assume in the family { kmi ∈ Z : i ∈ N }, for every m there are only a finite number of non-zero coefficients kmi . Then the spectrum of the operator ∞ X X ¢ ¡ ¢ ¡ kmi gi Dx (t) = am x t − m∈Z

i=1

on lq (Rd , E), Cq (R, E), and Lpq (R, E) is the set ∞ ´ ´ o [n ³ X ³ Y σ uki mi am : ui ∈ U . m∈Z i=1

(c) Let the family { gi ∈ R : i ∈ N } be linearly independent over the field Q. Assume in the family { rmi ∈ Q : i ∈ N }, for every m there are only a finite number of non-zero coefficients rmi . Then the spectrum of the operator ∞ X X ¡ ¢ ¡ ¢ Dx (t) = am x t − rmi gi m∈Z

i=1

on lq (Rd , E), Cq (R, E), and Lpq (R, E) is the set ∞ ´ ´ o [n ³ X ³Y σ χi (rmi ) am : χi ∈ X(Q) . m∈Z i=1

Remark. We note that the operator considered in (c) gives the most general example of a difference operator induced by a bounded discrete measure. But, although (b) is a special case of (c), the formula in (b) is more effective. Proof. These are consequences of 4.5.5(b, g) and the structure of the character spaces of the groups Z and Rd described P in the example in 4.2.1. (a) Clearly, D = Tµ , where µ = m∈Z am ⊗ δm . The case of lq (Z, E) follows immediately from the isomorphism X(Z) ' U (see the example 4.2.1(a)). The other cases can be regarded as special cases of (b) and (c). (c) Let H be the spanP of {gi } over the field Q. We recall that any element ∞ h ∈ H has the form h = i=1 ri gi , where only a finite number of coefficients ri are not zeroes. By the example 4.2.1(e) the restriction of any character of Rd to Q∞ χ (r H has the form χ(h) = i=1 i i ), where χi are arbitrary characters of Q and P∞ h = i=1 ri gi . P∞ P Clearly, D = Tµ , where µ = m∈Z am ⊗ δhm and hm = i=1 rmi gi . Therefore ∞ ´ X ³Y ¡ ¢ 1 ⊗ ζχ (µ) = χi (rmi ) am , m∈Z i=1

and it remains to refer to (2). (b) is a special case of (c): one should set ui = χi (1). ¤

4.5. INVERTIBILITY AND CAUSAL INVERTIBILITY

279

Example. Let L be Cq (R, C) or Lpq (R, C). (a) By the theorem σ(Sh ) = { u ∈ C : |u| = 1 }, where Sh is the shift operator on L by h 6= 0. ¡ ¢ (b) Consider the operator Dx (t) = x(t − 1) + x(t − 2). By the theorem (or by 1.4.10) σL (D) = { u + u2 : |u| = 1 }. It is interesting to note that R(D) = 2, but −3/2 belongs to the unbounded component (see 1.4.5) of the resolvent set %(D) = C \ σ(D). Thus (see 2.2.2) the operator 3/2 + D is causally invertible. ¡ (c) ¢ More generally, simple straightforward calculations shows that the operator Dx (t) = x(t) + bx(t − 1) + cx(t − 2), b, c ∈ R (!), is not invertible if and only if c = ±b − 1, or c = 1 and |b| ≤ 2. √ ¡ ¢ (d) The spectrum of the operator Dx (t) = 3x(t − 1) + 2x(t − 2) is the ring { u ∈ C : 1 ≤ |u| ≤ 5 }. ¡ ¢ (e) The spectrum of the operator Dx (t) = x(t − 1) + x(t + 1) is the segment { u ∈ R : |u| ≤ 2 }. 4.5.8. The causal spectrum of a difference operator Theorem. Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. Let am ∈ B, m = 0, 1, . . . , P∞ and m=0 kam k < ∞. (a) The causal spectrum of the operator ¡

∞ X

¢

Dx (t) = a0 x(t) +

am x(t − m)

m=1

on lq (Z, E), lq (Rd , E), Cq (R, E), and Lpq (R, E) is the set ∞ ´ o [n ³ X σ a0 + um am : u ∈ U+ . m=1

(b) Let the family { gi ∈ R : gi > 0, i ∈ N } be linearly independent over the field Q. Assume in the family { kmi ∈ Z : i ∈ N }, for every m there are only P∞a finite number of non-zero coefficients kmi , and the numbers hm = i=1 kmi gi are positive. Then the causal spectrum of the operator ¡

¢

Dx (t) = a0 x(t) + = a0 x(t) +

∞ X m=1 ∞ X m=1

am x(t − hm ) ∞ X ¡ ¢ am x t − kmi gi i=1

on lq (Rd , E), Cq (R, E), and Lpq (R, E) is the set ∞ ∞ ³Y ´ ´ o X [ Sn ³ −γhm σ a0 + e uki mi am : ui ∈ U, γ ∈ [0, +∞) . σ(a0 ) m=0

i=1

280

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

(c) Let the family { gi ∈ R : gi > 0, i ∈ N } be linearly independent over the field Q. Assume in the family { rmi ∈ Q : i ∈ N }, for every m there are only P∞a finite number of non-zero coefficients rmi , and the numbers hm = i=1 rmi gi are positive. Then the causal spectrum of the operator ∞ X ¡ ¢ Dx (t) = a0 x(t) + am x(t − hm )

= a0 x(t) +

m=1 ∞ X

∞ X ¡ ¢ am x t − rmi gi

m=0

(12)

i=1

on lq (Rd , E), Cq (R, E), and Lpq (R, E) is the set ∞ ∞ ³Y ´ ´ o [ Sn ³ X −γhm σ(a0 ) σ a0 + e χi (rmi ) am : χi ∈ X(Q), γ ∈ [0, +∞) . m=0

i=1

Proof. The proof is similar to that of 4.5.7 and follows from 4.5.6(b, g, h) and 4.3.1. (a) The case of lq (Z, E) follows immediately from 4.3.1(c). The other cases follow from (b) and (c). (c) According to 4.3.1(b), a non-trivial character of R+ d has the form χ(h) = e−γh χ0 (h), where χ is a character of R and γ ∈ [0, +∞). We represent χ0 as 0 d P∞ Q∞ χ0 (h) = i=1 χi (ri ) forPh = i=1 ri gi (see the proofQof 4.5.7 or the example ∞ ∞ 4.2.1(e)). Then for h = i=1 ri gi we havePχ(h) = e−γh i=1 χi (ri ). ∞ Clearly, D = Tµ , where µ = a0 ⊗ δ0 + m=1 am ⊗ δhm . Therefore ¡

∞ ∞ ³Y ´ X ¢ −γh 1 ⊗ ζχ (µ) = a0 + e χi (rmi ) am . m=1

i=1

As for the trivial character χ∞ we have ¡ ¢ 1 ⊗ ζχ∞ (µ) = a0 . It remains to refer to (7). (b) is a special case of (c). Indeed one should set ui = χi (1). ¤ Example. Let L be Cq (R, C) or Lpq (R, C). (a) By the theorem σ + (Sh ) = { u ∈ C : |u| ≤ 1 }, where Sh is the operator of the shift by h > 0 on L. ¡ ¢ (b) Consider the operator Dx (t) = x(t − 1) + x(t − 2). By the theorem σ + (D) = { u + u2 : |u| ≤ 1 }. ¡ (c) ¢ More generally, simple straightforward calculations shows that the operator Dx (t) = x(t) + bx(t − 1) + cx(t − 2), b, c ∈ R (!), is causally invertible if and only if |b| − 1 < c < 1. Cf. the example in 2.6.6. ¡ ¢ √ (d) The causal spectrum of the operator Dx (t) = 3x(t − 1) + 2x(t − 2) is the ball { u ∈ C : |u| ≤ 5 }.

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281

4.5.9. Differential operators. We consider the spaces (cf. 3.3.1) Cq = Cq (R, E),

Cq1 = Cq1 (R, E),

Cq−1 = Cq−1 (R, E),

Lpq = Lpq (R, E),

1 1 Wpq = Wpq (R, E),

−1 −1 Wpq = Wpq (R, E),

where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. And we assume that (X, Y ) is one of the following pairs (cf. 2.4.1): (Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ).

Let µ = µd + µac , where µd ∈ Md (R, B) and µac ∈ Mac (R, B), and let ν ∈ M(R, B). We consider the operators A = Tµ , B = Tν , and D = Tµd . If (X, Y ) d 1 , Lpq ) we define the operator L : X → Y to be L = A dt is (Cq1 , Cq ) or (Wpq + B. −1 And if (X, Y ) is (Cq , Cq−1 ) or (Lpq , Wpq ) we define the operator L : X → Y to be d A + B. It can be shown that since A is shift invariant it acts on each of the L = dt d d 1 −1 spaces Cq , Cq1 , Cq−1 , Lpq , Wpq , and Wpq , and in fact A dt = dt A. So in all cases L=A

d d + B = A + B. dt dt

We set ΦL (ω) = iω µ ˇ(ω) + νˇ(ω),

ω ∈ R,

WL (τ ) = iτ µ e(τ ) + νe(τ ),

τ ∈ C+ .

Note that these quantities can be defined from the identities L(χω ·e) = χω ΦL (ω)e and L(χτ · e) = χτ WL (τ )e, where χω (t) = eiωt and χτ (t) = eτ t , and e ∈ E. Theorem. (a) The operator L is invertible if and only if the operator D is invertible, and the operator ΦL (ω) is invertible for all ω ∈ R. (b) The operator L is invertible if and only if there exists ε > 0 such that °¡ ¡ ¢−1 ¢−1 ° ° < 1/ε for all ω ∈ R. ΦL (ω) exists, and (ω 2 + 1)1/2 ° ΦL (ω) + (c) Let µ, ν ∈ M . The operator L is causally invertible if and only if the operator D is causally invertible, and the operator WL (τ ) is invertible for all τ ∈ C+ . (d) Let µ, ν ∈ M+ . The operator L is causally invertible if and only if there ex°¡ ¡ ¢−1 ¢−1 ° ° < 1/ε ists ε > 0 such that WL (τ ) exists, and (τ 2 + 1)1/2 ° WL (τ ) + for all τ ∈ C . d Proof. (a, c) We make use of the isomorphism U = dt + 1 : X → Y form 2.3.4 1 and 2.3.9. Consider, for example, the case (X, Y ) = (Cq , Cq ). We consider the operator T = LU −1 . We note that T is (causally) invertible ¡ d ¢¡ d ¢−1 if and only if so is L. We observe that T = LU −1 = A dt = + B dt +1 ¡ d ¡d ¢¡ d ¢−1 ¢−1 −1 A( dt + 1) − A + B dt + 1 = A + (−A + B) dt + 1 = A + (B − A)U .

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From the explicit formula for U −1 it is evident that U −1 considered as acting from Cq into Cq belongs to Aac , i.e., it is induced by a measure lying in Mac . Since Aac is an ideal (see 4.4.12) Tµac + (B − A)U −1 ∈ Aac as well. To complete the proof it remains to apply 4.5.5(e) or 4.5.6(e), respectively. (b, d) One should apply 4.5.5(d) or 4.5.6(d) to T = LU −1 = U −1 L and observe that U −1 = Tη , where ηˇ(ω) = 1/(iω + 1) and ηe(τ ) = 1/(iτ + 1). ¤ Remark. (a) Let L be the operator considered in the theorem. In the pair (X, Y ) we consider the equation Lx = f. Assume that the operator L is causally invertible in (X, Y ). Then it follows immediately from 3.3.4 and 2.1.3 that the equation is input–output stable. The converse statement (i.e., if the equation is input–output stable then L is causally invertible) is also true. For the verification of this fact we shall use some results from ch. 5 (in particular, this illustrates their importance). First we consider the case of stability on (−∞, +∞). By 4.4.4 A, B ∈ s. Utilizing the idea of 5.5.11 we consider the operator T = LU −1 or T = U −1 L. Clearly, T = A + (B − A)U −1 = A + U −1 (B − A). Obviously, T ∈ s too. By 5.2.7 the invertibility of T does not depend on q; from the proof of 5.2.7 it is evident that the causal invertibility of T does not depend on q as well. Hence the causal invertibility of L does not depend on q, too. By 3.3.6 and 3.3.8 it follows that the causal invertibility of L is equivalent to input–output stability. Cf. 6.2.7. (b) We recall that by 3.5.9 the stability is exponential provided A, B ∈ e. (c) Assume A, B ∈ e. We recall from 3.6.4 that the equation Lx = f possesses exponential dichotomy in (X, Y ) if and only if the operator L is invertible. We notice that by the theorem invertibility (and hence the presence of dichotomy) does not depend on the pair (X, Y ). Example. (a) Consider the equation x(t) ˙ + bx(t − h) = f (t) with b ∈ R and h > 0. It can be shown (see, e.g., [Kol2 , ch. 1, §4, 5] or the explanation in (b) below) that the function W (τ ) = iτ + be−τ h does not vanish in C+ , i.e., the equation is stable (input–output and exponentially) if and only if 0 < bh < π/2. (b) Consider the equation x(t) ˙ + ax(t) + bx(t − h) = f (t) with a, b ∈ R and h > 0. It is easy to see that the function Φ(ω) = iω + a + be−ωh has a root in R if and only if a + b = 0, or a = −ω cos hω/ sin hω and b = ω/ sin hω for at least one ω ∈ [0, +∞). These equations determine a family of curves on the plane of the parameters (a, b). They divide the plane into components. In each component the number of roots is constant. By the theorem the component, in which the number of roots is zero, corresponds to stability (input–output and exponential). Simple reasoning (see, e.g., [Kol2 , ch. 1, §4, 6]) shows that stability holds if and only if the parameters (a, b) are contained in the open sector bounded by the curves a+b = 0, a ≥ −1/h, and a(ω) = −ω cos hω/ sin hω, b(ω) = ω/ sin hω, ω ∈ [0, π/h). Equivalently, this component can be described (see [Bel2 , theorem 13.8]) by the p 2 inequalities a > −1/h and −a < b < a + d2 /h2 , where d is the root of the equation hd = a tan d satisfying 0 < d < π. A lot of other examples of this kind can be found, e.g., in [Bel2 ] and [Pin].

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283

4.5.10. Sufficient conditions for causal invertibility. Let Σ ⊆ C be a compact set. We call the union of Σ and all bounded components of the complement of Σ the polynomial hull of Σ and denote it by the symbol po Σ, cf. 1.4.5. P∞ Corollary. Let am ∈ B, m ∈ N, and m=0 kam k < ∞. (a) Let the family { gi ∈ R : gi > 0, i ∈ N } be linearly independent over the field Q. Assume in the family { rmi ∈ Q : i ∈ N }, for every m there are only a finite number of non-zero coefficients rmi , and the numbers hm = P ∞ i=1 rmi gi are positive. Then the causal spectrum of the operator (12) on lq (Rd , E), Cq (R, E), and Lpq (R, E) is contained in the set (cf. 4.5.8(c)) po

∞ ³Y ∞ ´ ´ o [n ³ X σ a0 + χi (rmi ) am : χi ∈ X(Q) . m=0 i=1

(b) Let µ ∈ Md⊕ac (R, B). If the spectrum of Tµ does not intersect the imagd inary axis then the operator L = dt + Tµ is invertible. + (c) ([Sl1 ]) Let µ ∈ Md⊕ac (R, B). If the spectrum of Tµ does not intersect C+ d + Tµ is causally invertible. then the operator L = dt Proof. (a) follows from ¡ and¢ 4.5.7(b). S 1.4.5 (b) By 4.5.5(d) the set { σ µ ˇ(ω) : ω ∈ R } does not intersect imaginary axis. Hence by 4.5.9(c) L is invertible. ¢ S ¡ (c) By 4.5.6(d) the set { σ µ e(τ ) : τ ∈ C+ } does not intersect C+ . Hence by 4.5.9(d) L is causally invertible. ¤ 4.5.11. Periodic functional operators. Let 1 ≤ p ≤ ∞, and let L be the space C = C(Rn , E) or Lp∞ = Lp∞ (Rn , E). For u = (u1 , u2 , . . . , un ) ∈ Un and k = (k1 , k2 , . . . , kn ) ∈ Zn we set uk = k1 u1 + uk22 + · · · + uknn . (We recall that Zn and Un are mutually dual groups, and the function k 7→ uk is a character of Zn .) For any u ∈ Un we denote by Lu the subspace of L consisting of all functions x satisfying the identities Sk x = uk x,

k ∈ Zn ,

¡ ¢ where Sk x (t) = x(t − k). Clearly, it suffices to verify these identities for the basis (1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, 0, . . . , 1) of Zn . Note that L0 consists of all 1-periodic functions with respect to all variables, and each Lu is isomorphic to L0 . We say that an operator T ∈ B(L) is 1-periodic if T Sk = Sk T for all k ∈ Z n . Clearly, the subspaces Lu , u ∈ Un , are invariant under a 1-periodic operator T . Theorem. Assume T ∈ s(L) is 1-periodic, see 1.6.8 for the definition of s. Then the operator T is invertible on L if and only if its restrictions T : Lu → Lu are invertible for all u ∈ Un . Remark. (a) The consideration of the family of the operators T : Lu → Lu , u ∈ Un , may be simpler than the consideration of one operator T on the large

284

IV. SHIFT INVARIANT OPERATORS AND EQUATIONS

space L. For example, each operator T : Lu → Lu may be compact whereas T : L → L is not. (b) By 1.6.8 an operator T of the class s acts on Lpq (respectively, on Cq ) for all q. In 5.2.7 we shall show that the invertibility of T does not depend on q as well. Thus the index q = ∞ in this theorem can be changed to any other. Proof. First assume that L = Lp∞ (Rn , E). With each x ∈ Lp∞ we asson n ciate ¡the family xi (t) = ¢ x(t − i), t ∈ (0, 1] , i ∈ Z . Thus we represent Lp∞ n n as l∞ Z , Lp ((0, 1] , E) , cf. 1.6.3. By definition the operator T : Lp∞ → Lp∞ is n 1-periodic ¡ if nand ¢ only if it commutes with all shifts Sk : ¡l∞ → nl∞¢, k ∈ Z , i.e., if T ∈ A l∞¡(Z ) . Since ¢T ∈ s we can state that T ∈ Ad l∞ ) ¡. We denote by ¡ (Z ¢¢ n n n 1]n , E) { Tk ∈ B Lp ((0, 1] , E) : k ∈ Z } the family lying¡ in l1¡ Z , B Lp ((0,¢¢ which corresponds (see 4.4.9) to the measure µ ∈ M Zn , B Lp ((0, 1]n , E) inducing the operator T . In other words, { Ti−j : i, j ∈ Zn } is the matrix of T . Now, by 4.5.5(g) we obtain that T is invertible if and only if the operators X u−k Tk : Lp ((0, 1]n , E) → Lp ((0, 1]n , E) k∈Zn

are invertible for all u ∈ Un = X(Zn ). For each u ∈ Un ¡we consider the extension (Φu x)k = u−k x, k ∈ Zn , acting from ¢ Lp ((0, 1]n , E) to l∞ Zn , Lp ((0, 1]n , E) . Evidently, Φu establishes an isomorphism between Lp ((0, 1]n , E) and Lu . It is straightforward to P verify that the operators P u u u u k T = T : L → L and k∈Zn u Tk are similar, namely, k∈Zn uk Tk = Φ−1 u T Φu . Clearly, this completes the proof for the case L = Lp∞ . The proof for the case L = C is analogous. We represent C as l∞ (Zn , C♦ ), P where C♦ is C♦0 , see the remark in 1.6.10. In this case k∈Zn u−k Tk acts on C♦ . The extension (Φu x)k = u−k x is again an isomorphism between C♦ and Lu . Hence the above argument remains valid. ¤ 4.5.12. Periodic differential operators. Let 1 ≤ p ≤ ∞, and let L be 1 either C = C(R, E) or Lp∞ = Lp∞ (R, E). We denote by W 1 the space Wp∞ or 1 −1 −1 −1 C , and by W the space Wp∞ or C , respectively. For any u ∈ U we denote 1u −1 u 1 −1 by W (by W ) the subspace ¡ ¢ of W (of W ) consisting of all functions x such that S1 x = ux, where S1 x (t) = x(t − 1). Corollary. Let D, B ∈ s(L) be 1-periodic operators. d (a) The operator L = D dt + B : W 1 → L is invertible if and only if its restrictions L : W 1 u → Lu are invertible for all u ∈ U. d (b) The operator L = dt D + B : L → W −1 is invertible if and only if its restrictions L : Lu → W −1 u are invertible for all u ∈ U. Proof. (a) We consider the operator T = LU −1 : L → L, see 2.3.4 for the definition of U . Clearly, T = D + (B − D)U −1 . It is easy to verify that T ∈ s(L), see 5.5.11 for a more detailed explanation. We observe that S1 U = U S1 . Therefore U establishes an isomorphism between 1u W and Lu . Hence the operator L : W 1 u → Lu is invertible if and only if the operator T : Lu → Lu is invertible. It remains to refer to 4.5.11. (b) is proved in a similar way, cf. 2.5.10. ¤

CHAPTER V

OPERATORS WITH VARYING COEFFICIENTS In this chapter we establish some qualitative analogues of the results of the previous chapter for the not shift invariant case. It is convenient to formulate them in the language of full subalgebras. We recall from 1.4.4 that a subalgebra B of an algebra A with a unit is full if the invertibility in A of any B ∈ B implies B −1 ∈ B. We have already had some experience with full subalgebras. For example, we know that the subalgebra e of all operators with exponentially decreasing memory is full in the algebra of all bounded operators, see 3.4.2 and 3.5.2(d), but the subalgebra B+ of all causal operators is not, see the examples in 1.4.4, 2.1.2, 2.5.1, and 3.3.6. We take as a pattern the full subalgebras of shift invariant operators from 4.5.5(f) and 4.5.6(f). In this chapter we give some generalizations of those results. Full subalgebras arise explicitly or implicitly in various applications. We mention here some ideas. First, the narrower the class of operators or equations considered, the more informative the results one can obtain. If one works in a full subalgebra one may be sure that transformations of operators and equations will not lead to more difficult ones. Second, properties of equations are often connected with the relevant subalgebra being full. For example, in chapter 3 we have seen that the essential parts of the proofs of theorems on exponential stability and exponential dichotomy are reduced to the subalgebra e being full. Another example is the independence of properties of operators or equations from the functional spaces on which they are considered, see §6.3. The properties of the Green function are the properties of the inverse operator; the firsts will be directly deduced from the results on full subalgebras, see §6.4. Finally, we mention the reference to 5.2.7 in the proof of 4.5.5.

5.1. Multiplication operators Let G be a locally compact abelian group, and let X be a functional¡ space ¢ on G. We say that A ∈ B(X) is a multiplication operator if A has the form Ax (t) = a(t)x(t). Multiplication operators are important components of many classes of functional differential equations. Therefore they deserve special attention. In this section we discuss the relation between the preceding explicit definition of a multiplication operator ¡ and ¢ the following abstract definition. Assume that the oscillation operators Ψχ x (t) = χ(t)x(t), χ ∈ X(G), act continuously on X; here X(G) is the dual group of G, see 4.2.1 for details. We say that an operator A ∈ B(X) is oscillation invariant if it commutes with the oscillation operators, i.e., AΨχ = Ψχ A for all χ ∈ X(G).

285

286

V. OPERATORS WITH VARYING COEFFICIENTS

Clearly, a multiplication operator is an example of an oscillation invariant operator. The set of multiplication operators forms a very wide subclass in the set of oscillation invariant operators. Our aim in this section is to discuss when, conversely, an oscillation invariant operator is a multiplication operator. The main result is theorem 5.1.11. It is interesting that the Fourier transform takes shift operators Sh into oscillation operators Ψχ . Therefore the oscillation invariant operators are similar to the shift invariant operators (see 4.1.3). See §6.5 for the discussion of this point of view. 5.1.1. Multiplication operators. Let E be a Banach space and B be B(E). Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. For a locally compact topological space T with a positive measure λ we consider the spaces Lq = Lq (T, E) = Lq (T, E, λ), C = C(T, E), and C0 = C0 (T, E). And for a locally compact abelian group G with the Haar measure λ we consider the spaces Lpq = Lpq (G, E) = Lpq (G, E, λ) and Cq = Cq (G, E). For all these spaces we employ the brief notation L. Let ξ ∈ L∞ (G, B, λ). We call the operator ¡ ¢ Ψξ x (t) = ξ(t)x(t), (1) considered as acting on Lpq , a multiplication operator. We ¡ ¢ denote the set of all multiplication operators ¡ on Lpq¢(G, E) by X = X(Lpq G, E) . In a similar way one defines the set X = X Lq (T, E) . Let ξ ∈ C(G, B). We call operator (1), considered as acting on Cq (G, E), a multiplication operator. We on¢ ¡ ¢ denote the set of all multiplication operators ¡ Cq (G, E) by¡X = X(C¢q G, E) . In a similar way we define the sets X = X C(T, E) and X = X C0 (T, E) . We shall not deal much with multiplication operators on the spaces of continuous functions until §5.6. We have discussed the invertibility of multiplication operators in 2.6.1. Proposition. Let L be defined as above. (a) The mapping ξ 7→ Ψξ is a morphism from the algebra L∞ (G, B) (respectively, C(G, B)) into the algebra B(L). Moreover, kΨξ k = kξk for all ξ. Thus X(L) is a Banach algebra. (b) Let Ψ : Lpq → Lpq be a multiplication operator. For all p and q the subspace Lp0 q0 ⊆ L0pq (see 1.8.1) is invariant under Ψ0 and the restriction Ψ0 : Lp0 q0 → Lp0 q0 of Ψ0 to Lp0 q0 is a multiplication operator. A similar assertion holds for the spaces Lq (T, E, λ). (c) Multiplication operators on Lpq are continuous in the hLpq , Lp0 q0 i-topology (see 1.1.7 and 1.8.2). A similar assertion holds for the spaces Lq (T, E, λ). Proof. (a) and (b) are evident. (c) follows from 1.1.10 and (b). ¤ 5.1.2. Operators with zero memory. If ξ takes its values in C ¡we say that¢ (1) is a scalar multiplication operator. We say that an operator A ∈ B Lpq (G, E) has zero memory if it commutes with all scalar multiplication operators, i.e., AΨξ = Ψξ A

for all ξ ∈ L∞ (G, C).

5.1. MULTIPLICATION OPERATORS

287

In a similar way we define operators with zero memory on the space Lq (T, E). For L = Cq (G, E), or L = C(T, E), or L = C0 (T, E) we say that an operator A ∈ B(L) has zero memory if AΨξ = Ψξ A

for all ξ ∈ C(G, C).

We denote by Xz = Xz (L) the set of all operators A ∈ B(L) with zero memory. We shall not deal much with Xz on the spaces of continuous functions until 5.2.1. Clearly, X ⊆ Xz . Proposition. Xz (L) is a closed full subalgebra of B(L). Proof. This is a special case of 1.4.4(b). ¤ Remark. (a) The projector Pa : C −1 → C −1 has zero memory in the sense of 2.3.8. (b) The elements of the quotient algebra B+ / Rad B+ possess some properties of zero memory, see 2.2.9 and the definition of the semi-norm G. 5.1.3. A characterization of operators with zero memory Lemma. Let L = Lq or L = Lpq . An operator A ∈ B(L) belongs to Xz (L) if and only if it commutes with the operators ¡ ¢ PE x (t) = 1E (t)x(t) of the multiplication by characteristic functions 1E of all measurable subsets E ⊆ G (respectively, E ⊆ T ). Proof. It suffices to observe that the closure of the span of the collection {PM } is the set of all scalar multiplication operators. ¤ 5.1.4. The equality of X(lq ) and Xz (lq ) for a discrete group. Let G be a discrete abelian group. We recall from 1.6.7 that a matrix { Aij : i, j ∈ G } is called main diagonal if Aij = 0 for all i 6= j. ¡ ¢ Proposition.¡ Let G be a discrete abelian group. The algebra X l (G, E) z q ¢ coincides with X lq (G, E) and consists of all operators induced by main diagonal matrices. Proof. Clearly, an operator¡induced ¢ by a main diagonal matrix is a multiplication operator, i.e., it lies in X lq (G) . Let us prove the converse. We denote by Pi the operator of the multiplication by the characteristic function of a point i ∈ G. From the identity Pi Ax = APi x we see that (Ax)i depends on xi only. This implies that the matrix of A is main diagonal and A is restored by its matrix. ¤ 5.1.5. Operators with zero memory on a non-discrete group. Let G be a non-discrete locally compact We recall from 1.6.3 that the ¡ abelian group. ¢ spaces Lpq are defined to be lq I, Lp (Qi , E) and therefore one can use matrix representation for A ∈ B(Lpq ).

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¡ ¢ ¡ ¢ Corollary. An operator A ∈ B Lpq (G, E) ©belongs to¡ Xz Lpq (G, E) ifª and ¢ only if it is induced by a main diagonal matrix Aii ∈ Xz Lp (Qi , E) : i ∈ I . Proof. If A ∈ Xz then by 5.1.4 A is induced by a main diagonal matrix. Furthermore, it is easy to see that Aij commutes with Ψξ . The converse statement is evident. ¤ 5.1.6. A representation of operators A ∈ Xz Theorem. Let G be a locally compact abelian group. ¡ ¢ ¡ ¢ (a) If the space E is finite-dimensional then X L (G, E) = X L (G, E) . z pq pq ¡ ¢ Thus any operator A ∈ Xz Lpq (G, E) can be represented in the form ¡ ¢ Ax (t) = a(t)x(t), (2) where a ∈ L∞ (G, B). ¡ ¢ (b) Any operator A ∈ Xz Cq (G, E) can be represented in the form (2), where a : G → B is bounded ¡and strongly continuous, ¢ ¡ ¢ see 1.1.5 for the definition. In particular, Xz Cq (G, E) = X Cq (G, E) if the space E is finitedimensional. ¡ ¢ ¡ ¢ Proof. (a) By¡5.1.5 it suffices to prove that X L (Q , E) = X L (Q , E) . z p i p i ¢ Assume A ∈ Xz Lp (Qi , E) and E = C. We define a ∈ Lp (Qi , C) by the rule ¡ ¢ a(t) = Au (t), ¡ ¢ where u(t) = 1 for all t. We show that Ax (t) = a(t)x(t) for all x ∈ Lp (Qi , E). From 5.1.3 we have A1E = APE u = PE Au = a1E for any measurable set E. The estimate ka1E kLp ≤ kAk · k1E kLp shows that a ∈ L∞ (Qi , C). It remains to recall that the span of functions {1E } is dense in Lp (Qi , E). The case of an arbitrary finite-dimensional Banach space E is reduced to the preceding case by means of introducing of co-ordinates. One should choose dual 0 0 0 0 bases (see 1.7.1) e1 , e2 , . . . ¡, en ∈ E and ¢ e1 , e2 , . . . , en ∈ E and then consider0 the scalar operators Akm ∈ Xz Lp (Qi , C) defined as follows Akm x = hA(em x), ek i. (b) Let t0 ∈ G, e ∈ E, and let u ∈ Cq be a function such that u(t) = e in a neighbourhood U of t0 . We set ¡ ¢ a(t)e = Au (t), t ∈ U. Clearly, this rule defines a bounded function a : G → B correctly. From the continuity of Au it follows that a is strongly continuous. ¤ Remark. We give an example where assertion (a) is false for an infinitedimensional E. Let ¡ E = L1 (R, ¢ C). In the space L1 (R, E) (it is useful to recall from 1.5.9 that L1 R, L1 (R, C) ' L1 (R × R, C)) we consider the operator ¡¡ ¢ ¢ ¡ ¢ Au (t) (ω) = eiωt x(t) (ω). ¡ ¢ In other words, Ax (t) = a(t)x(t), where a(t) : E → E is the operator of the multiplication by the function ω 7→ eiωt . It is easy to verify that A acts on L1 (R, E) and is bounded. Nevertheless, the function a is not measurable (but is strongly continuous).

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289

5.1.7. The Bohr compactification. Let G be an arbitrary locally compact abelian group, X = X(G), and Gb = X(Xd ), i.e., Gb be the dual of the group X considered with the discrete topology. Clearly, G ⊆ Gb . By 4.2.4 Gb is a compact group. We recall from the Kronecker theorem that G is dense in Gb . Thus Gb is a compact extension of G. The group Gb is called the Bohr compactification of G. Let G and H be locally compact abelian groups. We say that ϕ : G → H is a morphism of topological groups if it is continuous and preserves the group operation, i.e., ϕ(t + s) = ϕ(t) + ϕ(s) for all t, s ∈ G. Proposition ([Dix, theorem 16.1.1]). (a) The natural embedding j : G → Gb is continuous. (b) Any morphism ϕ : G → K of a locally compact abelian group G into a compact abelian group K possesses a continuation to a morphism ϕ : Gb → K. Proof. First we prove an auxiliary statement. For t ∈ G and χ ∈ X(G) we denote χ(t) by the symmetric symbol ht, χi. Let G and H be locally compact abelian groups. Then any morphism ϕ : G → H induces the dual morphism ϕˆ : X(H) → X(G) defined from the identity ht, ϕ(κ)i ˆ = hϕ(t), κi,

t ∈ G, κ ∈ X(H).

In this assertion it only is not evident that ϕˆ is continuous. According to the definition of the¡ topology ¢ on the dual group (see 4.2.1) we consider the neighbourhood WK,ε ϕ(κ ˆ 0 ) = { χ : |ht, χi − ht, ϕ(κ ˆ 0 )i| < ε for all t ∈ K } of ϕ(κ ˆ 0 ), where ε > 0 and K ⊆ G is a compact set. Clearly, ϕˆ maps ¡ the set ¢ Wϕ(K),ε (κ0 ) = { κ : |hh, κi − hh, κ0 i| < ε for all h ∈ ϕ(K) } into WK,ε ϕ(κ ˆ 0 ) . Since ϕ(K) is compact Wϕ(K),ε (κ0 ) is a neighbourhood of κ0 . Thus ϕˆ is continuous. Evidently, ϕˆ ˆ = ϕ. ˆ where 1 : Xd → X is the identity mapping. (a) Indeed, j = 1, ϕ ˆ

1

(b) We consider the diagram X(K) − → X(G) − → Xd (G). Since X(K) and Xd are discrete groups the mapping 1ϕˆ is continuous (though 1 : X → Xd is not continuous). Clearly, its dual 1c ϕˆ : Gb → K gives the desired extension of ϕ. ¤ 5.1.8. The group of isometries of a compact set. Let K be a compact subset of a Banach space (or, more generally, of a compact metric space). We denote by A = A(K) the set of all invertible isometric mappings A : K → K. We endow A with the metric %(A, B) = max{ kAx − Bxk : x ∈ K }. We observe that A is a group with respect to the operation of composition. Lemma. A = A(K) is a compact (non-commutative) topological group. Proof. First we show that A is a topological group. Assume %(A, A0 ) < ε and %(B, B0 ) < ε. Then for any x ∈ K we have (we employ here the the property of

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A, B, A0 , and B0 being isometric) kABx − A0 B0 xk ≤ kABx − AB0 xk + kAB0 x − A0 B0 xk = kBx − B0 xk + kAx − A0 xk ≤ %(B, B0 ) + %(A, A0 ) < 2ε, −1 x − A0 A−1 kA−1 x − A−1 0 xk = kA0 A 0 xk

= kA0 A−1 x − xk = kA0 A−1 x − AA−1 xk = kA0 x − Axk ≤ %(A, A0 ) < ε, which imply that the operations (A, B) 7→ AB and A 7→ A−1 are continuous. Next, we show that A is a complete metric space. If An ∈ A is a Cauchy sequence then, evidently, the pointwise limit A of An is an isometry. On the other hand, from the last estimate it follows that A−1 n ∈ A is a Cauchy sequence, too. It is easy to see that its pointwise limit is the inverse of A. Therefore A is invertible and hence A ∈ A. By Ascoli’s theorem (see, e.g., [Bou1 , ch. 10, §2, 5]) A is compact. Thus A is a compact topological group. ¤ 5.1.9. Almost periodic functions. Let E be a Banach space, G be an arbitrary locally compact abelian group, and X = X(G). Let e1 , e2 , . . . , en ∈ E and χ1 , χ2 , . . . , χn ∈ X. A function x of the form x(t) =

m X

χk (t)ek

(3)

k=1

is called a trigonometric polynomial. We denote by AP = AP (G, E) the closure of the space of all trigonometric polynomials in C(G, E). The functions x ∈ AP (G, E) are called almost periodic. Theorem ([Dix, theorem 16.2.1]). For a function x ∈ C(G, E) the following assumptions are equivalent. (a) x is almost periodic. (b) x possesses an extension to a function x e ∈ C(Gb , E). (c) The set H = H(x) = { Sh x : h ∈ G } is conditionally compact in C(G, E). Proof. (a) ⇒ (c) Assume x is a trigonometric polynomial of the form (3). Then Pn the set H(x) is contained in the set M of all functions xu1 ,...,un (t) = k=1 uk χk (t)ek , where uk ∈ U = { u ∈ C : |u| = 1 }. We observe that M is compact since it is the image of the compact set U × U × · · · × U under the continuous mapping (u1 , . . . , un ) 7→ xu1 ,...,un , see 1.1.1.

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Assume a function z belongs to AP (G, E). For an arbitrary ε > 0 we choose a trigonometric polynomial x such that kx − zk < ε. To prove that H(z) is conditionally compact it suffices to observe that a finite ε-net for H(x) is a finite 2ε-net for H(z). (b) ⇒ (a) First we consider the case where E = C. We make use of the Stone–Weierstrass theorem. In the C ∗ -algebra C(Gb , C) we consider the set H of extensions of all trigonometric polynomials to Gb . Clearly, H is a ∗-subalgebra with a unit of the C ∗ -algebra C(Gb , C). By Pontrjagin’s theorem different points t1 , t2 ∈ Gb induce different characters of Xd (G). This means that H separates points. Hence H is dense in C(Gb , C). The case of an arbitrary E follows from 1.5.3. (c) ⇒ (b) We fix a function x satisfying (c). Clearly, for any h ∈ G the mapping Sh : C → C is an isometry and maps the set H = H(x) onto itself. Therefore it maps the closure H of H onto itself, too. We stress that H is compact (with respect to the metric induced by the embedding in C). We denote by A = A(H) the compact group of all invertible isometric mappings A : H → H. We endow A with the metric %(A, B) = max{ kAx − Bxk : x ∈ H }, see 5.1.8. For h ∈ G we denote by Ah ∈ A the mapping from H to H induced by the operator Sh . We show that for any x ∈ H the function h 7→ Ah x = Sh x is continuous in the metric of C. Clearly, it suffices to verify the continuity at the point h = 0. We take an arbitrary open neighbourhood U ⊆ H of x and consider the compact set H \ U . For any y ∈ H \ U there exists a point ty ∈ G such that y(ty ) 6= x(ty ). We set ε = εy = |y(ty ) − x(ty )|. Let Vy be an ε/3-neighbourhood of y in C. We choose a neighbourhood Wy ⊆ G of the zero of G so that |x(ty ) − x(ty − g)| < ε/3 for all g ∈ Wy . Then Sh x ∈ / Vy for all h ∈ Wy . Next, we choose a finite covering Vy1 , Vy2 , . . . , Vyn of H \ U . Finally, we set W = Wy1 ∩ Wy2 ∩ · · · ∩ Wyn . For h ∈ W we have Sh x ∈ / Vy1 ∪ Vy2 ∪ · · · ∪ Vyn , i.e., Sh x ∈ / H \ U or Sh x ∈ U , which means the continuity of the function x 7→ Ah x. Now we show that the mapping h 7→ Ah is continuous with respect to the metric %. Assume for a given ε > 0 a finite set { x1 , . . . , xm } is an ε-net for H. By what has been proved, for any k we can choose Vk such that kAh xk − xk k < ε for all h ∈ Vk . Now let x ∈ H and h ∈ V1 ∩· · ·∩Vm be arbitrary. We choose k so that kx− xk k ≤ ε. Then we have kAh x−xk ≤ kAh x−Ah xk k+kAh xk −xk k+kxk −xk < 3ε, which proves the continuity at h = 0. The case of an arbitrary h is easily reduced to that of h = 0. We denote by K the closure of { Ah : h ∈ G } in A. Obviously, K is a compact abelian group. Thus the mapping h 7→ Ah is a morphism of topological groups from G to K. By 5.1.7 it¡possesses ¢ an extension ¡ ¢ to the morphism ge 7→ Age from Gb to K.¡ We observe that A−h x (0) = S−h x (0) = x(h). Therefore the function ¢ ge 7→ A−eg x (0), ge ∈ Gb , is the required extension of x. ¤ 5.1.10. Oscillation invariant operators. Let G be an arbitrary locally compact abelian group. For any χ ∈ X(G) (see 4.2.1 for the definition of X(G))

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we consider the oscillation operator ¡

¢ Ψχ x (t) = χ(t)x(t).

Clearly, Ψχ acts isometrically on Lpq and Cq . Let X and Y be spaces of functions on G such that oscillation operators act continuously both on X and Y . We say that an operator A ∈ B(X, Y ) is oscillation invariant if it commutes with all oscillation operators, i.e., AΨχ = Ψχ A

for all χ ∈ X(G).

We denote by XΨ = XΨ (X, Y ) the collection of all oscillation invariant operators A ∈ B(X, Y ). As usual, if X = Y we employ the brief notation XΨ (X). Let E be an arbitrary Banach space, and let L = L(G, E) be one of the spaces Lpq (G, E) or Cq (G, E), 1 ≤ p ≤ ∞, and ¡ 1≤q≤ ¢ ∞ or q = 0. By the definition of an almost periodic function, A ∈ XΨ L(G, E) commutes with operators of the multiplication by scalar almost periodic functions: AΨϑ x = Ψϑ Ax

for all ϑ ∈ AP (G, C).

It is easy to see that X ⊆ Xz ⊆ XΨ . ¡ ¢ ¡ ¢ Proposition. XΨ L(G, E) is a closed full subalgebra of B L(G, E) . Proof. This is a special case of 1.4.4(b). ¤ 5.1.11. The equality of XΨ and Xz Theorem.

¡ ¢ ¡ ¢ (a) If q 6= ∞ then XΨ Lpq (G, E)¡ = Xz Lpq ¢(G, E) . (b) Assume an operator A ∈ XΨ Lp∞ (G, E) is continuous with ¡ ¢ respect to the hLp∞ , Lp0 1 i-topology, cf. 5.1.1. Then A ∈ Xz Lp∞ (G, E) .

Proof. We break the argument into five steps. (i) Assume N ⊆ G is a compact set. We note that since the natural embedding j : G → Gb is continuous (see 5.1.7(a)), by 1.1.1 the topologies induced on N by embeddings in G and Gb coincide. Therefore by Urysohn’s theorem any continuous function ϑ : N → C possesses an extension to a continuous function ϑ : Gb → C and consequently, by 5.1.9, to an almost periodic function ϑ : G → C. (ii) We show that if a function x ∈ Lpq (G, E) is supported in a compact set N then AΨζ x = Ψζ Ax for all ζ ∈ C(G, C). Indeed, let ϑ be an almost periodic function which coincides with ζ on N . Then AΨζ x = AΨϑ x = Ψϑ Ax.

5.1. MULTIPLICATION OPERATORS

293

Next, we observe that without loss of generality we may assume that ϑ is equal to a given constant on a given compact set R that does not intersect N . The arbitrariness of R and the independence of the right hand side of the last identity from ϑ imply that Ax vanishes outside N . Therefore AΨζ x = Ψϑ Ax = Ψζ Ax. (iii) We show that if a function x ∈ Lpq (G, E) is supported in a compact set N then Ψξ Ax = AΨξ x for all ξ ∈ L∞ (G, C). We choose a sequence ζn ∈ C(G, C) such that Z kζn kC ≤ kξkL∞ and lim |ζn − ξ| dλ = 0. n→∞

N

We observe that for any function z ∈ L∞ (G, E) supported in N the sequence Ψζn z converges to Ψξ z in the L1 -norm and consequently in the Lpq -norm. Now let x ∈ Lpq (G, E) be an arbitrary function supported in N . We take a sequence zk ∈ L∞ (G, E) (note L∞ , not Lpq !) which is supported in N and converges to x in the Lpq -norm. For all n and k we have the estimate kΨζn x − Ψξ xk ≤ kΨζn (x − zk )k + kΨζn zk − Ψξ zk k + kΨξ (zk − x)k ≤ kζn kL∞ · kx − zk k + kΨζn zk − Ψξ zk k + kξkL∞ · kzk − xk. Given any ε > 0, first we pick k such that kζn kL∞ · kx − zk k + kξk · kzk − xk < ε (we recall that kζn k ≤ kξk) and then n such that kΨζn zk − Ψξ zk k < ε. From the preceding estimate we obtain kΨζn x − Ψξ xk < 3ε, which means that the sequence Ψζn x converges to Ψξ x in the Lpq -norm for any x ∈ Lpq (G, E) supported in N . By (ii) the function Ax is also supported in N and AΨζn x = Ψζn Ax. Passing to the limit we obtain AΨξ x = Ψξ Ax. (iv) Assume q 6= ∞. Then the set of all compactly supported functions lying in Lpq is dense in Lpq (G, E). Therefore from (iii) we obtain Ψξ A = AΨξ for all ξ ∈ L∞ (G, C). (v) Assume q = ∞. From (iii) we obtain Ψξ Ax = AΨξ x for all ξ ∈ L∞ (G, C) and x ∈ Lp0 ⊆ Lp∞ . By 1.8.2 Lp0 is dense in Lp∞ in the hLp∞ , Lp0 1 i-topology. By assumption and 5.1.1 the operators A and Ψξ are hLp∞ , Lp0 1 i-continuous. Therefore Ψξ Ax = AΨξ x for all ξ ∈ L∞ (G, C) and x ∈ Lp∞ . ¤ Example. We give an example where XΨ 6= Xz . Let l∞ = l∞ (Z, C). For convenience we shall write elements of l∞ as functions. We consider the family of subsets Un,ω,ε = { k ∈ Z : k > n, |eiωk − 1| < ε } of Z depending on the parameters n ∈ N, ω ∈ [0, 2π), and ε > 0. First we show that any finite intersection ∩N j=1 Unj ,ωj ,εj is non-empty. Indeed, PN we consider the almost periodic function ϑ(k) = j=1 eiωj k , k ∈ Z. We take a sequence nm ∈ N such that km = nm+1 − nm → +∞ as m → ∞. By the definition of an almost periodic function, without loss of generality we may assume that the

294

V. OPERATORS WITH VARYING COEFFICIENTS

sequence Snm ϑ is a Cauchy sequence in l∞ . In particular, ¡ kSnm+1 ϑ − Snm¢ϑk → 0 or, equivalently, kSnm+1 −nm ϑ − ϑk → 0. Substituting in Snm+1 −nm ϑ − ϑ (k) the point km = nm+1 −nm we obtain |ϑ(0)−ϑ(nm+1 −nm )| → 0. Evidently, ϑ(0) = N . Thus ϑ(km ) = ϑ(nm+1 − nm ) → N , which is possible only if eiωj km → 1 for all j. Therefore for any nj and εj the intersection of the sets Unj ,ωj ,εj is non-empty. We recall from [Bou1 , ch. 1, §6] that a collection ∇ of subsets of a set Z is called a filter on Z if (i) U ∈ ∇ and V ⊇ U imply V ∈ ∇, (ii) U, V ∈ ∇ implies U ∩V ∈ ∇, and (iii) ∅ ∈ / ∇. A filter is called an ultrafilter if it can not be extended to a wider filter. It is easy to show that a filter ∇ is an ultrafilter if and only if for any subset W ⊆ Z either W or Z \ W belongs to ∇. From Zorn’s lemma it follows that any filter can be extended to an ultrafilter. We consider an ultrafilter ∇ on Z which contains all the sets Un,ω,ε , where n ∈ N, ω ∈ [0, 2π), and ε > 0. For any x ∈ l∞ we set \ f (x) = { x(V ) : V ∈ ∇ }, where the bar means the closure. Since ∇ is an ultrafilter this intersection consists of one point. Indeed, suppose the contrary, i.e., f (x) contains two distinct points t1 and t2 . Let O1 and O2 be disjoint neighbourhoods of t1 and t2 , respectively. Then the collection ∇1 = { U ⊆ Z : x−1 (O1 ) ∪ U ∈ ∇ } is a filter; it is wider than ∇ because x−1 (C \ O1 ) ∈ ∇1 , but x−1 (C \ O1 ) ∈ / ∇; consequently ∇ is not an ultrafilter. Thus we can interpret f (x) as a complex number. It is straightforward to verify that the mapping f : x 7→ f (x) is a linear bounded functional (and, moreover, a character of l∞ ). We observe that by virtue of the choice of Un,ω,ε , for all x ∈ l∞ we have f (Ψω x) = f (x), ω ∈ [0, 2π), ¢ where Ψω x (k) = eiωk x(k) plays the role of Ψχ on the group Z. We set ¡ ¢ Ax (k) = f (S−k x), ¡

cf. the example in 1.6.4. We show that A ∈ XΨ . Indeed, ¡ ¢ AΨω x (k) = f (S−k Ψω x) = f (eiωk Ψω S−k x) = eiωk f (Ψω S−k x) = eiωk f (S−k x) ¡ ¢ = eiωk Ax (k) ¡ ¢ = Ψω Ax (k). Finally, we show that A ∈ / Xz . Clearly, the operator A is equal to zero on the subspace l0 ⊆ l∞ and consequently has the zero matrix. If A were in Xz then by 5.1.5 A would be restored by its matrix, which in turn would imply A = 0. But A 6= 0; namely, A takes the function x(k) = 1, k ∈ Z, into itself. (In particular, by 1.6.6(c) this implies that A ∈ / t.) It is interesting to note that A ∈ A, i.e., Sk AS−k = A for all k ∈ Z, too.

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295

5.1.12. The algebras x, xz , and xψ . Let us fix a representation of G in ¡ ¢ c the form G = R × V (see 1.6.1) and recall that Lpq (G, E) is lq I, Lp (Qi , E) , and Cq (G, E) can be represented as lq (I, C♦i ).¡ Formally, ¢the definitions of X, Xz , and XΨ makes no sense on the spaces lq I, Lp (Qi , E) and lq (I, C♦i ) since Lp (Qi , E) and C♦i depend on i. But they may be modified simply for this case. For any element d ∈ D we fix a point d¯ ∈ d. Then Qi = (k + [0, 1]c ) × d, c ¯ + Q, and one can identify i= as (k, d) ¡ (k, d) ∈ I ¢= Z × D, ¡ can be represented ¢ lq I, Lp (Qi , E) with lq I, Lp (Q, E) , see 1.6.3 for details; similarly,¡we identify ¢ L (G, E) , C♦i with C , see the remark in 1.6.10. We define classes x = x ♦ pq ¡ ¡ ¢ ¢ xz = xz Lpq (G, E) , and x¡ψ = xψ Lpq¢ (G, E) to be the corresponding¡ classes X, ¢ Xz , and¡ XΨ on lq¢(I) = lq I, Lp (Q, E) . And we define classes x = x C (G, E) , q ¡ ¢ xz = xz Cq (G, E) , and xψ = xψ Cq (G, E) to be the corresponding classes X, Xz , and XΨ on lq (I) =¡ lq (I, C♦ ). ¢Thus the results of this section hold for the discrete representations lq I, Lp (Q, E) of Lpq (G, E) and lq (I, C♦ ) of Cq (G, E). The following proposition shows that the definitions of x, xz , and xψ do not depend on the choice of d¯ ∈ d. Hence ¡the results of ¢ this section also make sense for the usual discrete representation lq I, Lp (Qi , E) of Lpq and lq (I, C♦i ) of Cq . Proposition.

¡ ¢ ¡ ¢ (a) We denote by Θ : lq I, Lp (Q, E) → lq I, Lp (Qi , E) (by Θ : lq (I, C♦ ) → lq (I, C♦i )) the isomorphism induced by the choice of d¯ ∈ d. We claim that the isomorphism Θ is induced by a main diagonal matrix. (b) The algebra x = xz¢ consists of all bounded linear operators acting on ¡ Lpq = lq I, Lp (Qi , E) (respectively, on Cq ' lq (I, C♦i )) induced by main diagonal matrices. (c) For any κ ∈ X(I) we define the operators (ψκ x)i = hi, κixi ,

i ∈ I,

¡ ¢ ¡ ¢ (here hi, κi means κ(i)) on the spaces lq I, Lp (Q, E) and lq I, Lp (Qi , E) (respectively, on the spaces lq (I, C♦ ) and lq (I, C♦i )). These operators coincide as operators on Lpq (respectively, on Cq ); namely, Θψκ Θ−1 = ψκ . (d) The algebra xψ consists of ¡ ¢ ¡ all operators ¢ commuting with all operators ψκ : lq I, Lp (Qi , E) → lq I, Lp (Qi , E) (respectively, with all operators ψκ : lq (I, C♦i ) → lq (I, C♦i )), κ ∈ X(I). Proof. (a) follows immediately from the definition of Θ. ¡ ¢ (b) By definition x = xz (see 5.1.4) consists of all operators on lq I, Lp (Q, E) or lq (I, C♦ ), respectively, induced by main diagonal matrices. By (a) Θ preserves this property. (c) follows from (a). (d) ¡By definition of all ¢ xψ consists ¡ ¢ operators which commute with operators ψκ : lq I, Lp (Q, E) → lq I, Lp (Q, E) (respectively, ψκ : lq (I, C♦ ) → lq (I, C♦ )), κ ∈ X(I). It remains to quote (c). ¤

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V. OPERATORS WITH VARYING COEFFICIENTS

5.2. Difference operators

¡ ¢ P∞ We call an operator of the form Dx (t) = m=1 am (t)x(t − hm ) a difference operator. Difference operators and difference equations Dx = f are of interest in themselves. Furthermore, difference operators are the usual parts of differential difference equations. Various assumptions about the convergence of the series generate distinct classes of difference operators. On the other hand, if in ¡ the¢definition of a difference operator we change the multiplication operators Am x (t) = am (t)x(t) into operators with zero memory or oscillation invariant operators we also obtain modifications of the definition. Some natural classes of such operators are investigated in this section. The main result is theorem 5.2.5. 5.2.1. Difference operators with summable memory. Let E be a Banach space. We denote by L = L(G, E) one of the spaces Lpq (G, E) or Cq (G, E), where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. For any h ∈ G we consider the shift operator ¡ ¢ Sh x (t) = x(t − h). We recall from 1.6.12 that Sh are uniformly bounded. We denote by SΨ =¡SΨ (L) (respectively, by S = S(L) and Sz = Sz (L)) the set ¢ of all operators D ∈ B L(G, E) of the form D=

∞ X

Am Shm ,

(3)

m=1

where P∞ hm ∈ G and Am ∈ XΨ (L) (respectively, Am ∈ X(L) or Am ∈ Xz (L)), with m=1 kAm Shm k < ∞. Instead of (3) we shall often employ the notation D=

X

Ah S h ,

(4)

h∈G

P where h∈G kAh Sh k < ∞ and, in particular, only a countable number of the coefficients Ah can be non-zero. Clearly, (see 4.5.5 for the definition of Ad ) Ad ⊆ S ⊆ Sz ⊆ SΨ . We call the operators D ∈ S difference operators with summable memory. We note that by the definition of X any difference operator can be represented by the explicit formula ∞ X ¡ ¢ Dx (t) = am (t)x(t − hm ), (5) m=1

P∞ where am ∈ L∞ (G, B) or am ∈ C(G, B) and m=1 kam k < ∞. Let us fix a representation of G in the form Rc × V. For any element d ∈ D¢ ¡ ¯ we fix a point d ∈ d. Then we represent the space Lpq as lq (I) = lq I, Lp (Q, E) and we represent the space Cq as lq (I) = lq (I, C♦ ), see 1.6.3, 1.6.10, and 5.1.12

5.2. DIFFERENCE OPERATORS

297

for details. We define s(L), sz (L), and sψ (L) to be the corresponding classes S, Sz , and Sψ associated with the discrete P∞ group I. Thus s, sz , and sψ consist of all operators D of the form D = ¡ m=1 Am sj¢m , where to x, xz , or ¡ Am belongs ¢ xψ , respectively, jm ∈ I, and sj : lq I, Lp (Q, E) → lq I, Lp (Q, E) or, respectively, sj : lq (I, C♦ ) → lq (I, C♦ ) is the operator (sj x)i = xi−j of the shift by j ∈ I. We stress that the operator sj , j ∈ I, depends on the choice of d¯ ∈ d and may not coincide with anyone of operators Sh : L → L, h ∈ G. Since x = xz (see 5.1.4 for¡ this equality) we have s = sz . Clearly, s = sz is ¢ the set of all operators on lq I, Lp (Q, E) (or lq (I, C♦ ), respectively) induced by matrices with ¡ summable ¢ memory. By 5.1.12(a) the same description of s = sz holds on lq I, Lp (Qi , E) and lq (I, C♦i ). Thus our new definition of s is in agreement with that in 1.6.8. In particular, from this description it is evident that the definition of s = sz does not depend on the choice of d¯ ∈ d. Proposition. (a) If A belongs to XΨ (X or Xz ) then for any h ∈ G the operator S−h ASh belongs to XΨ (X or Xz ), too. (b) The sets S, Sz , SΨ , s = sz , and sΨ are algebras with a unit. (c) The definition of sψ does not depend on the choice of d¯ ∈ d. ¡ ¢ (d) Let D ∈ S Lpq (G, E) . For all p and q the subspace Lp0 q0 ⊆ L0pq (see 1.8.1) is invariant under D0 and the restriction D0 : Lp0 q0 → Lp0 q0 of D0 to Lp0 q0 belongs to S, too. Proof. (a) and (b) follows immediately from definitions. (c) We consider, for example, the case L = Cq . Assume we have two variants of the choice: d¯ ∈ d and e d¯ ∈ d. We consider the corresponding isomorphisms e : lq (I, C♦ ) → lq (I, C♦i ), and the induced autoΘ : lq (I, C♦ ) → lq (I, C♦i ) and Θ e : lq (I, C♦ ) → lq (I, C♦ ). morphism Υ = Θ−1 Θ P∞ Assume D belongs to sψ according to the choice of d¯ ∈ d and D = m=1 Am sjm is the corresponding representation. We show that Υ−1 DΥ belongs to sψ with respect to the choice e d¯ ∈ d. Indeed, Υ−1 Am Υ ∈ sψ by 5.1.12(d). On the other −1 hand, Υ sjm Υ ∈ s since sjm ∈ s and the definition of s does not depend on the choice of d¯ ∈ d. It remains to recall that s ⊆ sψ and that sψ is an algebra. (d) follows from definition (4), 5.1.1, and assertion (a). ¤ 5.2.2. The equality of S, Sz , and SΨ . We recall from 1.6.8 and 5.2.1 that we denote by s the algebra of all operators with summable memory. Proposition. ¡ ¢ ¡ ¢ (a) Let G be a discrete abelian ¡group. Then S l (G, E) = s l (G, E) z q z q ¢ ¡ ¢ coincides with the algebra S l (G, E) = s l (G, E) . If q 6= ∞ then q q ¡ ¢ ¡ ¢ SΨ lq (G, E) = sψ lq (G, E) coincides with them, too. (b) Let G be an arbitrary locally compact abelian group. If the space E is finitedimensional then ¡ S(L) = ¢Sz (L) ¡for L = Lpq ¢ (G, E) and L = Cq (G, E). If q 6= ∞ then Sz Lpq (G, E) = SΨ Lpq (G, E) . Proof. (a) follows from 5.1.4 and 5.1.11, (b) follows from 5.1.6 and 5.1.11. ¤

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5.2.3. The mean value of a character. Let G be an arbitrary locally R compact abelian group. We normalize the Haar measure on Xb so that Xb dκ = 1. Lemma. For any κ ∈ Xb (G) one has ½ Z 1 hh, κi dκ = 0 Xb

for h = 0, for h 6= 0.

R Proof. If h = 0 then hh, κi = 1 for all h, and hence hh, κi dκ = 1. Assume measure, for any κ0 ∈ Xb R h 6= 0. From Rthe shift invariance of the Haar R we have hh, κi dκ = hh, κ + κ0 i dκ = hh, κ0 i hh, κi dκ. Since h 6= 0, by RPontrjagin’s theorem we have hh, κ0 i 6= 1 for at least one κ0 ∈ Xb , which implies hh, κi dκ = 0. ¤ 5.2.4. The crossed product S(G, L). In this subsection we define the algebra S(G, L), an analogue of the algebra of matrices for operators of the class S. Let L be either Lpq = Lpq (G, ¡ E) or C¢q = Cq (G, E). We denote by SΨ = SΨ (G, L) the subspace of l1 = l1 Gd , B(L) consisting of all families of the form { Ah Sh : h ∈ G }, where Ah ∈ XΨ (L). In other words, SΨ (G, L) consists of all families { Ah Sh : h ∈ G }, where Ah ∈ XΨ (L), bounded by the norm { Ah Sh : h ∈ G } =

X

kAh Sh k.

h∈G

Clearly, SΨ is a closed subspace of l1 . We note that the multiplication in SΨ induced by the convolution in l1 is defined by the rule n ³X ´ o { Ah Sh : h ∈ G } · { Bg Sg : g ∈ G } = Ah (Sh Bf −h S−h ) Sf : f ∈ G . h∈G

We recall from 5.2.1 that Sh Bf −h S−h ∈ XΨ (L). Therefore the product of elements of SΨ lies in SΨ . Thus SΨ (G, L) is a Banach algebra. The algebra SΨ = SΨ (G, L) is called the crossed product of the group {Sh } and the algebra XΨ (L). In a similar way we define S = S(G, L) and Sz = Sz (G, L). Proposition. P (a) The natural mapping { Ah Sh : h ∈ G } 7→ ¡ h∈G Ah S ¢ h is a morphism of algebras with a unit from¡SΨ (G, L) ¢ onto SΨ L(G, E) . (b) For any operator D ∈ SΨ L(G, E) of the form (4) one has kAh Sh k ≤ kDk. In particular, for the operator (5) one has kam kL∞ ≤ kDk ¡ · kS−h¢m k. Consequently the natural morphism from SΨ (G, L) onto SΨ L(G, E) is injective. (c) The summands Ah Sh in the representation (4) of D ∈ SΨ are determined uniquely.

5.2. DIFFERENCE OPERATORS

299

Proof. (a) is evident. (b) For h ∈ G and κ ∈ Xb (G) = X(Gd ) we denote κ(h) by hh, κi. Let D ∈ SΨ (L) be an operator of the form (4). For any κ ∈ Xb (G) we set Dκ =

X

hh, κiAh Sh .

(6)

h∈G

We observe that for χ ∈ X(G) ⊆ Xb (G) Dχ = Ψχ DΨ−1 χ ,

(7)

¡ ¢ where Ψχ x (t) = χ(t)x(t), see 5.1.10. Clearly, kDχ k = kDk for χ ∈ X. Since X is dense in Xb , from the continuity of κ 7→ Dκ we have kDκ k = kDk for κ ∈ Xb , too. From (6) and 5.2.3 it is evident that for any h0 ∈ G Z Xb

h−h0 , κiDκ dκ = Ah0 Sh0 .

It remains to note that the norm of the left side is less than or equal to kDk. The inequality kam kL∞ ≤ kDk · kS−hm k follows from the estimate kam kL∞ = kAm k = kAm Shm S−hm k ≤ kAm Shm k · kS−hm k ≤ kDk · kS−hm k. P (c)P Suppose an operator D ∈ SΨ has two representations D = h∈G Ah Sh and P 0 0 D = h∈G Ah Sh . Clearly, the sum h∈G (Ah − Ah )Sh is a representation of the zero operator D −D. Then from (b) it follows that k(Ah −A0h )Sh k = 0, h ∈ G. ¤ 5.2.5. The subalgebra SΨ is full. Let E be a Banach space and L = L(G, E) be either Lpq (G, E) or Cq (G, E), where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Theorem. (a) The subalgebra SΨ (L) is full in the algebra B(L). (b) The subalgebra sψ (L) is full in the algebra B(L). Proof. (a) Consider an operator D ∈ SΨ (L) represented in the form (4). For any κ ∈ Xb (G) we consider the operator (6). We assume that the operator D is invertible in the algebra B(L). Then from (7) it is evident that operators Dχ , χ ∈ X, are also invertible and k(Dχ )−1 k = kD−1 k. Since X is dense in Xb , by −1 −1 1.4.2 the operators Dκ are invertible for all ¡ κ ∈ Xb ¢too, with ¡ k(Dκ ) ¢k = kD k. Let us consider the Banach algebra l1 Gd , B(L) ' Md Gd , B(L) . We recall from 4.5.2(g) (see also 4.4.9(b)) that an element { Th ∈ B(L) : h ∈ G } of this

300

V. OPERATORS WITH VARYING COEFFICIENTS

P algebra is invertible if and only if the operators Tκ = h∈G hh, κiTh are invertible for all κ ∈ Xb . ¡ ¢ We consider the family { Ah Sh ∈ B(L) : h ∈ G } ∈ l1 Gd , B(L) associated with the operator D. As we have seen above, the operators Dκ are invertible. Hence ¡ ¢ ¡ ¢ the family {Ah Sh } is invertible in l1 Gd , B(L) . Assume {R } ∈ l G , B(L) is h 1 d P the inverse of it. Then the function κ 7→ h∈G hh, κiRh is the pointwise inverse of the function κ 7→ Dκ . Thus the operators (Dκ )−1 can be represented in the form X (Dκ )−1 = hh, κiRh , h∈G

P

for some Rh ∈ B(L), with h∈G kRh k < ∞. We show that Rh has the form Rh = Bh Sh , where Bh ∈ XΨ (L). This will imply that D−1 ∈ SΨ (L) and the proof will be complete. We begin with the simplest case h = 0. First we note the following auxiliary identity which follows directly from the preceding representation for (Dκ )−1 and 5.2.3: Z Z −1 (Dκ ) dκ = R0 or more generally h−h, κi(Dκ )−1 dκ = Rh dκ. Xb

Xb

From the evident identity Dκ+χ = Ψχ Dκ Ψ−1 χ ,

χ ∈ X, κ ∈ Xb ,

(8)

we obtain the similar identity for (Dκ )−1 : (Dκ+χ )−1 = Ψχ (Dκ )−1 Ψ−1 χ ,

χ ∈ X, κ ∈ Xb .

Taking the integral over Xb (and using the shift invariance of the Haar measure) we obtain Z Z ³Z ´ −1 −1 (Dκ ) dκ = (Dκ+χ ) dκ = Ψχ (Dκ )−1 dκ Ψ−1 χ . Xb

In view of the formula

R Xb

Xb

Xb

−1

(Dκ )

dκ = R0 the last identity implies

R0 = Ψχ R0 Ψ−1 χ ,

for all χ ∈ X,

which, by definition, means that R0 ∈ XΨ (L). To consider the case of an arbitrary h one should, instead of from (8), begin with the identity Dκ+χ S−h = h−h, κiΨχ Dκ S−h Ψ−1 χ ,

χ ∈ X, κ ∈ Xb ,

−1 which in turn follows from (8) and the identity S−h Ψ−1 χ = h−h, χiΨχ S−h . A simple modification of the above reasoning yields

Rh S−h = Ψ−1 χ Rh S−h Ψχ ,

for all χ ∈ X,

which, by definition, means that Bh = Rh S−h ∈ XΨ (L). (b) is a special case of (a). ¤ Remark. A direct application of the Bochner–Phillips theorem enables one to prove the following generalization of the theorem: if an operator D ∈ SΨ (L) is left (right) invertible in B(L) then it has a left (right) inverse in SΨ (L), too.

5.2. DIFFERENCE OPERATORS

301

5.2.6. The subalgebra s is full Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) Let G be a discrete abelian group and lq = lq (G, E). Then the subalgebra s(lq ) = sz (lq ) is full in the algebra B(lq ). (b) Let G be an arbitrary locally compact abelian group and L be one of the spaces Lpq (G, E) or Cq (G, E). Then the subalgebra s(L) = sz (L) is full in the algebra B(L). Proof. (a) The case q 6= ∞ follows from 5.2.5 and 5.2.2(a). So it remains to consider the case q = ∞. Assume D ∈ s(l∞ ). Clearly, the subspace l0 ⊆ l∞ is invariant under D. There0 fore (see 1.2.9 and 1.2.2) the subspace l0⊥ ⊆ l∞ is invariant under the operator 0 0 0 0 0 D : l∞ → l∞ . We recall from 1.8.3 that l∞ = l0 ⊕ l0⊥ . Since D acts on l0 it follows that l00 is also invariant under D0 . Assume that D is invertible. Then D0 is invertible as well. Moreover, since both l00 and l0⊥ are invariant under D0 they are invariant under (D0 )−1 , too. Assume D is represented in the form (4) with Ah ∈ xz . By 5.2.5 P the operator −1 R = D belongs to sΨ (l∞ ), i.e., it possesses the representation R = h∈G Bh Sh , where Bh ∈ xΨ . We must show that Bh ∈ xz . For all κ ∈ Xb we consider the operators X X Dκ = hh, κiAh Sh and Rκ = hh, κiBh Sh . (9) h∈G

h∈G

Clearly, Rκ = (Dκ )−1 , κ ∈ Xb . 0 is invariant under operators For the same reason as above, the subspace l00 ⊆ l∞ 0 (Rκ ) , κ ∈ Xb . According to (9) we represent the operators Bh in the form ³Z ´ Bh = h−h, κiRκ dκ S−h . Xb

This representation shows that the subspace l00 ' l1 (G, E0 ) is invariant under the operators Bh0 , too. By 1.1.10 this implies that the operators Bh are continuous with respect to the hl∞ , l1 i-topology, which, in turn, by 5.1.11 implies Bh ∈ xz . (b) is a special case of (a). ¤ 5.2.7. The invertibility of D ∈ s is independent from q. We recall from 1.6.8 that an operator belonging to the class s acts on lq (G, E) for all q. Corollary. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) Let G be a discrete abelian group and lq = lq (G, E). If an operator D ∈ s is invertible on lq for some q then it is invertible on lq for all q. (b) Let G be an arbitrary locally compact abelian group and Lpq = Lpq (G, E) and Cq = Cq (G, E). If an operator T ∈ s is invertible on Lpq (on Cq ) for some q then it is invertible on Lpq (on Cq ) for all q. Proof. (a) Assume D ∈ s is invertible. Then by 5.2.6 D−1 ∈ s as well, i.e., by the definition of the class s, the operator D−1 is induced by a matrix of the class s. Clearly (see 1.4.6), this matrix induces the inverse of D on all spaces. (b) is a special case of (a). ¤

302

V. OPERATORS WITH VARYING COEFFICIENTS

5.2.8. The subalgebras Sz (Lpq ) and S(Lpq ) are full Corollary. (a) The subalgebra Sz (Lpq ) is full in the algebra B(Lpq ). (b) Let the space E be finite-dimensional. Then the subalgebra S(Lpq ) is full in the algebra B(Lpq ). Proof. (a) The case q 6= ∞ follows from 5.2.5 and 5.2.2(a). Assume D ∈ Sz (Lp∞ ) is invertible. By 5.1.4 and 1.6.12 Sz ⊆ s. Then by 5.2.7 D is invertible on Lp0 ; and by what has been proved, D−1 : Lp0 → Lp0 belongs to Sz . Consequently D−1 : Lp∞ → Lp∞ belongs to Sz , too. (b) follows from 5.2.2(b). ¤ 5.2.9. The algebras D, Dz , and DΨ . Let L be either Lpq = Lpq (G, E) or Cq = Cq (G, E). We denote by Df = Df (L) (Dz f = Dz f (L) or DΨ f = DΨ f (L), respectively) the subalgebra of B(L) consisting of all operators D of the form PM D = m=1 Am Shm with Am ∈ X(L) (Am ∈ Xz (L) or Am ∈ XΨ (L), respectively). We denote by D = D(L) (Dz = Dz (L) or DΨ = DΨ (L), respectively) the closure of Df (Dz f or DΨ f , respectively) or, equivalently, the closure of S (Sz or SΨ , respectively) in norm. We call operators D ∈ D difference operators with uniformly fading memory. Let us fix a representation of G in the form Rc × V. For any element d ∈ D¢ ¡ we fix a point d¯ ∈ d. Then we represent the space Lpq as lq (I) = lq I, Lp (Q, E) and the space Cq as lq (I) = lq (I, C♦ ), see 1.6.3, 1.6.10, and 5.1.12 for details. We define df (L) and d(L), and dz f (L) and dz (L), and dψ f (L) and dψ (L) to be the corresponding subalgebras Df , D, Dz f , Dz , Dψ f , and Dψ associated with the discrete group I. In particular, df , dz f , and dψ f consist of all operators D PM of the form D = m=1 Am sjm , where jm ∈ I and Am belong to x, xz , or xψ , respectively. Since x = xz (see 5.1.4) we have df = ¡ dz f and d ¢ = dz . Clearly, df = dz f is the set of all bounded operators on lq I, Lp (Q, E) (or lq (I, C♦ ), respectively) induced by matrices with a finite number of By 5.1.12(a) the ¡ non-zero diagonals. ¢ same description of df = dz f holds on lq I, Lp (Qi , E) and lq (I, C♦i ). Thus our new definitions of df and d are in agreement with those in 1.6.7. In particular, from this description it is evident that the definitions of df = dz f and d = dz do not depend on the choice of d¯ ∈ d. Remark. By 5.1.4 and 1.6.12 D and Dz are contained in d, but DΨ is not, see the example in 5.1.11. Corollary. Let L = L(G, E) be either Lpq = Lpq (G, E) or Cq = Cq (G, E). (a) The subalgebra DΨ (L) is full in the algebra B(L). (b) The subalgebra Dz (Lpq ) is full in the algebra B(Lpq ). (c) Let the space E be finite-dimensional. Then the subalgebra D(Lpq ) is full in the algebra B(Lpq ). (d) The definitions of dψ f (L) and dψ (L) do not depend on the choice of d¯ ∈ d. (e) The subalgebras d(L) = dz (L) and dψ (L) are full in the algebra B(L).

303

5.2. DIFFERENCE OPERATORS

Proof. (a) follows from 5.2.5. (b) and (c) follow from 5.2.8. (d) The case of dψ (L) follows from 5.2.1(c). The case of dψ f (L) is handled in a way similar to that of 5.2.1(c). (e) is a special case of (b) and (a). ¤ Remark. (a) Since each D ∈ D (D ∈ Dz or D ∈ DΨ , respectively) can be represented as a limit of a sequence Dk ∈ Df (Dk ∈ Dz f or Dk ∈ DΨ f , respectively), one may associate with the operator D the formal series X Ah S h , D∼ h∈G

with corresponding Ah . We discuss such series in 6.2.6. (b) We shall return to the discussion of properties of algebras DΨ and dψ in 6.5.3 and 6.5.4. 5.2.10. The invertibility of D ∈ S(Lpq ) is independent from p and q. We recall that an operator belonging to the class S acts on Lpq (G, E) for all p and q. Corollary. Let the space E be finite-dimensional, and let Lpq = Lpq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. If an operator D ∈ S is invertible on Lpq for some p and q then it is invertible on Lpq for all p and q. Proof. The proof is similar to that of 5.2.7 and based on 5.2.8(b) and 5.2.4.

¤

5.2.11. The subgroup generated by the set of shifts. Let E be a Banach space and L = L(G, E) be one of the spaces Lpq (G, E) or Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let H be a (not obligatorily closed) subgroup of the group G. We denote by SH Ψ = SH Ψ (L) (respectively, by¡ SH = S¢H (L) and SH z = SH z (L)) the algebra consisting of all operators D ∈ B L(G, E) of the form X D= Ah Sh , (10) h∈H

where Ah ∈ XΨ (L) (Ah ∈ X(L) or Ah ∈ Xz (L)) and

P h∈H

kAh Sh k < ∞.

Theorem. The subalgebra SH Ψ (L) (respectively, SH (L) and SH z (L)) is full in the algebra SΨ (L) (respectively, S(L) and Sz (L)). Example. Assume G = R and consider an operator D ∈ SΨ (L) of the form (3). We denote by H the subgroup of G generated by the family of the shifts {hm } in (3). Clearly, H is countable, cf. the remark in 4.5.7. Hence H 6= G. The theorem (together with 5.2.5) states that D−1 ∈ SH Ψ provided that D−1 exists. Proof. Since SH = S ∩ SH Ψ and SH z = Sz ∩ SH Ψ it only suffices to prove that the subalgebra SH Ψ (L) is full in the algebra SΨ (L). We assume is P invertible and represent D and D−1 in the P that D ∈ SH Ψ (L) forms D = h∈H Ah Sh and D−1 = h∈G Bh Sh . For all κ ∈ Xb (G) we consider the operators (9). It is easy to verify that (cf. the proof of 5.2.6) Rκ = (Dκ )−1 .

304

V. OPERATORS WITH VARYING COEFFICIENTS

We consider the group H⊥ = { κ ∈ Xb (G) : κ(h) = 1 for all h ∈ H }. Clearly, Dκ+ϕ = Dκ

for all κ ∈ Xb and ϕ ∈ H⊥ .

Therefore the operators Rκ = (Dκ )−1 possess the same property: Rκ+ϕ = Rκ

for all κ ∈ Xb and ϕ ∈ H⊥ .

Using this identity, the shift invariance of the Haar measure, representation (9), and 5.2.3, for all ϕ ∈ H⊥ we obtain Z Bh Sh = h−h, κiRκ dκ Xb Z = h−h, κ + ϕiRκ+ϕ dκ Xb Z = h−h, ϕi h−h, κiRκ+ϕ dκ Xb Z = h−h, ϕi h−h, κiRκ dκ Xb

= h−h, ϕiBh Sh . Suppose Bh Sh 6= 0. Then h−h, ϕi = 1 for all ϕ ∈ H⊥ , i.e., h ∈ (H⊥ )⊥ . We show that it follows that h ∈ H. Indeed, assume the contrary: let h ∈ / H. Then by Pontrjagin’s theorem there exists a character ϕ of Gd /Hd such that hh, ϕi = 6 0. But ⊥ by 4.2.1 ϕ can be identified with an element of H . This is a contradiction. ¤ 5.2.12. Difference operators with exponential memory. Let L be one of the spaces Lpq (G, E) or Cq (G, E), and let H be a subgroup of G represented in the form H ' Ra × Zb × K, where a and b are non-negative integers, and K is a compact abelian group, cf. 1.6.1. According to this representation, for h = (t1 , t2 , . . . , ta ; n1 , n2 , . . . , nb ; k) ∈ Ra × Zb × K ' H we set |h| = |t1 | + |t2 | + · · · + |ta | + |n1 | + |n2 | + · · · + |nb |. We denote by EH Ψ = EH Ψ (L) (respectively,¡ by EH = ¢ EH (L), EH z = EH z (L)) the algebra consisting of all operators D ∈ B L(G, E) of the form (10), where Ah ∈ XΨ (L) (respectively, Ah ∈ X(L) and Ah ∈ Xz (L)) and there exists γ > 0 such that X eγ|h| kAh Sh k < ∞. h∈H

Clearly (cf. 1.6.9), this estimate is equivalent to the existence of N < ∞ and η > 0 such that kAh Sh k ≤ N e−η|h| for all h ∈ H. If G = Ra × Zb × K itself we denote briefly the algebra EG Ψ by EΨ , and so on. We note that for G = Zn the algebra E = EG coincides with the algebra e considered earlier in 1.6.9 and 3.4.1.

5.2. DIFFERENCE OPERATORS

305

Theorem. The subalgebra EH Ψ (L) (respectively, EH (L) and EH z (L)) is full in the algebra SΨ (L) (respectively, in S(L) and Sz (L)). Remark. We note that theorem 3.4.2 is a special case of this theorem. ¡ ¢ Example. Let us consider an operator D ∈ S Lpq (R, C) defined by the rule ¡

+∞ X

¢

Dx (t) =

+∞ X

aij (t)x(t − ih1 − jh2 ),

i=−∞ j=−∞

where h1 and h2 are linearly independent over the field Q of rational numbers. Assume D is invertible. By 5.2.11 the operator D−1 can be represented in the similar form: +∞ X ¡ −1 ¢ D x (t) =

+∞ X

bij (t)x(t − ih1 − jh2 ).

i=−∞ j=−∞

The last theorem allows one to obtain two corollaries. If one has (this corresponds to H generated by h1 and h2 ) X eγ(|i|+|j|) kaij kL∞ < ∞, h∈H

a similar estimate holds for the coefficients bij of the operator D−1 . If one has (and this corresponds to H = R) X eγ|ih1 +jh2 | kaij kL∞ < ∞, h∈H

a similar estimate holds for the coefficients bij of the operator D−1 , too. Proof. It suffices only to prove that the subalgebra EH Ψ (L) is full in the algebra SΨ (L), since EH = S ∩ EH Ψ and EH z = Sz ∩ EH Ψ . We assume that D ∈ EH Ψ (L) is invertible. By 5.2.11 R =P D−1 belongs to P SH Ψ (L). We represent D and R as D = h∈H Ah Sh and R = h∈H Bh Sh , and consider the operators X X ehh,ϑi Bh Sh , Dϑ = ehh,ϑi Ah Sh and Rϑ = h∈H

h∈H

where hh, ϑi = t1 ϑ1 + t2 ϑ2 + · · · + ta ϑa + n1 ϑa+1 + n2 ϑa+2 + · · · + nb ϑa+b and ϑ = (ϑ1 , ϑ2 , . . . , ϑa , ϑa+1 , ϑa+2 , . . . , ϑa+b ). By the definition of the class EH Ψ the operators Dϑ make sense for all ϑ ∈ Ca+b with | Re ϑj | ≤ γ, and by 5.2.5 and 5.2.11 the operators Rϑ make sense at least for ϑ ¡∈ iRa+b .¢ According to the isomorphism SΨ (G, L) ' SΨ L(G, E) (see 5.2.4) we endow SΨ (L) and SH Ψ ⊆ SΨ with the norm X X kCh Sh k for T = Ch S h . T = h∈H

h∈H

306

V. OPERATORS WITH VARYING COEFFICIENTS

It is easy to verify that the function ϑ 7→ Dϑ is holomorphic on | Re ϑj | < γ with respect to the norm · on SΨ (L). We need only elementary properties of holomorphic functions of several variables which are simple analogues of the corresponding properties of holomorphic functions of one variable, see, e.g., [Mal]. To avoid additional references we shall not use explicitly the theory of holomorphic functions of several variables. In particular, by saying that ϑ 7→ Dϑ is holomorphic we mean that it is differentiable with respect to each ϑ1 , ϑ2 , . . . , ϑa+b separately. By the definition of multiplication we have the equality DR =

³X

Ah Sh

´³X

XX

Bg Sg

g∈H

h∈H

=

´

Ah Sh Bg Sg

h∈H g∈H

=

XX

Ah Sh Bf −h Sf −h

h∈H f ∈H

=

XX

Ah (Sh Bf −h S−h )Sh Sf −h

f ∈H h∈H

=

X³X

´ Ah (Sh Bf −h S−h ) Sf .

f ∈H h∈H

By virtue of the identity DR = 1 and 5.2.4, this yields X

½ Ah (Sh Bf −h S−h ) =

h∈H

1

for f = 0,

0

for f 6= 0.

A similar equality follows from the identity RD = 1. From these two equalities it is easy to see that Dϑ Rϑ = Rϑ Dϑ = 1

for all ϑ ∈ iRa+b .

Consequently Dϑ are invertible for ϑ ∈ iRa+b with (Dϑ )−1 = Rϑ = R . By the definition of the class EH Ψ we can choose η > 0 such that R ·

X

(eη|h| − 1)kAh Sh k ≤ 1/2.

h∈H

Note that for ϑ = ν + iω, with ω, ν ∈ Ra+b and |νj | ≤ η, we can write the estimate P Dν+iω − Diω ≤ h∈H (eη|h| − 1)kAh Sh k. Therefore by 1.4.2, for | Re ϑj | ≤ η, the elements Dϑ are invertible with (Dϑ )−1 ≤ M , where M = 2 R . Clearly (see the example in 1.4.9), the function ϑ 7→ (Dϑ )−1 is holomorphic on | Re ϑj | < η. We recall that (Dϑ )−1 = Rϑ for ϑ ∈ iRa+b . Thus the function ϑ 7→ Rϑ possesses the holomorphic continuation to | Re ϑj | < η. We show that P this continuation coincides with the function ϑ 7→ h∈H ehh,ϑi Bh Sh .

307

5.3. SMOOTHING OPERATORS

For fixed ϑ2 , ϑ3 , . . . , ϑa+b with Re ϑj = 0, the function ϑ1 7→ (Dϑ )−1 is holomorphic on the open strip | Re ϑ1 | < η. From the uniqueness of the holomorphic of a function of one variable we obtain the equality (Dϑ )−1 = P continuation hh,ϑi Bh Sh for | Re ϑ1 | < η and ϑ2 , ϑ3 , . . . , ϑa+b with Re ϑj = 0. Next, we h∈H e fix ϑ1 with | Re ϑ1 | < η and ϑ3 , ϑ4 , . . . , ϑa+b with Re ϑj = 0. Now the function ϑ2 7→ (Dϑ )−1 is holomorphic on the open strip | Re ϑ2 | < η. Again, from the uniqueness of the holomorphic continuation of a function variable it follows P of onehh,ϑi −1 that the function ϑ2 7→ (Dϑ ) coincides with ϑ2 7→ h∈H e Bh Sh . And so on. P As a consequence, we obtain that the series h∈H ehh,ϑi Bh Sh is bounded in the norm · for all | Re ϑj | < η. Let Λ denote the finite set of all ϑ ∈ Ra+b with |ϑj | = η/2, cf. the proof of 3.4.1. Clearly, for any h ∈ H we have e|h|η/2 kBh Sh k ≤ max

nX

ehh,ϑi kBh Sh k : ϑ ∈ Λ

h∈H

= max

n X

o

o ehh,ϑi Bh Sh : ϑ ∈ Λ ,

h∈H

which means that R ∈ EH Ψ .

¤

5.3. Smoothing operators An integral operator with locally bounded kernel can be defined equivalently as an operator which acts from L1 to L∞ locally. We call this kind of operators smoothing operators. We adhere to the latter description of smoothing operators because it is more suitable from the point of view of operator theory. In this section we discuss two algebras of smoothing operators. 5.3.1. Universal operators with summable memory. Let G be a nondiscrete locally compact abelian group. We note that since λ(Qi ) = 1, L∞ (Qi , E) ⊆ Lp (Qi , E) ⊆ L1 (Qi , E)

for all 1 ≤ p ≤ ∞.

We denote by V = V(G, E) the set of all matrices V = { Vij : i, j ∈ I } consisting of operators Vij which act from Lp (Qj , E) to Lp (Qi , E) for all 1 ≤ p ≤ ∞; and for all 1 ≤ p ≤ ∞ satisfy the estimate X h∈I

sup kVij : Lp (Qj , E) → Lp (Qi , E)k < ∞.

i−j=h

By saying that an operator acts on several spaces we mean that it is defined by the same rule on their common part. We stress that the smallest space L∞ (Qi , E) is dense in Lp (Qi , E) for all p. By the Riesz–Thorin theorem the validity of the above estimate for p = 1 and p = ∞ implies its validity for all p. Thus one may imagine V(G, E) as the

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V. OPERATORS WITH VARYING COEFFICIENTS

¡ ¢ ¡ ¢ intersection of s I, L1 (Qi , E) and s I, L∞ (Qi , E) . Clearly, V is an algebra. We call matrices V ∈ V universal matrices with summable memory. ¡ ¢ Obviously, a matrix V ∈ V induces an operator V ∈ B L (G, E) of the class pq ¡ ¢ s = s L¡ pq (G, E) ¢for all 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by V = V Lpq (G, E) the algebra of all such operators. We call operators V ∈ V universal operators with summable memory. Example. By 1.6.12 S ⊆ V, see 5.2.1 for the definition of S. Theorem. ¡ ¢ (a) If an operator V ∈ V Lpq (G, E) is invertible on Lpq for some q (and fixed p) then it is invertible on Lpq for all q (and the same p).¡ ¢ (b) Let E be a finite-dimensional Banach space, and let V ∈ V Lpq (G, E) . For all p and q the subspace Lp0 q0 ⊆ L0pq (see 1.8.1) is invariant under V 0 and the restriction V 0 : Lp0 q0 → Lp0 q0 of V 0 to Lp0 q0 belongs to V, too. Proof. (a) is a¡ special case of¢¢ 5.2.7. ¢ ¡ ¡ 0 0 I, Lp (Qi , E) ⊆ (b) Since V ⊆ s lq I, Lp (Qi , E) , by ¡5.2.1(d) the subspace l q ¢¢ ¡ L0pq is invariant under V 0 and V 0 ∈ s lq0 I, Lp (Qi , E)0 . If p 6= ∞, by 1.8.1 ¡ ¢ Lp (Qi , E)0 = Lp0 (Qi , E0 ); thus lq0 I, Lp (Qi , E)0 = Lp0 q0 . Suppose p = ∞. We consider a matrix entry Vij of V . Since Vij maps L1 (Qj , E) to L1 (Qi , E) its conjugate (Vij )0 maps L∞ (Qi , E0 ) to L∞ (Qj , E0 ). On the other hand, (Vij )0 maps L∞ (Qi , E)0 to L∞ (Qj , E)0 . We recall that L1 (Qi , E0 ) can be considered as a subspace of L∞ (Qi , E)0 . Since (Vij )0 : L∞ (Qi , E)0 → L∞ (Qj , E)0 maps L∞ (Qi , E0 ) ⊆ L1 (Qi , E0 ) to L∞ (Qj , E0 ) ⊆ L1 (Qj , E0 ) it also maps the closure L1 (Qi , E0 ) of L∞ (Qi , E0 ) to the closure L1 (Qj , E0 ) of L∞ (Qj , E0 ). Thus the matrix 0 entry (Vij )0 maps L1 (Qi , E0 ) to ¡L1 (Q j , E ). ¡ ¢¢ ¡ ¢ 0 0 0 0 I, L∞ (Qi , E) 0 I, L1 (Qi , E ) Consequently since V ∈ s l , it maps l ⊆ q q ¡ ¢ ¡ ¡ ¢¢ 0 0 0 lq0 I, L∞ (Qi , E) into itself. Moreover, V ∈ s l¡q0 I, L1 (Qi , E ¢) . ¡ ¢ By the definition of V as the intersection of s I, L1 (Qi , E) and s I, L∞ (Qi , E) we obtain that V 0 ∈ V. ¤ ¡ ¢ Example. If G is non-discrete then the invertibility of V ∈ V L (G, C) can pq ¡ ¢ ¡ ¢ depend on p and the subalgebra V Lpq (G, C) is not full in B Lpq (G, C) . These facts follow from 4.2.12. Indeed, by 4.4.1(b) the operator T x = ξ ∗ x of the convolution with the measure ξ from 4.2.12 belongs to V. It can be shown that ˇ : χ ∈ X(G) }; this fact can be the spectrum of T on L2 is the closure the set { ξ(χ) obtained easily from the observation that the operator F−1 T F, where F : L2 → L2 ˇ At the same time, is the Fourier transform, is the operator of multiplication by ξ. from the example in 5.6.9 below if follows that the spectrum of T on L1 , L∞ , and C coincides with the spectrum of T in the algebras At (L1 ), At (L∞ ), and At (C), respectively, which is equal to the spectrum of ξ in the algebra M of all bounded ˇ : χ ∈ X(G) } = σL (ξ). Thus the measures. By 4.2.12 i ∈ σM (ξ), but i ∈ / { ξ(χ) 2 operator V = i − T is invertible on L2 , but¡not invertible on L , L∞ , and C. It 1 ¢ remains to if the subalgebra V Lpq (G, C) were full the invertibility ¡ observe that ¢ of V ∈ V Lpq (G, C) would not depend on p, cf. the proof of 5.2.7.

5.3. SMOOTHING OPERATORS

309

5.3.2. Smoothing operators with summable memory. We denote by N∞ = N∞ (G, E) the set of all matrices N = { Nij : i, j ∈ I } consisting of operators Nij which act from L1 (Qj , E) to L∞ (Qi , E) and satisfy the estimate X h∈I

sup kNij : L1 (Qj , E) → L∞ (Qi , E)k < ∞.

i−j=h

We call the matrices N ∈ N∞ smoothing matrices¡ with summable memory. We ¢ note that with the notation of 1.6.8 N∞ (G, E) is s I, {L1 (Qj , E)}, {L∞ (Qi , E)} . Clearly, N∞ ¢⊆ V. Therefore¡ a matrix¢ N ∈ N∞ induces an operator ¡ N ∈ B Lpq (G, E) of the class s = s ¡Lpq (G, E) ¢ for all 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by N∞ = N∞ Lpq (G, E) the set of all such operators. We call the operators N ∈ N∞ smoothing operators with summable memory. ¡ ¢ ¡ ¢ Proposition. N∞ Lpq (G, E) is a proper two-sided ideal in V Lpq (G, E) . Proof. Clearly, N∞ is a two-sided ideal. We show that N∞ is proper. Assume the contrary: let the identity operator belong to N∞ . Then the identity mapping from L1 (Qi , E) to L∞ (Qi , E) is a topological isomorphism. But this is impossible by virtue of 1.6.2(d). Indeed, if 0 < λ(E) < ε, E ⊆ Qi then one has k1E kL∞ = 1, but k1E kL1 < ε. ¤ 5.3.3. Integral operators with bounded kernels. Let X, Y , and Z be Banach spaces. A mapping B : X × Y → Z is called bilinear if the mappings B(x, ·) and B(·, y) are linear for all fixed x ∈ X and y ∈ Y . A bilinear mapping B is called continuous if there exists K < ∞ such that kB(x, y)k ≤ K · kxk · kyk for all x and y. We denote by kBk the smallest K satisfying this property. ¯ be locally compact topological spaces with measures λ Lemma. Let Q and Q ¯ respectively; and let E be a finite-dimensional Banach space. and λ, ¯ 0 (a) For any continuous bilinear ¡ mapping B ¢ : L1 (Q, E) × L1 (Q, E ) → C there ¯ B(E) such that for all x ∈ L1 (Q, E) and exists a function n ∈ L∞ Q × Q, 0 ¯ E ) one has y ∈ L1 (Q, ZZ B(x, y) =

­ ® ¯ n(t, s)x(s), y(t) dλ(t)dλ(s).

Moreover, kBk = knkL∞ .¡ ¢ ¯ E) there exists a function n ∈ (b) For ¡any operator¢N ∈ B L1 (Q, E), L∞ (Q, ¯ B(E) such that for all x ∈ L1 (Q, E) one has L∞ Q × Q, ¡ ¢ N x (t) =

Z n(t, s)x(s) dλ(s).

In this case kN : L1 → L∞ k = knkL∞ . (Clearly, by Fubini’s theorem the function s 7→ n(t, s)x(s) is integrable for locally almost all t.)

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V. OPERATORS WITH VARYING COEFFICIENTS

Remark. Here and below we define products of the kind hn(t, s)x(s), y(t)i and n(t, s)x(s) to be zeros if one of the factors is zero and the other(s) is undefined. Proof. We restrict ourselves to the proof for the case E = C. Its generalization to an arbitrary finite-dimensional E is trivial, cf. the proof of 5.1.6. (a) Evidently, the formula K K ´ X ³X B(xk , yk ) xk ⊗ yk = b k=1

k=1

¯ It is easy to see that b is continuous defines a linear functional b on L1 (Q)⊗L1 (Q). ¯ Namely, kbk ≤ kBk. with respect to the cross-norm π (see 1.7.3) on L1 (Q)⊗L1 (Q). ¯ ' L1 (Q× Q), ¯ Therefore b possesses the extension by continuity to L1 (Q) ⊗π L1 (Q) ¯ is induced by a see 1.7.4. By 1.8.1 any linear bounded functional on L1 (Q × Q) ¯ function n ∈ L∞ (Q × Q). In particular, we have ZZ ¯ n(t, s)x(s)y(t) dλ(t)dλ(s).

B(x, y) = b(x ⊗ y) =

This representation yields the estimate kBk ¡ ≤ knk = kbk. Thus¢ kBk = knk. ¯ E) induces the bilin(b) We observe that an operator N ∈ B L1 (Q, E), L∞ (Q, R ¯ ear form B(x, y) = hN x, yi dλ on L1 (Q) × L1 (Q). Clearly, we have the estimate kBk ≤ kN : L1 → L∞ k. We make use of (a). Let n be the kernel of B. We consider the operator ¡ ¢ N1 x (t) =

Z n(t, s)x(s) dλ(s).

¯ we have hN x, yi = hN1 x, yi, Clearly, N1 acts from L1 to L∞ , and for all y ∈ L1 (Q) which implies N = N1 . The inequality kN : L1 → L∞ k ≤ knk = kBk is plain. ¤ 5.3.4. An integral representation for N ∈ N∞ Proposition. Assume there exist measurable functions β : G → [0, ∞) and n : G × G → B(E) such that the function β belongs to L∞1 , and the function n satisfies the estimate kn(t, s)k ≤ β(t − s)

for all t, s ∈ G.

(For simplicity we suppose that β and n are defined everywhere. See also §5.4 below for a generalization.) Then the formula ¡

¢ N x (t) =

Z n(t, s)x(s) dλ(s)

¡ ¢ defines an operator N ∈ N∞ Lpq (G, E) .

(1)

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5.3. SMOOTHING OPERATORS

Assume additionally ¡ ¢that E is finite-dimensional. Then, conversely, any operator N ∈ N∞ Lpq (G, E) can be represented in the form (1) with such an n. Proof. The first assertion is evident. We prove the second one. By the definition of the class N∞ we have X (N x)i = Nij xj .

(2)

j∈I

We stress that the sum contains at most a countable sum¡ number of non-zero ¢ ei × Q e j , B(E) such that mands. By 5.3.3, for all i, j ∈ I there exists nij ∈ L∞ Q Z ¡ ¢ e i. Nij xj (t) = nij (t, s)xj (s) dλ(s), t∈Q (3) ej Q

ei Without loss of generality we may assume that nij (t, s) is defined for all t ∈ Q e j , and knij (t, s)k ≤ αi−j for all t ∈ Q e i and s ∈ Q e j , where and s ∈ Q e j , E) → L∞ (Q e i , E)k, αh = sup kNij : L1 (Q

h ∈ I.

i−j=h

In particular, we have nij (t, s) = 0 if αi−j = 0. e j , we Substituting (3) into (2) and taking into account that xj (t) = x(t), t ∈ Q obtain ¡ ¢ ¡ ¢ N x (t) = N x i (t) XZ e i. = nij (t, s)x(s) dλ(s), t∈Q j∈I

ej Q

e i , E). Since V ⊆ s the series in (2) converges absolutely in the norm of Lq (Q Therefore by 1.5.5 the sum in the last formula converges absolutely for almost e i or, equivalently, for locally almost all t ∈ G. We define the function all t ∈ Q e i and s ∈ Q e j . Then the n : G × G → B(E) by the rule n(t, s) = nij (t, s) for t ∈ Q last formula can be rewritten as (1). For g ∈ G we set β(g) = sup{ α(m,e) : m ∈ Zc , e ∈ D, g ∈ (m + (−1, 1)c ) × e }. Then we have knij (t, s)k ≤ αi−j ≤ β(t − s) for i = (k, d) and j = (l, b), and t ∈ (k + (0, 1]c ) × d and s ∈ (l + (0, 1]c ) × b. Finally, we show that β ∈ L∞1 . For g = (r, v) ∈ Rc × V ' G we denote by ∆ = ∆(g) the set of all h = (m, e) ∈ Zc × K ' I such that e = v + K and r ∈ m + [−1, 1)c , i.e., m1 − 1 ≤ r1 < m1 + 1, . . . ,P mc − 1 ≤ rc < mc + 1. Clearly, β(g) that: (i) P ≤ max{ αh : h ∈ ∆ }. Consequently β(g) ≤ h∈∆ αh . We observe c α < ∞ by the definition of N , (ii) ∆ contains at most 2 elements and ∞ h∈I h P e i . Therefore kβkL ≤ 2c (iii) β is a constant on each set Q ∞1 h∈I αh . ¤ g 5.3.5. The invertibility of N ∈ N ∞ is independent from p and q. We de¡ ¢ ¡ ¢ g g note by N∞ = N L (G, E) the algebra of all operators α + N ∈ B L (G, E) , ∞ pq pq ¡ ¢ where N ∈ N∞ Lpq (G, E) and α ∈ C.

312

V. OPERATORS WITH VARYING COEFFICIENTS

g Theorem. If an operator T ∈ N ∞ is invertible on Lpq for some p and q then it is invertible on Lpq for all p and q. Proof. The independence of invertibility from q follows from 5.3.1. We prove the independence of invertibility from p. Let α ∈ C and N ∈ N∞ . We assume that the operator T = α + N is invertible on Lpq . First we show that α 6= 0. Assume the contrary. Then since N∞ is a proper ideal (see 5.3.2), the product N N −1 belongs to N∞ and can not be the identity operator 1. Thus we arrive to a contradiction. Let p1 6= p. We observe that the operator (α + N )−1 on Lp1 q can be defined by the following rule N 1 − (α + N )−1 α α 1 N = − (α + N )−1 α α

(α + N )−1 =

for p1 < p,

(α + N )−1

for p1 > p.

Actually, the operator N acts from L1q to L∞q . Hence if the operator (α + N )−1 in the right hand side is considered as acting on Lpq then the whole right hand side defines the operator which maps Lp1 q into itself. Finally, trivial straightforward ¡1 ¢ 1 −1 calculations show that, e.g., the identities (α + N ) − (α + N ) N = 1 and α α ¡1 ¢ 1 −1 (α + N ) α − α N (α + N ) = 1 really hold. ¤ Remark. The theorem remains valid for N ∈ V such that N ∈ / N∞ , but some k its power N belongs to N∞ . The proof is based on the identity (α + N )−1 =

1 N N2 N k−1 Nk − 2 + 3 − · · · + (−1)k−1 k + (−1)k k (α + N )−1 . α α α α α

g 5.3.6. The subalgebra N ∞ is full g Theorem. The subalgebra N ∞ (Lpq ) is full in the algebra B(Lpq ). Proof. Let λ ∈ C and N ∈ N∞ . Assume the operator T = α+N is invertible. −1 g Since N ∈ s. On the other hand, by 5.3.5 the matrix entries ∞ ⊆ s, by 5.2.6 T −1 (T )ij of the operator T −1 act from Lp (Qj , E) to Lp (Qi , E) for all p. In other words, T −1 ∈ V. Let us consider the quotient algebra V/N∞ , see 5.3.2. Clearly, the natural projection of T = α + N coincides with the scalar element α ∈ V/N∞ . Therefore the natural projection of T −1 must coincide with 1/α (in particular, this implies that α 6= 0). But this means that T −1 has the form α + N1 with N1 ∈ N∞ . ¤ 5.3.7. Universal operators with exponential memory. From now on, in this section we assume that G is a locally compact compactly generated abelian group. According to 1.6.1 we identify G with Ra × Zb × K. We denote by I e i = (k + (0, 1]a ) × {l} × K for the subgroup Za × Zb × {0} ⊆ G and we put Q i = (k, l, 0) ∈ I. Then for g = (t1 , . . . , ta ; n1 , . . . , nb ; k) ∈ Ra × Zb × K we set |g| = |t1 | + |t2 | + · · · + |ta | + |n1 | + |n2 | + · · · + |nb |.

313

5.3. SMOOTHING OPERATORS

In particular, |h| makes sense for h ∈ I ' Za × Zb × {0}. We denote by W = W(G, E) the set of all matrices W = { Wij : i, j ∈ I } possessing the following properties: operators Wij act from Lp (Qj , E) to Lp (Qi , E) for all 1 ≤ p ≤ ∞; and for any 1 ≤ p ≤ ∞ there exists γ > 0 such that X eγ|h| sup kWij : Lp (Qj , E) → Lp (Qi , E)k < ∞. h∈I

i−j=h

Clearly (cf. 1.6.9, 5.2.12), this estimate holds if and only if there exist N < ∞ and η > 0 such that kWij : Lp (Qj , E) → Lp (Qi , E)k < N e−η|i−j|

for all i, j ∈ I.

By the Riesz–Thorin theorem the validity of these estimates for p = 1 and p = ∞ implies ¡their validity¢ for all¡ p. Thus one ¢ can imagine W(G, E) as the intersection of e I, L1 (Qi , E) and e I, L∞ (Qi , E) . Clearly, W is an algebra and W ⊆ V. We call matrices W ∈ W universal matrices with exponential memory. ¡ ¢ Evidently, a matrix W ∈ W induces an operator W ∈ B L (G, E) of the pq ¡ ¢ class e = e ¡Lpq (G, E) ¢ for all 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by W = W Lpq (G, E) the set of all such operators. We call operators W ∈ W universal operators with exponential memory. Example. By 1.6.12 E ⊆ W, see 5.2.12 for the definition of E. Theorem. ¡ ¢ (a) If an operator W ∈ W Lpq (G, E) is invertible on Lpq for some q (and fixed p) then it is invertible on Lpq for all q (and the same p).¡ ¢ (b) Let E be a finite-dimensional Banach space, and let W ∈ W Lpq (G, E) . For all p and q the subspace Lp0 q0 ⊆ L0pq (see 1.8.1) is invariant under W 0 and the restriction W 0 : Lp0 q0 → Lp0 q0 of W 0 to Lp0 q0 belongs to W, too. Proof. (a) is a special case of 5.3.1. The proof of (b) is similar to that of 5.3.1.

¤

5.3.8. Smoothing operators with exponential memory. We denote by M∞ = M∞ (G, E) the set of all matrices M = { Mij : i, j ∈ I } possessing the following properties: operators Mij act from L1 (Qj , E) to L∞ (Qi , E); and there exists γ > 0 such that X eγ|h| sup kMij : L1 (Qj , E) → L∞ (Qi , E)k < ∞. h∈I

i−j=h

Evidently, M∞ ⊆ N∞ . We call the matrices M ∈ M∞ smoothing matrices with exponential memory. We note ¢that with the notation of 1.6.9 M∞ (G, E) is the set ¡ e I, {L1 (Qj , E)}, {L∞ (Qi , E)} . ¡Clearly, M ¢ ∞ ⊆ W. Therefore ¡ a matrix¢ M ∈ M∞ induces an operator M ∈ B Lpq (G, E) of the class e = e Lpq ¡ (G, E) for ¢ all 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by M∞ = M∞ Lpq (G, E) the set of all such operators. We call the operators M ∈ M∞ smoothing operators with exponential memory.

314

V. OPERATORS WITH VARYING COEFFICIENTS

Example. The operator ¡ −1 ¢ U f (t) =

Z

+∞

e−s f (t − s) ds

0

Z

t

=

e−(t−s) f (s) ds

−∞

(see 2.3.4 and 2.3.9), considered as acting from Lpq (R, E) into itself, belongs to M∞ . See the explanation in the proof of 2.3.4. ¡ ¢ ¡ ¢ Proposition. M∞ Lpq (G, E) is a proper two-sided ideal in W Lpq (G, E) . Proof. The proof is similar to that of 5.3.2. ¤ 5.3.9. The integral representation for M ∈ M∞ Proposition. Assume there exist measurable functions β : G → [0, ∞) and m : G × G → B(E) (defined everywhere), and γ > 0 such that the function h 7→ eγ|h| β(h) belongs to L∞1 , and the function m satisfies the estimate km(t, s)k ≤ β(t − s) Then the formula

¡ ¢ M x (t) =

for all t, s ∈ G.

Z m(t, s)x(s) dλ(s)

(4)

¡ ¢ defines an operator M ∈ M∞ Lpq (G, E) . Assume additionally that E is finite-dimensional. Then, conversely, any opera¡ ¢ tor M ∈ M∞ Lpq (G, E) can be represented in the form (4) with such an m. We note that the validity of the estimate km(t, s)k ≤ β(t − s) is equivalent to the existence of N and η > 0 such that km(t, s)k ≤ N e−η|t−s| , cf. 1.6.9 and 5.2.12. Proof. The proof is a modification of that of 5.3.4. The first assertion is evident. We prove the second one. Let M ∈ M∞ . The existence of the representation (4) with γ = 0 follows immediately from 5.3.4. e j , E) → L∞ (Q e i , E)k; and for g ∈ G For h ∈ I we set αh = supi−j=h kMij : L1 (Q we set β(g) = max{ α(m,n,0) : (m, n, 0) ∈ I, g ∈ (m + (−1, 1)) × n × K }. As in the proof of 5.3.4, we have kn(t, s)k ≤ β(t − s). It remains to show that the function h 7→ eγ|h| β(h) belongs to L∞1 for some γ > 0. For g = (r, n, d) ∈ Ra × Zb × K ' G we denote by ∆ = ∆(g) the set of all h = (m, n, 0) ∈ Za+b ×{0} = I such that r ∈ m+[−1, 1)a , i.e., m1 −1 ≤ r1 < m1 +1, . . . , ma − 1 ≤ raP < ma + 1. Clearly, β(g) ≤ max{ αh : h ∈ ∆ }. By the definition of M∞ we have h∈I eγ|h| αh < ∞. Hence the function h 7→ eγ|h| r(h) belongs to L∞1 , too. ¤ g 5.3.10. The subalgebra M ∞ is full. By analogy with 5.3.5 we denote by ¡ ¢ ¡ ¢ ] ] M∞ = M¡∞ Lpq (G, E) the algebra of all operators α + M ∈ B L (G, E) , where pq ¢ M ∈ M∞ Lpq (G, E) and α ∈ C.

5.4. INTEGRAL OPERATORS

315

] Theorem. The subalgebra M ∞ (Lpq ) is full in the algebra B(Lpq ). Proof. The proof is similar to that of 5.3.6. Assume an operator T = α + M , −1 ] where α ∈ C and M ∈ M∞ , is invertible. Since M ∈ e. ∞ ⊆ e, by 5.2.12 T On the other hand, by the embedding M∞ ⊆ N∞ and 5.3.5 the matrix entries (T −1 )ij of the operator T −1 act from Lp (Qj , E) to Lp (Qi , E) for all p. In other words, T −1 ∈ W. We consider the quotient algebra W/M∞ , see 5.3.8. Clearly, the natural projection of T = α + M coincides with the scalar element α ∈ W/M∞ . Therefore the natural projection of T −1 must coincide with 1/α (in particular, this implies that α 6= 0). But this means that T −1 has the form α + M1 with M1 ∈ M∞ . ¤

5.4. Integral operators This section is close to the previous one. Here we discuss two algebras of integral operators dominated by convolutions with functions of the class L1 . 5.4.1. The class N1 of kernels. Let G be a locally compact non-discrete abelian group, and let E be a Banach space. We denote by N1 = N1 (G, E) the set of all measurable functions n : G × G → B(E) satisfying the two following assumptions: (i) there exists a σ-compact set K ⊆ G such that n(t, s) = 0 for all t − s ∈ / K (clearly, if G is σ-compact itself, this assumption is superfluous); (ii) there exists an integrable function β : G → [0, +∞) such that kn(t, s)k ≤ β(t − s)

for locally almost all (t, s) ∈ G × G.

(1)

We allow a function n ∈ N1 to be undefined on a locally null set contained in a set of the form { (t, s) : t − s ∈ K }, where K ⊆ G is σ-compact. We say that two functions from N1 are equivalent if they differ only on a locally null set. Thus, to be more correct, N1 consists of equivalence classes. We define the norm knk = knkN1 to be the infimum of kβkL1 over all β satisfying (1). If n ∈ N1 we say that the kernel n is summable. We note that the set { (t, s) : t − s ∈ N }, where N is null, is locally null. Therefore assumption (1) depends on the equivalence class of β only. In particular, β may be defined only almost everywhere. Proposition. (a) For any n ∈ N1 there exists the smallest (in the sense of the ordering in ¯ L . L1 (G, R)) β¯ satisfying (1). Obviously, knkN1 = kβk 1 (b) The class N1 = N1 (G, E) forms a Banach space with respect to the natural linear operations and the norm k · kN1 . Proof. (a) We fix n ∈ N1 and denote by B the set of all β satisfying (1). Thus knk = inf{ kβkL1 : β ∈ B }. Clearly, if β1 , β2 ∈ B then min{ β1 , β2 } ∈ B, too. We choose a sequence βk ∈ B such that kβk kL1 < knk + 1/k. Replacing βk to min{ β1 , β2 , . . . , βk }, without loss of generality we may assume that βk+1 ≤ βk . By 1.5.5 the sequence βk converges pointwise to a function β¯ ∈ L1 . Clearly, ¯ ≤ knk and β¯ satisfies (1). Therefore kβk ¯ = knk. kβk

316

V. OPERATORS WITH VARYING COEFFICIENTS

¯ } were Let β ∈ B. Clearly, we have min{ β, β¯ } ∈ B. If the set { t : β(t) < β(t) ¯ not null the L1 -norm of the function min{ β, β } ∈ B would be less than knk; but ¯ = knk. Hence β¯ is the smallest element in B. this contradicts the identity kβk (b) Clearly, n 7→ knkN1 is a semi-norm. Assume that knkN1 = 0. Then inequality (1) holds for the zero function β in L1 , which implies that n is locally null. Thus k · kN1 is a norm. The completeness of N1 follows easily from 1.1.2. ¤ 5.4.2. The multiplication in N1 . For n, m ∈ N1 we define the product by the rule Z ¡ ¢ n ? m (t, s) = n(t, ξ)m(ξ, s) dλ(ξ). (2) In this formula we set n(t, ξ)m(ξ, s) = 0 if one of the factors is undefined, but the other is equal to zero. Proposition. The multiplication is well defined. Namely, the function ξ 7→ n(t, ξ)m(ξ, s) is integrable for locally almost all (t, s) ∈ G × G, and formula (2) defines the function n ? m of the class N1 . Moreover, if n and m satisfy (1) with βn and βm then n ? m satisfies (1) with βn ∗ βm . Consequently the class N1 = N1 (G, E) is a Banach algebra. Proof. Assume n, m ∈ N1 . We take σ-compact sets Kn and Km such that n(t, s) = 0 for all t − s ∈ / Kn and m(t, s) = 0 for all t − s ∈ / Km , respectively. Then the integrand (t, ξ, s) 7→ n(t, ξ)m(ξ, s) in (2) is equal to zero if t − ξ ∈ / Kn or ξ − s ∈ / Km or, equivalently, t ∈ / ξ + Kn or s ∈ / ξ − Km . Consequently n(t, ξ)m(ξ, s) = 0 if t − s ∈ / Kn + Km . (Indeed, t ∈ ξ + Kn and s ∈ ξ − Km implies t − s ∈ Kn + Km if and only if t − s ∈ / Kn + Km implies t ∈ / ξ + Kn or s ∈ / ξ − Km .) Clearly, the set Kn + Km is σ-compact. Thus condition (i) in the definition of the class N1 holds for the function n ? m. Let U ⊆ G be a compact set. We consider the integrand (t, ξ, s) 7→ n(t, ξ)m(ξ, s) on the set U × G × U ⊆ G × G × G. Clearly, it is measurable. Furthermore, n(t, ξ) = 0 if t − ξ ∈ / Kn or, equivalently, ξ ∈ / t − Kn . And m(ξ, s) = 0 if ξ−s ∈ / Km or, equivalently,¡ ξ ∈ / s + Km . Therefore ¢ ¡ the integrand is equal ¢ to zero outside the σ-compact set U × (U − Kn ) × G ∩ G × (U + Km ) × U . On the other hand, for locally almost all (t, ξ, s) ∈ U × G × U we have the estimate kn(t, ξ)m(ξ, s)k ≤ βn (t − ξ)βm (ξ − s),

(3)

where βn and βm satisfy (1) with n and m, respectively. Since the left hand side is zero outside a σ-compact set, in fact estimate (3) holds almost everywhere. Therefore (we consider (t, s) as a single variable) ¶ Z ∗ µZ ∗ kn(t, ξ)m(ξ, s)k dλ(ξ) dλ(s) dλ(t) U ×U µZ ¶ Z ≤ βn (t − ξ)βm (ξ − s) dλ(ξ) dλ(s) dλ(t) U ×U Z ¡¡ ¢ ¢ = βn ∗ βm (t − s) dλ(s) dλ(t) < ∞. U ×U

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317

Applying Fubini’s theorem we obtain that integral (2) exists for almost all (t, s) ∈ U × U . From (3) it follows that estimate (1) holds for n ? m with βn?m = βn ∗ βm . In particular, this implies the inequality kn ? mk ≤ knk · kmk. ¤ 5.4.3. Operators of the class N1 . We discuss the operator Z ¡ ¢ N x (t) = n(t, s)x(s) dλ(s)

(4)

with n ∈ N1 (G, E). In this formula we set n(t, s)x(s) = 0 if one of the factors is undefined, but the other is equal to zero. We say that N ∈ N1 is an integral operator with summable memory. Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let n ∈ N1 (G, E). (a) For any x ∈ Lp (G, E) the function s 7→ n(t, s)x(s) is integrable for almost all t ∈ G if p 6= 0, ∞, and for locally almost ¡ all t ∈ G ¢ if p = 0, ∞. In all cases formula (4) defines an operator N ∈ B Lp (G, E) with kN k ≤ kβkL1 . (b) For any x ∈ Lpq (G, E) the function s 7→ n(t, s)x(s) is integrable¡ for locally¢ almost all t ∈ G, and formula (4) defines an operator N ∈ s Lpq (G, E) with kN k ≤ 2c kβkL1 . Proof. We take K and β from the definition of N1 . We recall from 4.4.1 that the operator of convolution Z ¡ ¢ Tβ x (t) = β(t − s)x(s) dλ(s) c acts both on Lp (G, ¡ C) with kT ¢ β k ≤ kβkL1 , and on Lpq (G, C) with kTβ k ≤ 2 kβkL1 . Moreover, Tβ ∈ s Lpq (G, C) . Let x ∈ Lp or x ∈ Lpq . To show that formula (4) makes sense we make use of Fubini’s theorem. Clearly, the function (t, s) 7→ n(t, s)x(s) is measurable. Let U ⊆ G be a compact set. By the definition of the class N1 we have n(t, s) = 0 for t − s ∈ / K or, equivalently, s ∈ / t − K. Thus the restriction of the function (t, s) 7→ n(t, s)x(s) to U × G is equal to zero outside the σ-compact set U × (U − K). On the other hand, almost everywhere on U × (U − K) we have |n(t, s)x(s)| ≤ β(t − s)|x|(s). Therefore we arrive at the estimate Z ∗Z ∗ Z Z |n(t, s)x(s)| dλ(s) dλ(t) ≤ β(t − s)|x|(s) dλ(s) dλ(t) U G U G Z ¡ ¢ = Tβ |x| (t) dλ(t) U

= k 1U Tβ |x| kL1 < ∞, which shows that the integral (4) exists for almost all t ∈ U , and the function 1U N x is integrable. Consequently the function N x is defined locally almost everywhere and measurable. We also note that the estimate Z ∗ ¯ ¯ ¡ ¢ ¯N x¯(t) ≤ |n(t, s)x(s)| dλ(t) ≤ Tβ |x| (t)

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holds for locally almost all t ∈ G. This estimate implies immediately that N x ∈ Lpq if x ∈ Lpq . Now assume that x ∈ Lp . From the last estimate it is easy to see that N x ∈ Lp . By 1.6.2(e) the set F = { t ∈ G : x(t) 6= 0 or x(t) is undefined } is σ-compact. For any t ∈ G we have n(t, s) = 0 if t − s ∈ / K or, equivalently, t ∈ / s + K. On the other hand, x(s) = 0 if s ∈ / F . Therefore / K + F then either n(t, s) = 0 or ¡ ¢ if t ∈ x(s) = 0 for each s. Consequently N x (t) = 0 for t ∈ / K + F . Since K + F is σ-compact it follows that actually N x ∈ Lp . The last estimate also shows that kN k ≤ kTβ k, and kNij k ≤ k(Tβ )ij k for the entries Nij of the matrix of N¢. Moreover, it follows that N is restored by its ¡ matrix. Thus N ∈ s Lpq (G, E) . ¤ 5.4.4. The algebra N1 of operators. Let E be a Banach space,¢ and let ¡ 1 ≤ p ≤ ∞, and 1 ≤ q ¡≤ ∞ or q = ¢ 0. We denote by N1 = N1 Lpq (G, E) the set of all operators N ∈ B Lpq (G, E) induced by kernels n ∈ N1 (G, E). Proposition. n 7→ N is a morphism¡ of Banach¢ algebras from ¡ The mapping ¢ N1 (G, E) to B Lpq (G, E) . Consequently its image N1 Lpq (G, E) is an algebra. Proof. It only is not completely evident that n 7→ N preserves multiplication. Let n, m ∈ N1 (G, E). We denote by N , M , and Π the operators induced by n, m, and n ? m, respectively. First we show that for each pair x ∈ L∞ (G, E) and y ∈ L1 (G, E0 ) one has hΠx, yi = hN M x, yi or, in detailed form, ¶ Z DZ µZ E n(t, ξ)m(ξ, s) dλ(ξ) x(s) dλ(s), y(t) dλ(t) µZ ¶ Z DZ E = n(t, ξ) m(ξ, s)x(s) dλ(s) dλ(ξ), y(t) dλ(t). By assumptions, n(t, ξ) = 0 for t − ξ ∈ / Kn , m(ξ, s) = 0 for ξ − s ∈ / Km , and y(t) = 0 for t ∈ / F , where Kn , Km , and F are σ-compact sets. Therefore the function (t, ξ, s) 7→ hn(t, ξ)m(ξ, s)x(s), y(t)i is equal to zero outside the σ-compact set F × (F − Kn ) × (F − Kn − Km ). Clearly, this function is measurable and estimated by the function (t, ξ, s) 7→ βn (t − ξ)βm (ξ − s)|x|(s)|y|(t) locally almost everywhere. Therefore it belongs to L1 (G × G × G). Thus the identity follows from Fubini’s theorem. The proved identity shows that Πx = N M x for all x ∈ L∞ (G, E). It remains to note that L∞ ∩ Lpq = L∞q is dense in Lpq . ¤ 5.4.5. N∞ is dense in N1 . We denote by N∞ = N∞ (G, E) the subclass of N1 (G, E) consisting of all¢kernels n : G × G → B(E) satisfying the estimate (1) for ¡ some β ∈ L∞1 G, [0, ∞) locally almost everywhere, cf. 5.3.4. Clearly, without

5.4. INTEGRAL OPERATORS

319

loss of generality we may assume that β is defined everywhere and is equal to zero outside a σ-compact set. Moreover, the equivalence class of n ∈ N∞ (G, E) contains a function that is also defined everywhere and satisfy (1) for all t and s. Thus this definition is in agreement with 5.3.4. ¡ ¢ Remark. We do not state that any operator N ∈ N∞ Lpq (G, E) is induced by a kernel n ∈ N∞ (G, E) if E is infinite-dimensional. Proposition. For an arbitrary n ∈ N1 (G, E) and any ε > 0 there exists n ¯ ∈ N∞ (G, E) such that kn − n ¯ kN1 < ε. Proof. Clearly, L∞1 (G, R) is dense in L1 (G, R). (We recall that in L∞1 we use the identification almost everywhere and in L1 only everywhere. The phrase ‘L∞1 (G, R) is dense in L1 (G, R)’ means literally that L∞1 ∩L1 is dense in L1 .) We ¯ L < ε. Evidently, without loss of generality choose β¯ ∈ L∞1 (G, R) so that kβ − βk 1 we may assume that n, β, and β¯ are defined everywhere, estimate (1) holds for all ¯ t and s, and 0 ≤ β(h) ≤ β(h) for all h. We set  ¯  β(t − s) β(t − s) n ¯ (t, s) = n(t, s) ·  0

if β(t − s) 6= 0, if β(t − s) = 0.

¯ − s) and 0 ≤ n(t, s) − n ¯ − s). ¤ Clearly, n ¯ (t, s) ≤ β(t ¯ (t, s) ≤ β(t − s) − β(t 5.4.6. N∞ is an ideal in N1 Proposition. The set N∞ (G, E) is a two-sided ideal in the algebra N1 (G, E). ¡ ¢ Proof. ¡By virtue of 5.4.2 it suffices to show that L G, C is an ideal in the ∞1 ¢ ¡ ¢ algebra L11 G, C ' L1 G, C . But this is a reformulation of 4.4.11(b, d). ¤ f1 is full. We denote by N f1 (G, E) the algebra 5.4.7. The subalgebra N ¡ ¢ f N1 (G, E) with ¡ an adjoint ¢ unit, and by N1 ¡Lpq (G, E)¢ the algebra of all operators α + N ∈ B Lpq (G, E) , where N ∈ N1 Lpq (G, E) and α ∈ C. Clearly, the morphism n 7→ N defined in 5.4.4 possesses the natural continuation to these algebras. ¡ ¢ f1 Lpq (G, E) Theorem. Let E be ¡a finite-dimensional Banach space. Then N ¢ is a full subalgebra of B Lpq (G, E) . Proof. Let n ∈ N1 and N be the operator (4). Assume α ∈ C and the operator α + N is invertible.¡ ¢ We note that by 5.4.5 N L (G, E) is contained in the 1 pq ¡ ¢ ¡ ¢ closure of the proper (see 5.3.2) ideal N∞ Lpq (G, E) . Therefore N1 Lpq (G, E) is proper itself. Hence α 6= 0. According to 5.4.5 we choose n ¯ ∈ N∞ such that kn − n ¯ kN1 < α/2. Then by f1 . Therefore by 1.4.6 1.4.2 the kernel α + (n − n ¯ ) is invertible in the algebra N

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V. OPERATORS WITH VARYING COEFFICIENTS

f1 , too. We consider the ¯ ) is invertible in N the corresponding operator α + (N − N kernel ¡ ¢−1 k = α + (n − n ¯) (α + n) ¡ ¢−1 ¡ ¢ = α + (n − n ¯) α + (n − n ¯) + n ¯ ¡ ¢−1 ¡ ¢ ¡ ¢−1 = α + (n − n ¯) α + (n − n ¯ ) + α + (n − n ¯) n ¯ ¡ ¢−1 ¯) = 1 + α + (n − n n ¯ ¡ ¢ ¯ ) −1 N ¯ . By 5.4.4 the operator and the corresponding operator K = 1+ α+(N − N K is invertible as a product of two invertible operators. Since n ¯ ∈ N∞ and ¡ ¢−1 ¡ ¢−1 f α + (n − n ¯) ∈ N1 , by 5.4.6 we have α + (n − n ¯) n ¯ ∈ N∞ . Therefore by −1 g 5.3.6 K ∈ N ∞ . From the representation ¡ ¢ ¯ ) −1 (α + N )−1 = K −1 α + (N − N f1 . it is clear that (α + N )−1 ∈ N

¤

f1 is independent from p and q 5.4.8. The invertibility of N ∈ N Corollary. Let E be a finite-dimensional Banach space and Lpq = Lpq (G, E), f1 is invertible on Lpq 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. If an operator N ∈ N for some p and q then it is invertible on Lpq for all p and q. Proof. The proof is similar to that of 5.2.7 and based on 5.4.7. ¤ 5.4.9. The algebra M1 of kernels. Let G be compactly generated, i.e., G ' R × Zb × K, where a and b are non-negative integers, and K is a compact abelian group. According to this isomorphism, for h = (t1 , t2 , . . . , ta ; n1 , n2 , . . . , nb ; k) ∈ Ra × Zb × K we set a

|h| = |t1 | + |t2 | + · · · + |ta | + |n1 | + |n2 | + · · · + |nb |. For η > 0 we denote by M1 η = M1 η (G, E) the set of all measurable functions m : G × G → B(E) such that there exists an integrable function β : G → [0, +∞) such that km(t, s)k ≤ e−η|t−s| β(t − s)

for locally almost all (t, s) ∈ G × G.

(5)

(We note that G is σ-compact. Hence an assumption of the kind (i) in 5.4.1 is superfluous.) We denote by M1 = M1 (G, E) the union of M1 η over all η > 0. Clearly, M1 η ⊆ M1 γ if η > γ, and M1 ⊆ N1 . If m ∈ M1 we say that the kernel m is exponentially summable. For η > 0 and n ∈ M1 η we denote by knkη = knkM1 η the infimum of kβkL1 over all β satisfying (5). Clearly, knkM1 γ ≤ knkM1 η if η > γ.

5.4. INTEGRAL OPERATORS

321

Proposition. For all η > 0, M1 η (G, E) is a Banach algebra. Consequently M1 (G, E) is an algebra, too. Proof. By virtue of 5.4.2 it is sufficient to prove the following: assume β1 , β2 ∈ L1 and β˜1 (h)¢ = e−η|h| β1 (h) and β˜2 (h) = e−η|h| β2 (h); then the function β12 (h) = ¡ eη|h| β˜1 ∗ β˜2 (h) belongs to L1 and kβ12 kL1 ≤ kβ1 kL1 kβ2 kL1 . Indeed, Z ¢ ¡ kβ12 kL1 = eη|h| β˜1 ∗ β˜2 (h) dλ(h) Z ³Z ´ η|h| ˜ ˜ = e β1 (s) · β2 (h − s) dλ(s) dλ(h) µ ¶ Z Z ¡ −η|s| ¢ ¡ −η|h−s| ¢ η|h| = e e β1 (s) · e β2 (h − s) dλ(s) dλ(h) ¶ Z µZ ¡ ¢ ¡ ¢ ≤ β1 (s) · β2 (h − s) dλ(s) dλ(h) = kβ1 kL1 kβ2 kL1 .

¤

5.4.10. M∞ is dense in M1 . For η > 0 we denote by M∞ η = M∞ η (G, E) the subclass of M1 η (G, E) consisting m : G×G → B(E) which satisfy ¡ of all kernels ¢ the estimate (5) for some β ∈ L∞1 G, [0, ∞) . We denote by M∞ = M∞ (G, E) the union of M∞ η over all η > 0. Clearly, M∞ η ⊆ M∞ γ if η > γ, and M∞ ⊆ M1 . It is easy to see (cf. 5.4.5) that this definition is in agreement with 5.3.8. Proposition. For arbitrary η, ε > 0 and m ∈ M1 η (G, E) there exists m ¯ ∈ M∞ η (G, E) such that km − mk ¯ M1 η < ε. Proof. The proof is similar to that of 5.4.5. ¤ 5.4.11. M∞ is an ideal in M1 Proposition. For any η > 0 the set M∞ η (G, E) is a two-sided ideal in the algebra M1 η (G, E). Consequently M∞ (G, E) is a two-sided ideal in the algebra M1 (G, E). Proof. By virtue of 5.4.2 it suffices to prove the following: assume β1 ∈ L1 , and β2 ∈ L∞1 ¡, and β˜1 (h) = e−η|h| β1 (h) and β˜2 (h) = e−η|h| β2 (h); then the function ¢ β12 (h) = eη|h| β˜1 ∗ β˜2 (h) belongs to L∞1 . Indeed, ¡ ¢ β12 (h) = eη|h| β˜1 ∗ β˜2 (h) Z η|h| =e β˜1 (s) · β˜2 (h − s) dλ(s) Z ¡ η|s| ¢ ¡ ¢ η(|h|−|s|−|h−s|) =e e β˜1 (s) · eη|h−s| β˜2 (h − s) dλ(s) Z ¡ ¢ ¡ ¢ η(|h|−|s|−|h−s|) =e β1 (s) · β2 (h − s) dλ(s). It remains to observe that eη(|h|−|s|−|h−s|) < 1 and, by 4.4.11 (cf. the proof of 5.4.6), the convolution β1 ∗ β2 belongs to L∞1 .

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g1 is full. We denote the class of all operators 5.4.12. M ¡ The subalgebra ¢ M ¡∈ B Lpq (G, ¢ E) induced by kernels m ∈ M1 (G, E) by the symbol M1 = M1 Lpq (G, E) . We say that M ∈ M1 is an integral operator with exponential memory. By 5.4.9 and 5.4.4 M1 is an algebra. g1 (G, E) the algebra M1 (G, E) with an adjoint unit, and we We denote by M ¡ ¢ ¡ ¢ g1 Lpq (G, E) the algebra of all operators α + M ∈ B Lpq (G, E) , denote by M ¡ ¢ where M ∈ M1 Lpq (G, E) and α ∈ C. ¡ ¢ ¡ ¢ ] ] We define M1 η Lpq (G, E) , M 1 η (G, E), and M1 η Lpq (G, E) , η > 0, similarly. ¡ ¢ g1 Lpq (G, E) Theorem. Let E be ¡a finite-dimensional Banach space. Then M ¢ is a full subalgebra of B Lpq (G, E) . Proof. The proof is a modification of that of 5.4.7. Let m ∈ M1 and M be the operator induced by m. Assume α ∈ C and the operator α + M is invertible. As we have noted in the proof of 5.4.7, α 6= 0. By definition m ∈ M1 η for some η > 0. By 5.4.10 we choose m ¯ ∈ M∞ η such that km − mk ¯ M1 η < α/2. Then by 1.4.2 the kernel α + (m − m) ¯ is invertible in ¯ ] the algebra M 1 η . Therefore by 1.4.6 the corresponding operator α + (M − M ) is ] invertible in M 1 η , too. We consider the kernel (cf. the proof of 5.4.7) ¡ ¢−1 k = α + (m − m) ¯ (α + m) ¡ ¢−1 = 1 + α + (m − m) ¯ m ¯ ¡ ¢ ¯ ) −1 M ¯ . We observe that by and the corresponding operator K = 1 + α + (M − M 5.4.4 the operator K is invertible as a product of two invertible operators. Since ¡ ¢−1 m ¯ ∈ M∞ , by 5.4.11 we have α + (m − m) ¯ m ¯ ∈ M∞ . Therefore by 5.3.10 ¡ ¢ −1 −1 ¯ ) −1 it is ] K ∈ M∞ . From the representation (α + M ) = K −1 α + (M − M g1 . ¤ clear that (α + M )−1 ∈ M

5.5. Operators with locally fading memory This section is devoted to the class t introduced in 1.6.6. We modify slightly that definition to make it more convenient for further exposition (see §6.1). The class t is the widest class of operators described in terms of matrices we consider. It seems that the majority of operators arising in applications belong to t. It is important that the investigation of operators of the class t on l∞ , Lp∞ , and C∞ to large extent can be reduced to the investigation of operators of the class t on l0 , Lp0 , and C0 , respectively. 5.5.1. Directions on a Banach space. Let X be a Banach space, and let ∆ be a fixed infinite set. We say that a family Λ+ = {X α : α ∈ ∆} of closed subspaces X α ⊆ X is a +-direction on X if ∀α, β ∈ ∆

∃γ ∈ ∆

Xα + Xβ ⊆ Xγ.

5.5. OPERATORS WITH LOCALLY FADING MEMORY

323

Let {X \α : α ∈ ∆} be another family of closed subspaces X \α ⊆ X (formally, \α is just a symbol different from α, which allows one to differ X α and X \α ). We say that the family Λ− = {Xα : α ∈ ∆} of quotient spaces Xα = X/X \α is a −-direction on X if ∀α, β ∈ ∆

∃γ ∈ ∆

X \γ ⊆ X \α ∩ X \β

or, in other words, Xγ can be considered as a quotient space both of Xα and Xβ . e are equivalent if We say that +-directions {X α : α ∈ ∆} and {X αe : α e ∈ ∆} e such that X α ⊆ X αe , and conversely, for for any α ∈ ∆ there exists α e ∈ ∆ e there exists α ∈ ∆ such that X αe ⊆ X α . We say that −-directions any α e ∈ ∆ e are equivalent if for any α ∈ ∆ there exists α e {Xα : α ∈ ∆} and {Xαe : α e ∈ ∆} e∈∆ \α \e α e there exists α ∈ ∆ such that such that X ⊇ X , and conversely, for any α e∈∆ \e α \α X ⊇X . Assume we have ±-directions on a Banach space X. By 1.2.9 a +-direction on X induces the −-direction (X 0 )α = X 0 /(X α )⊥ ' (X α )0 , α ∈ ∆, on X 0 ; and a −-direction on X induces the +-direction (X 0 )α = (X \α )⊥ ' (Xα )0 , α ∈ ∆, on X 0 . Unless otherwise specified, we assume that X 0 is endowed with these directions. Example. (a) Let I be a set (the main example is I = Z) and Ei , i ∈ I, be Banach spaces. We consider the space X = lq (I) = lq (I, Ei ), 1 ≤ q ≤ ∞ or q = 0 (see 1.5.7 for the definition). Let α be a subset of I; we denote by X α the subspace of X consisting of all {xi } such that xi = 0 for i ∈ / α. Let ∆ be the α class of all finite subsets α ⊆ I. Then the family {X : α ∈ ∆} is a +-direction, and the family {Xα = X/X \α : α ∈ ∆} is a −-direction on X; here \α means the set-theoretical complement of α to I. Clearly (in this example), the spaces X α and Xα are naturally isomorphic. These directions on lq will be referred to as standard. We recall from 1.8.1 that if q 6= ∞ the conjugate of lq (I, Ei ) can be identified with lq0 (I, E0i ). Clearly, the directions on lq0 (I, E0i ) induced by the standard directions on lq coincide with the standard ones. (b) Let E be a Banach space, and ¡let 1 ≤ p ≤ ∞, ¢ and 1 ≤ q ≤ ∞ or q = 0. We identify the space Lpq (G, E) with lq I, Lp (Qi , E) , and the space Cq (G, E) with lq (I, C♦i ), see 1.6.10. We call the directions on Lpq and Cq induced by these isomorphisms according to (a), the standard directions. (c) A closed subset α ⊆ G is called regular (see [Sik, §1] for details) if the closure of the interior of α coincides with α. For a regular α we denote by \α the closure of the set-theoretical complement of α; clearly, \α is regular, too. Let X = Lpq or X = Cq . For a regular α ⊆ G we denote by X α the subspace of all x ∈ X that are equal to zero outside α. Let ∆ be the class of all compact regular subsets α ⊆ G. Then the family {X α : α ∈ ∆} is a +-direction, and the family {Xα = X/X \α : α ∈ ∆} is a −-direction on X. We call these directions natural. Note that these examples are trivial if I is finite or G is compact. Proposition. The standard and natural directions on Lpq (G, E) and Cq (G, E) are equivalent. Proof. The proof follows from the definitions, see 1.6.10. ¤

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5.5.2. The classes tf and t. The classes tf and t have been defined in 1.6.6. Here we generalize slightly their definitions to make them consistent with our new notation. Let (X, Y ) be a pair of Banach spaces with fixed directions. We say that an operator T ∈ B(X, Y ) has locally bounded memory if for any α ∈ ∆ there exists β ∈ ∆ such that T Xα ⊆ Y β,

(1)

T X \β ⊆ Y \α .

(2)

We denote the class of all such operators by tf = tf (X, Y ), and its closure by t = t(X, Y ). If X = Y we employ the shorten notation tf (X) and t(X). If T ∈ t we say that T has locally fading memory. Clearly, d(L) ⊆ t(L) for L = Lpq and L = Cq . It is easy to see that t is essentially wider than d, see also the remark in 6.5.5 below. Remark. For a causal operator T on (−∞, τ ] the assumption (1) is always true; and for a causal operator T on [τ, +∞) the assumption (2) is always true. We stress that (2) is equivalent (see 1.2.5) to the existence of the operator T : Xβ → Yα .

(20 )

Proposition. Let (X, Y ) be a pair of Banach spaces with directions. (a) The classes tf (X, Y ) and t(X, Y ) are subspaces of B(X, Y ). If A, B ∈ tf then AB ∈ tf . If A, B ∈ t then AB ∈ t. In particular, tf (X, Y ) and t(X, Y ) are algebras with a unit provided X = Y . (b) Assume T ∈ B(X, Y ). Then T ∈ tf if and only if T 0 ∈ tf . And T ∈ t if and only if T 0 ∈ t. (c) Consider the spaces lq = lq (I, Ei ), 1 ≤ q ≤ ∞ or q = 0, with standard directions. The class tf (lq ) consists of all operators T ∈ B(lq ) defined by matrices {Tij } such that any column Tj = {Tij : i ∈ I} and any row Ti = {Tij : j ∈ I} of {Tij } contain only a finite number of non-zero members. Thus our new definitions ¡ ¢of tf and t are in agreement with that in 1.6.6. (d) Operators T ∈ t lq (I, Ei ) , 1 ≤ q ≤ ∞ or q = 0, are hlq , lq0 i-continuous. (e) Two equivalent directions on (X, Y ) define the same classes tf (X, Y ) and tf (X 0 , Y 0 ) and consequently the same classes t(X, Y ) and t(X 0 , Y 0 ). Proof. (a) From (1) and (2) it follows clearly that tf is a subspace. The set t is also a subspace because it is a closure of tf . Assume A, B ∈ tf . Properties (1) and (2) imply that AB ∈ tf . Since t is the closure of tf it possesses the same property. (b) Let T ∈ tf . For α, β in (1) we consider the diagram j

q

j

q

0 −−−−→ X α −−−−→ X −−−−→ X\α = X/X α −−−−→ 0       Ty Ty Ty 0 −−−−→ Y β −−−−→ Y −−−−→ Y\β = Y /Y β −−−−→ 0.

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325

Evidently, its rows are exact. By (1) the left square is commutative. Then by 1.2.4 one can define the right T so that the right square become commutative, too. Hence the conjugate diagram (see 1.2.9) q

j

q

j

0 ←−−−− (X 0 )α ←−−−− X 0 ←−−−− (X 0 )\α ←−−−− 0 x x x    0 0 0 T  T  T  0 ←−−−− (Y 0 )β ←−−−− Y 0 ←−−−− (Y 0 )\β ←−−−− 0 has also exact rows (see 1.2.6) and commutative squares. The commutativity of the right square means that assumption (2) holds for T 0 . Conversely, let assumption (2) hold for T 0 . Then the left square of the latter diagram is commutative. Arguing as above (see 1.2.9) one shows that T 00 (X α )⊥⊥ ⊆ (Y β )⊥⊥ . Clearly, in order to prove that T X α ⊆ Y β , it suffices to verify that, e.g., X ∩ (X α )⊥⊥ = X α . The inclusion X α ⊆ (X α )⊥⊥ is evident. We prove the converse. Suppose x ∈ X ∩ (X α )⊥⊥ , but x ∈ / X α . Then by the Hanh–Banach theorem there exists x0 ∈ (X α )⊥ such that hx, x0 i 6= 0, which contradicts the assumption x ∈ (X α )⊥⊥ . Now for α, β in (2) we consider the diagram j

q

j

q

0 −−−−→ X \β −−−−→ X −−−−→ Xβ −−−−→ 0       Ty Ty Ty 0 −−−−→ Y \α −−−−→ Y −−−−→ Yα −−−−→ 0. Evidently, its rows are exact. By (2) the left square is commutative. Then by 1.2.4 one can define the right T so that the right square become commutative, too. Hence the conjugate diagram q

j

q

j

0 ←−−−− (X 0 )\β ' X 0 /(X 0 )β ←−−−− X 0 ←−−−− (X 0 )β ←−−−− 0 x x x    0 0 0 T  T  T  0 ←−−−− (Y 0 )\α ' Y 0 /(Y 0 )α ←−−−− Y 0 ←−−−− (Y 0 )α ←−−−− 0 has also exact rows (see 1.2.6) and commutative squares. The commutativity of the right square means that assumption (1) holds for T 0 . Conversely, let assumption (1) hold for T 0 . Then the right square of the latter diagram is commutative, which means that T 0 maps (Y 0 )α ' (Y \α )⊥ into (X 0 )β ' (X \β )⊥ . Arguing as above we see that T maps X \β into Y \α . Then from the initial diagram we obtain that (2) holds for T . Clearly, the isometric isomorphism between tf (X, Y ) and tf (Y 0 , X 0 ) can be extended by continuity to the isomorphism between t(X, Y ) and t(Y 0 , X 0 ). (c) We fix j ∈ I and set α = {j}. By the definition of the standard direction, X α consists of all x = {xk } such that xk = 0 for k 6= j. Assumption (1) implies

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that there exists β = {i1 , i2 , . . . , im } such that (T x)k = 0 for k ∈ / β. This means that in the column Tj only the entries Ti1 j , Ti2 j , . . . , Tim j can be non-zero. Next, we fix i ∈ I and set α = {i}. We recall that X \α consists of all {xk } such that xk = 0 for k = α. By virtue of (2) there exists β = {j1 , . . . , jm } such that xj1 = · · · = xjm = 0 imply (T x)i = 0. Thus the row Ti contains only a finite number of non-zero entries. Moreover, this implies that (T x)i is determined by the values of xj1 , . . . , xjm only. In particular, T is induced by its matrix in the sense of 1.6.4. The converse assertion is evident. (d) We make use of 1.1.10. The case q 6= ∞ follows from 1.8.1. Assume q = ∞. Clearly, it suffices to consider the case of T ∈ tf (l∞ ). By (b) and (1), T 0 maps l00 ⊆ (l∞ )0 into itself. Hence it maps l1 ⊆ (l∞ )0 into itself, too. (e) is evident. ¤ 0 Remark. Let us consider the space l∞ = l∞ (I, Ei ) and its conjugate l∞ . We 0 0 ⊥ 0 α represent l∞ as l0 ⊕ l0 , see 1.8.3. It is easy to see that the subspaces (l∞ ) , 0 0 α α ∈ ∆, are contained in l0 ' l1 (I, Ei ) and naturally isomorphic to (l1 ) ; and the 0 0 \α ) , α ∈ ∆, contain l0⊥ ; thus (l∞ )α are naturally isomorphic to the subspaces (l∞ 0 0 0 quotient spaces (l0 )α ' (l1 )α of l0 ' l1 (I, Ei ). S 0 Assume T ∈ t(l∞ ). We observe that the subspace l00 = α (l1 )α is dense in l00 ' l1 (I, E0i ). By (1) we have T l00 ⊆ l00 , which implies T l00 ⊆ l00 , cf. (d). The 0 described in the above paragraph shows that the structure of directions on l∞ membership of T in t is the property of the restriction T : l00 → l00 of T to l00 .

5.5.3. A two-diagonal representation for T ∈ tf (lq ). Below we shall need an auxiliary construction described in the following lemma. ¯ i , i ∈ I, be Banach spaces. Consider the spaces Let I be a set, and let Ei and E ¯ X = lq (I, Ei ) and Y = lq (I, Ei ), 1 ≤ q ≤ ∞ or q = 0, with standard directions. ¡ ¢ ¯ i ) . The set I can be represented Lemma. Let T ∈ tf = tf lq (I, Ei ), lq (I, E both as the union of pairwise disjoint sets αδ , δ ∈ Λ, and as the union of pairwise disjoint sets βδ , δ ∈ Λ, such that the following conditions (a) and (b) hold. (a) T X αδ ⊆ Y βδ and T X \αδ ⊆ Y \βδ for all δ ∈ Λ. Thus X is naturally isomorphic to lq (Λ, X αδ ) and Y is naturally isomorphic to lq (Λ, Y βδ ), and in this representation T is induced by a main diagonal matrix with diagonal entries Tδδ : X αδ → Y βδ . But αδ and βδ are not obligatorily finite. (b) Each αδ , δ ∈ Λ, can be represented as the union of pairwise disjoint finite sets αδk ⊆ I, k ∈ N, and each βδ , δ ∈ Λ, can be represented as the union of pairwise disjoint finite sets βδk ⊆ I, k ∈ N, such that T X αδk ⊆ Y βδk and T X \αδk ⊆ Y \βδk−1 for all k ∈ N. Thus X αδ is naturally isomorphic to lq (N, X αδk ) and Y βδ is naturally isomorphic to lq (N, Y αδk ), and in this representation the operator Tδδ : X αδ → Y βδ is induced by a two-diagonal matrix. Namely, Tij can be non-zero only if i = j or i = j − 1. Finally, X is naturally isomorphic to lq (Λ × N, X αδk ) and Y is naturally isomorphic to lq (Λ × N, Y αδk ), and in this representation T is induced by a matrix in which Tδi,σj can be non-zero only if δ = σ, and i = j or i = j − 1.

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327

Proof. We take for α ¯ 1 ⊆ I an arbitrary non-empty finite subset. According ¯ to (1) we pick a finite subset β¯1 ⊆ I such that T X α¯ 1 ⊆ Y β1 . And according to ¯ (2) we pick a finite subset α ¯2 ⊇ α ¯ 1 such that T X \α¯ 2 ⊆ Y \β1 . Then we pick finite subsets β¯k ⊇ β¯k−1 , k = 2, 3, . . . , so that ¯

T X α¯ k ⊆ Y βk ,

(3)

and finite subsets α ¯k ⊇ α ¯ k−1 , k = 2, 3, . . . , so that ¯

T X \α¯ k ⊆ Y \βk−1 .

(4)

For completeness we set α ¯ 0 = β¯0 = ∅. Finally, we set αk = α ¯k \ α ¯ k−1 and ¯ ¯ βk = βk \ βk−1 . As usual, for k ∈ N we denote by X αk the subspace of all {xi } ∈ X such that xi = 0 for i ∈ / αk , and by Y βk we denote the subspace of all {yi } ∈ Y such that yi = 0 for i ∈ / βk . We consider the spaces X1 = lq (N, X αk ) and Y1 = lq (N, Y βk ). Clearly, X1Sand Y1 are isometrically isomorphic to the subspaces X α and Y β , S where α = k αk and β = k βk . From the matrix definition of T T it follows that T X α ⊆ Y β . On the other hand, T \α \αk \β from X = k X and Y = k Y \βk , and (4) it follows that T X \α ⊆ Y \β . Clearly, X can be represented as X α ⊕ X \α with lq -norm and so can Y ; note that the matrix of T associated with this representation is main diagonal. We consider the matrix of T : X1 → Y1 . By (3), Tij = 0 for i > j. And by (4), Tij = 0 for i < j − 1. In other words, Tij can be non-zero only if i = j or i = j − 1. Next, we pick a new non-empty finite subset α ¯ 1 ⊆ I \ α and as above construct β¯1 ⊆ I \ β, α ¯ 2 ⊆ I \ α, β¯2 ⊆ I \ β, . . . satisfying (3) and (4). Finally, we obtain new subspaces X α and Y β with the same properties and so on. Proceeding in the same way and using Zorn’s lemma we can construct families αδ , βδ ⊆ I, δ ∈ Λ, and subspaces X αδ and Y βδ satisfying the desired properties. ¤ 5.5.4. The subalgebra t(lq ) is full Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. ¯ (a) Let I be spaces. If an operator ¡ a set, and let E¢i and Ei , i ∈ I, be Banach ¡ ¢ −1 ¯ ¯ i ), lq (I, Ei ) . T ∈ t lq (I, Ei ), lq (I, Ei ) is ¡invertible¢ then T ∈ t lq (I, E ¡ ¢ Specifically, the subalgebra t l (I, E ) is full in the algebra B l (I, E ) . q i q i ¡ ¢ ¡ ¢ (b) The subalgebra t L (G, E) is full in the algebra B L (G, E) ; and the pq ¢ pq ¡ ¡ ¢ subalgebra t Cq (G, E) is full in the algebra B Cq (G, E) . Proof. (a) It suffices to show that if T ∈ tf (lq , lq ) is invertible in B(lq , lq ) then T ∈ t. So we assume that T belongs to tf and is invertible. We make use of 5.5.3. We fix δ ∈ Λ. Since X αδ ⊕X \αδ = X and Y βδ ⊕Y \βδ = Y , from 5.5.3(a) it follows that Tδδ : X αδ → Y βδ is invertible and k(Tδδ )−1 k ≤ kT −1 k. According to 5.5.3(b) we represent X αδ as lq (N, X αδk ) and Y βδ as lq (N, Y βδk ). Clearly, the operator Tδδ : lq (N, X αδk ) → lq (N, Y βδk ) belongs to df ⊆ e ⊆ t, see −1

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1.6.9 and 3.4.1 (3.4.4) for the definition of e. Therefore¢ according to 3.4.2 (3.4.4) ¡ −1 βδk we can assert that (Tδδ ) ∈ e lq (N, Y ), lq (N,¡X αδk ) . ¢ We fix an arbitrary ε > 0. Since (Tδδ )−1 ∈ e lq (N, Y βδk ), lq (N, X αδk ) we can ¡ ¢ pick Rδδ ∈ tf lq (N, Y βδk ), lq (N, X αδk ) such that kRδδ − (Tδδ )−1 k ≤ ε. Note that one may assume that ε does not depend on δ. We consider the operator ¡ ¢ R ¯ induced by the matrix {Rδδ }. It is easy to see that R ∈ tf lq (I, Ei ), lq (I, Ei ) and kR − T −1 k ≤ ε. Thus T −1 ∈ t. (b) is a special case of (a). ¤ 5.5.5. The isomorphism t(l∞ ) ' t(l0 ). Let I be a set, and let Ei , i ∈ I, be Banach spaces. We consider the spaces lq = lq (I, Ei ), q = ∞ or q = 0, with standard directions. Proposition (cf. 3.2.6). (a) The algebras tf (l∞ ) and tf (l0 ) are naturally isometrically isomorphic. (b) The algebras t(l∞ ) and t(l0 ) are naturally isometrically isomorphic. (c) For all T ∈ t(l∞ ) (see 1.3.1 for the definitions of | · |+ and | · |− ) |T : l∞ → l∞ |+ = |T : l0 → l0 |+ , |T : l∞ → l∞ |− = |T : l0 → l0 |− . Proof. (a) Assume T ∈ tf (l∞ ). We observe that for any α ∈ ∆ the subspace S α X ⊆ l∞ is contained in l0 , and l00 = α X is dense in l0 . By (1) we have T l00 ⊆ l00 , which implies T l0 ⊆ l0 . Conversely, assume T ∈ tf (l0 ). From (2) it follows that for any i ∈ I there exists a finite β ⊆ I such that the value (T z)i , z ∈ l0 , is determined by the values zj , j ∈ β. Assume x ∈ l∞ . We define (T x)i to be (T z)i , where z ∈ l0 coincides with x on β. Thus we obtain the extension T : l∞ → l∞ . It is easy to see that the extended operator belongs to tf (l∞ ). It is also easy to verify that this extension is the inverse operation of the restriction defined in the previous paragraph. We show that the isomorphism between tf (l0 ) and tf (l∞ ) is isometric. Clearly, it suffices to verify that kT : l∞ → l∞ k ≤ kT : l0 → l0 k. We fix ε > 0 and pick x ∈ l∞ , kxk = 1, such that kT xk > kT : l∞ → l∞ k − ε. After that, we choose i ∈ I such that |(T x)i | > kT : l∞ → l∞ k − 2ε. According to (2) we assign to α = {i} a finite set β. Next, we define z ∈ l0 by the rule: zk = 0 for k ∈ / β, and zk = xk for k ∈ β. Then z ∈ l0 and kzk ≤ kxk = 1. But by (2) we have (T z)i = (T x)i . Hence kT : l0 → l0 k ≥ kT zk > kT : l∞ → l∞ k − 2ε. (b) follows from (a) and the definition of t as the closure of tf . (c) We make use of 5.5.3. The inequality |T : l∞ → l∞ |+ ≤ |T : l0 → l0 |+ follows from the definition of | · |+ . Let us prove the opposite inequality. Assume that |T : l∞ → l∞ |+ < κ. We fix x ∈ l∞ , kxk = 1, such that kT xk < κ. We set X = Y = l∞ . Considering the representations of X and Y as l∞ (Λ, X αδ ) and l∞ (Λ, Y βδ ) from 5.5.3, it is easy to show that without loss of generality we may assume that x ∈ X αδ for some δ. Thus the proof is reduced to the case where X = l∞ (N, X αk ), Y = l∞ (N, Y βk ), and T is defined by a two-diagonal matrix. α

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329

For any ν > 0 we consider the operator (Ψν z)k = e−νk zk , cf. 3.4.1. Evidently, Ψν acts from l∞ (N, X αk ) into itself and from l∞ (N, Y βk ) into itself. We consider xν = Ψν x. Clearly, xν belongs to l0 and kxν k → kxk as ν → 0 (but xν 6→ x). On the other hand, since the matrix of T is two-diagonal we have kT Ψν − Ψν T k → 0 as ν → 0. Consequently for sufficiently small ν we have that kT xν k = kT Ψν xk is close to kΨν T xk ≤ kT xk < κ. These imply |T : l0 → l0 |+ ≤ κ. Next, we prove that |T : l∞ → l∞ |− ≤ |T : l0 → l0 |− . We set X = Y = l∞ . Applying 5.5.3 it is easy to reduce the problem to the case where X = l∞ (N, X αk ) and Y = l∞ (N, Y βk ), and T is defined by a two-diagonal matrix. We assume that |T : l∞ → l∞ |− > κ and show that κBl0 ⊆ T Bl0 , which means |T : l0 → l0 |− ≥ κ. By 1.3.3 it suffices to verify that for any ε > 0 and y ∈ l0 , kyk ≤ κ, there exists x ∈ l0 , kxk ≤ 1, such that ky − T xk < ε. By assumption we have κBl∞ ⊆ T Bl∞ . Therefore for a given y ∈ l0 , kyk ≤ κ, there exists x ∈ l∞ , kxk = 1, such that y = T x. We define Ψν as above and consider xν = Ψν x. Since kT Ψν − Ψν T k → 0 it follows that kT xν − Ψν yk → 0. And since y ∈ l0 we have kΨν y − yk → 0. Thus kT xν − yk → 0. It remains to observe that xν ∈ l0 and kxν k ≤ 1. We turn to the proof of the inequality |T : l∞ → l∞ |− > |T : l0 → l0 |− . We assume |T : l0 → l0 |− > κ and show that κBl∞ ⊆ T Bl∞ , i.e., |T : l∞ → l∞ |− ≥ κ. By 1.3.3 it suffices to verify that for any ε > 0 and y ∈ l∞ , kyk ≤ κ, there exists x ∈ l∞ , kxk ≤ 1, such that ky − T xk < ε. We set X = Y = l∞ . As usual we may assume that X = l∞ (N, X αk ) and Y = l∞ (N, Y βk ), and T is defined by a two-diagonal matrix. We fix a natural K. For each m = 0, 1, 2, . . . we define the operator Υm acting from l¢∞ (N, X αk ) into l0 (N, X αk ) and from l∞ (N, Y βk ) into l0 (N, Y βk ) by the rule ¡ Υm z i = umi zi , where ½ 1 − |i − mK − 1|/K for |i − mK − 1| ≤ K, umi = 0 for |i − mK − 1| ≥ K. P∞ Clearly, m=0 umi = 1 for all i ∈ I; and for each i the series contains at most two non-zero summands. For any entry Tij of the matrix of T , evidently, we have that kTij k ≤ kT k. Straightforward calculations show that kT Υm − Υm T k ≤ kT k/K. For y ∈¡l∞ , kyk m ∈ l0 , kxm k ≤ 1, ¢ ≤ κ, mentioned above and m ∈ N we pick xP ∞ such that T xm i = yi for |i − mK − 1| ≤ K + 1. We set x = m=0 Υm xm (the series converges componentwise). It is easy to see that kxk ≤ 1. Hence we have °¡ ¢ ° °¡ ¢° ° T Υm xm − Υm y ° = ° T Υm xm − Υm T xm ° ≤ kT k/K i i for |i − mK − 1| ≤ K + 1;

¡ ¢ T Υm xm − Υm y i = 0

for |i − mK − 1| > K + 1.

Consequently we have ∞ ´° °¡ ¢ ° °³ X ° ° Tx − y ° = ° T Υm xm − Υm y ° ≤ 3kT k/K ° i m=0

i

for all i.

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Thus the required x ∈ l∞ is constructed.

¤

5.5.6. The equivalence of the invertibility in t(l0 ) and t(l∞ ) Corollary. (a) Let I be a set, and let Ei , i ∈ I, be Banach spaces. Consider the spaces lq = lq (I, Ei ), q = ∞ or q = 0, with standard directions. An operator T ∈ t(l∞ ) is invertible if and only if the corresponding operator T ∈ t(l0 ) is invertible. ¡ ¢ (b) Let 1 ≤ p ≤ ∞. Let E be a Banach space. An operator T ∈ ¡t Lp∞ (G, E) ¢ is invertible if and only if the corresponding ¡ ¢ operator T ∈ t Lp0 (G, E) is invertible. An operator T ∈ ¡t C∞ (G, E) ¢ is invertible if and only if the corresponding operator T ∈ t C0 (G, E) is invertible. Proof. (Cf. 5.2.7.) (a) Assume T ∈ t(l∞ ) is invertible. Then by 5.5.4, T −1 belongs to t(l∞ ), too. By 5.5.5 the operator T −1 ∈ t(l∞ ) induces the operator T −1 ∈ t(l0 ). Since the algebras t(l∞ ) and t(l0 ) are isomorphic T −1 ∈ t(l0 ) is the inverse of T ∈ t(l0 ). The converse assertion is proved in a similar way. (b) is a special case of (a). ¤ 5.5.7. Consistent directions. We say that a +-direction and a −-direction on X are consistent if for any α ∈ ∆ there exists β ∈ ∆ such that |1 : X α → Xβ |+ = 1

and

|1 : X β → Xα |− = 1.

Here 1 : X → X is the identity operator. Proposition. (a) If directions on X are consistent then the induced directions on X 0 are consistent, too. (b) Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Then the natural and standard directions on lq (I), Lpq , and Cq are consistent. Proof. (a) follows from 1.3.4 and definitions. (b) is also evident. ¤ 5.5.8. The Fourier transform maps L1 into C0 Lemma. Let G be a non-discrete locally compact abelian group. Then for any ¡ ¢ g ∈ L1 (G, C) its Fourier transform gˆ and cotransform gˇ belong to C0 X(G), C . Proof. We recall from 4.2.2 that the Fourier (co)transform of g ∈ L1 (G, C) can be identified with its Gel0 fand transform. Hence the Fourier (co)transform acts into C. On the other hand, since G is non-discrete, L1 (G, C) is an algebra without a unit. Therefore the Gel0 fand transform maps L1 (G, C) into C0 . ¤ 5.5.9. Compactly supported sets. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by L one of the spaces Lpq (G, E) or Cq (G, E), or (Lpq )0 or (Cq )0 . We say that a subset Γ ⊆ L is compactly supported if there exists α ∈ ∆ such that Γ is contained in the subspace Lα . Here we consider Lpq and Cq with the natural (or, equivalently, standard) directions, and (Lpq )0 and (Cq )0 with the induced ones.

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331

Proposition. An operator T ∈ B(L) belongs to tf (L) if and only if T and T 0 take compactly supported sets into compactly supported ones. Proof. Clearly, assumption (1) holds if and only if T takes compactly supported sets into compactly supported ones. It remains to recall from the proof of 5.5.2(b) that (2) for T is equivalent to (1) for T 0 . ¤ 5.5.10. Ψ-bounded sets. Let E be a Banach space, and let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by L = L(G, E) one of the spaces Lpq (G, E) or Cq (G, E), or (Lpq )0 or (Cq )0 . We consider Lpq and Cq with the natural (or, equivalently, standard) directions, and (Lpq )0 and (Cq )0 with the induced directions. We recall that on Lpq and Cq the oscillation operators are defined as follows ¡ ¢ Ψχ x (t) = χ(t)x(t),

χ ∈ X(G).

We define Ψχ on (Lpq )0 and (Cq )0 as the conjugate of Ψχ on Lpq and Cq , respectively. (Cf. the definition of Sh on (Lpq )0 and (Cq )0 in 1.6.12.) Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let L be Lpq or Cq ; or let 1 ≤ p ≤ ∞, and 1 ≤ q < ∞ or q = 0, but q 6= ∞, and let L be (Lpq )0 or (Cq )0 . We say that a bounded subset Π ⊆ L is Ψ-bounded if the function χ 7→ Ψχ is uniformly continuous on Π, i.e., for every η > 0 there exists a neighbourhood W of the unit of the group X(G) such that sup{ kΨχ x − xk : x ∈ Π, χ ∈ W } ≤ η. Remark. In this definition we do not consider the cases of (Lp∞ )0 and (C∞ )0 ¡ ¢0 because of the following reason. There exists non-zero f ∈ l∞ (Z) such that f is equal to zero on l0 (i.e., f ∈ l0⊥ ) and f (Ψχ x) = f (x). Essentially an example of such an f has been constructed in the example in 5.1.11. Clearly, the one point set {f } becomes Ψ-bounded if one extends the preceding definition to the case of 0 l∞ word for word. Such examples must be excluded (otherwise the proposition below will be false); it may be done, e.g., if one demands additionally that Ψ¡ ¢ 0 0 0 bounded sets in (Lp∞ ) and (C∞ ) are contained in l1 I, Lp (Qi , E) ⊆ (Lp∞ )0 and ¡ ¢ l1 I, (C♦i )0 ⊆ (C∞ )0 , respectively, cf. the second remark in 5.5.2. We do not want to complicate our exposition with such an exception. Statements 5.5.5 (see above) and 5.6.5 (see below) will allow us to avoid this generalization. We recall from 1.1.1 that a set Γ ⊆ L is called an ε-net for Π ⊆ L if for any x ∈ Π there exists y ∈ Γ such that kx − yk ≤ ε. Proposition. Let G be a non-compact locally compact abelian group. (a) Let L be Lpq or Cq , 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0; or let L be (Lpq )0 or (Cq )0 , 1 ≤ p ≤ ∞, and 1 ≤ q < ∞ or q = 0, but q 6= ∞. Then a bounded set Π ⊆ L is Ψ-bounded if and only if is has a compactly supported ε-net for any ε > 0. (b) Let L be Lpq or Cq , 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Assume T ∈ t(L). Then T takes Ψ-bounded sets into Ψ-bounded sets. If q 6= ∞ then T 0 takes Ψ-bounded sets into Ψ-bounded sets, too.

332

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Proof. We note that since q 6= ∞, (Lpq )0 and (Cq )0 can be represented as lq0 . (a) First we observe that the operators Ψχ : Lα → L, L = Lpq , Cq , α ∈ ∆, depend on χ continuously, i.e., the unit ball of Lα is Ψ-bounded. In a similar way the operators Ψχ : L → Lα , L = Lpq , Cq , α ∈ ∆, depend on χ continuously as α well. The latter implies that the operators (Ψχ )0 : (L)0 = (Lα )0 → L0 depends on α χ continuously, i.e., the unit ball of (L0 ) is Ψ-bounded, too. Now it is clear that if Π has a compactly supported ε-net for any ε > 0 then Π is Ψ-bounded. Conversely, assume that a bounded set Π is Ψ-bounded and show that Π has a compactly supported ε-net for any ε > 0. We take η and W from the definition of a Ψ-bounded set. Let us consider a continuous function g : X(G) → [0, 1] supported in W such R that g(1) = 1 (here 1 ∈ X(G) is the unit character) and X g(χ) dχ = 1. We consider the operator Ψgˇ : L → L defined by the rule Z Ψgˇ = g(χ)Ψχ dχ. The meaning of this integral is not evident. This integral converges on compact sets. Namely, as we have seen, on the subspaces Lα , α ∈ ∆, the function χ 7→ Ψχ is continuous in norm. Therefore on Lα there is no problems with the interpretation of the integral. It remains to observe that the projections of x ∈ L onto Lα parallel to L\α determine the function x completely. (Here we consider the standard +-direction and use the assumption that q 6= ∞ if L is (Lpq )0 or (Cq )0 .) From the definition of Ψχ it follows that Ψgˇ is the multiplication operator ¡ ¢ Ψgˇ x (t) = gˇ(t)x(t), R where gˇ(t) = hχ, tig(χ) dχ is the Fourier cotransform of g. By 5.5.8 (and 4.2.4) gˇ ∈ C0 (G, C). Consequently the image Ψgˇ Π of the bounded set Π under Ψgˇ has a compactly supported ε-net, say Γ, for any ε > 0. By the assumption of the Ψ-boundedness of Π we have kΨχ x−xk ≤ η for x ∈ Π and χ ∈ W . Hence we obtain kΨgˇ x − xk ≤ η for x ∈ Π. The last inequality means that the set Ψgˇ Π is an η-net for Π. Clearly, if Γ is an ε-net for Ψgˇ Π then Γ is an (ε + η)-net for Π. (b) We fix δ > 0 and take R ∈ tf such that kT − Rk ≤ δ. We assume that Π is Ψ-bounded and Γ is a compactly supported ε-net of it. Then by 5.5.9 RΓ is compactly supported. Without loss of generality we may assume that Π is contained in the unit ball of the space L. Then, clearly, RΠ is a δ-net for T Π. On the other hand, RΓ is an (ε · kRk)-net for RΠ. Hence RΓ is an (ε · kRk + δ)-net for T Π. Thus T Π is Ψ-bounded. The assertion for T 0 is proved in a similar way. ¤ 5.5.11. Differential operators. Assume that G = R. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let Xq be one of our main spaces Cq = Cq (R, E),

Cq1 = Cq1 (R, E),

Cq−1 = Cq−1 (R, E),

Lpq = Lpq (R, E),

1 1 Wpq = Wpq (R, E),

−1 −1 Wpq = Wpq (R, E).

333

5.6. OPERATORS WITH CONTINUOUS COEFFICIENTS

We recall from 3.1.2 that any of these Xq can be represented as lq (Z, X♥i ). Thus one can easily define classes e = e(X, Y ), s = s(X, Y ), tf = tf (X, Y ), and t = t(X, Y ) using matrix representation provided (X, Y ) is one of the pairs (Cq , Cq ),

(Cq , Cq1 ),

(Lpq , Lpq ), 1 (Wpq , Lpq ),

1 (Lpq , Wpq ),

(Cq1 , Cq ), −1 (Lpq , Wpq ),

(Cq , Cq−1 ),

(Cq−1 , Cq ),

−1 (Wpq , Lpq ).

Since the isomorphism U is induced by a main diagonal matrix (see 3.1.2) we have that U −1 is also induced by a main diagonal matrix. Thus U, U −1 ∈ df . This enables one to reduce the investigation of the invertibility of an operator L : X → Y of the class e (s, tf , or t) to the investigation of the invertibility of an operator of the same class acting from X to X or from Y to Y . We have used this trick for many times. For example, if L acts from Cq to Cq−1 then T = U −1 L acts from Cq to Cq , and L ∈ t if and only if T ∈ t. We do not enumerate now all the corollaries of the results of this chapter arising in this way. Remark. We recall that along with the representation Cq ' lq (Z, X♥i ) we have considered the representation Cq ' lq (Z, X♦i ), see 1.6.10, 1.6.11. Similar 1 1 representations are possible for the spaces Wpq and Cq1 . Namely, if X = Wpq 1 one may consider the subspaces X♦i of Wpq consisting of all functions which are supported in [i, i + 2] and linear on [i + 1, i + 2]. And if X = Cq1 one may consider the subspaces X♦i of Cq1 consisting of all functions which are supported in [i, i + 2] and are polynomials of degree 3 on [i + 1, i + 2]. (Note that for X = Lpq and X = Cq−1 the spaces X♦i defined according to this principle coincide with X♥i .) These modifications lead to the same classes e ⊆ s ⊆ t (but not tf ). The proof is based on the following simple observation: the composition of isomorphisms lq (Z, X♥i ) → Xq → lq (Z, X♦i ) and its inverse belong to the class e.

5.6. Operators with continuous coefficients Up to this point we substantially have not consider full subalgebras of operators on the spaces of continuous functions Cq . In this section we fill this gap. We describe an abstract class of operators possessing some properties of operators with continuous coefficients. Operators of this class act simultaneously on L∞q and Cq , and their properties on L∞q and Cq are the same. 5.6.1. Operators with uniformly continuous coefficients. Let E be a Banach space, and G be a locally compact abelian group, and 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by L = L(G, E) one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), or (Lpq )0 or (Cq )0 . For any T, S ∈ B(L) we call the operator £ ¤ T, S = T S − ST the commutator of T and S.

334

V. OPERATORS WITH VARYING COEFFICIENTS

We recall that the shift operator on Lpq and Cq is defined by the rule ¡ ¢ Sh x (t) = x(t − h), h ∈ G. We also recall from 1.6.12 that the family {Sh } is uniformly bounded. We recall that Sh on (Lpq )0 and (Cq )0 is defined to be the conjugate of S−h on Lpq and Cq . We say that£ an operator T ∈ B(L) has uniformly continuous coefficients if the ¤ function h 7→ T, Sh (or, equivalently, the function h 7→ Sh T S−h ) is continuous in norm. (This terminology comes from the example, considered below in 5.6.8.) Clearly, if these functions are continuous at h = 0 then they are continuous everywhere. We denote by Cu = Cu (L) the set of all operators with uniformly continuous coefficients. Proposition. Cu (L) is a closed full subalgebra in the algebra B(L). Proof. The proof is evident, cf. 5.6.3 below. ¤ 5.6.2. Operators with continuous coefficients. Let L be as in 5.6.1. We consider Lpq and Cq with the standard (or, equivalently, natural) directions (see 5.5.1), and (Lpq )0 and (Cq )0 with the directions induced by ones on Lpq and Cq . We say that an operator T ∈ t(L) (see 5.5.2£ for the ¤ definition of the class t) has continuous coefficients if the function h 7→ T, Sh : Lα → L (or, equivalently, the function h 7→ Sh T S−h : Lα → L) is continuous in norm for any α ∈ ∆. (This terminology comes from the example, considered below in 5.6.8.) Clearly, if these functions are continuous at h = 0 then they are continuous everywhere. We denote by C = C(L) the set of all operators with uniformly continuous coefficients. Remark. The definitions of Cu and C are not completely analogous: we do not require Cu ⊆ t. We shall not use the class Cu systematic. We mostly consider it as an illustration of the main idea in the simplest situation. Proposition. Let L be Lpq or Cq , 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0; or let L be (Lpq )0 or (Cq )0 , 1 ≤ p ≤ ∞, and 1 ≤ q < ∞ (but q 6= ∞) or q = 0. Then an operator T ∈ t(L) belongs to C(L) if and only if for any Ψ-bounded set Π and any ε > 0 there exists a neighbourhood V of the unit of the group G such that °£ ¤ ° ° T, Sh x° ≤ εkxk or, equivalently, kSh T S−h x − T xk ≤ εkxk, x ∈ Π, h ∈ V. Proof. Let Π be a Ψ-bounded set, and let ε > 0 be arbitrary. According to 5.5.10(a) we choose a compactly supported ε-net Γ for Π. Without loss of generality we may assume that Π and Γ are contained in the unit ball of L. By the definition of C we choose a neighbourhood V so that kSh T S−h y − T yk ≤ εkyk for y ∈ Γ and h ∈ V . Let us consider an arbitrary x ∈ Π. We pick y ∈ Γ so that kx − yk ≤ ε. Then we have kSh T S−h x − T xk ≤ kSh T S−h x − Sh T S−h yk + kSh T S−h y − T yk + kT y − T xk ≤ εkSh T S−h k + εkyk + εkT k ≤ εkSh T S−h k + ε + εkT k, which can be done arbitrary small. The converse statement is obvious. ¤

5.6. OPERATORS WITH CONTINUOUS COEFFICIENTS

335

5.6.3. The subalgebra C is full Theorem. Let L be Lpq or Cq , 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0; or let L be (Lpq )0 or (Cq )0 , 1 ≤ p ≤ ∞, and 1 ≤ q < ∞ (but q 6= ∞) or q = 0. Then the set C(L) is a closed full subalgebra in the algebra B(L). £ Proof. ¤ Let £T1 , T2 ¤∈ C. £ To ¤show that T1 T2 ∈ C, by virtue of the identity T1 T2 , Sh = T1 T2 , Sh + T1 , Sh T2 it suffices to refer to 5.6.2 and 5.5.10(b). Assume T belongs to the closure of C. We show that in fact T belongs to C. Indeed, given ε > 0 we pick R ∈ C such that kT − Rk < ε. Then in the estimate °£ ¤ ° °£ ¤ ° °£ ¤ ° ° T, Sh x° ≤ ° R, Sh x° + ° T − R, Sh x° the first summand can be done sufficiently small provided h and x are chosen according to the definition of C; and the second summand is less than or equal to 2kT − Rk · kSh k · kxk which is less than 2εkSh k · kxk. Finally, we assume that T ∈ C is£invertible. Then by 5.5.4¤ T −1 ∈ t. Applying ¤ £ 5.6.2 and 5.5.10(b) to the identity T −1 , Sh = −T −1 T, Sh T −1 one can easily obtain that T −1 ∈ C. ¤ 5.6.4. The conjugate of an operator of the class C Proposition. Let L be Lpq or Cq , 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. If T ∈ C(L) then T 0 ∈ C(L0 ). Proof. We fix T ∈ C(L) and ε > 0, and choose R ∈ tf such that kT − Rk < ε. Then since kSh k ≤ 2c (see 1.6.12) we have °£ ¤ £ ¤° ° R, Sh − T, Sh ° < 2c+1 ε. From the definition of the natural direction it is easy to see that the operators Sh , h ∈ V , belong to tf in the following uniform sense: for any neighbourhood V of the zero of G and α ∈ ∆ there exists β ∈ ∆ such that Sh Lα ⊆ Lβ and Sh L\β ⊆ L\α for all h ∈ V simultaneously. By the definition of tf for an arbitrary neighbourhood V of the zero of G and α ∈ ∆ we choose β ∈ ∆ such RSh and Sh R map £ that¤ for all h ∈ V \βthe operators \α L\β into L\α .£ In this case R, S also maps L into L , and hence (see 1.2.5) h ¤ the operator R, Sh : Lβ → Lα is well defined, and consequently, by 1.2.9, the £ ¤ £ ¤0 operator R0 , Sh0 = − R, Sh maps (L0 )α into (L0 )β . For an arbitrary y ∈ (L0 )α we have °£ 0 0 ¤ ° °£ ¤ ° ° R , Sh y ° = ° R, Sh 0 y ° °£ ° ¤0 ≤ ° R, Sh : (L0 )α → (L0 )β ° · kyk °£ ° ¤ = ° R, Sh : Lβ → Lα ° · kyk. According to 5.5.7, we choose Lγ , γ ∈ ∆, such that |1 : Lγ → Lβ |− = 1. We consider the commutative (not exact) diagram j

q

Lγ −−−−→ L −−−−→  [R,S ] h y q

Lβ  [R,S ] h y

L −−−−→ Lα .

336

V. OPERATORS WITH VARYING COEFFICIENTS

By 1.3.7 we have ° °£ ° °£ ¤ ¤ ° R, Sh : Lβ → Lα ° = ° R, Sh : Lβ → Lα ° · |1 : Lγ → Lβ |− °£ ° ¤ ≤ ° R, Sh : Lγ → Lα ° °£ ° ¤ ≤ ° R, Sh : Lγ → L°. °£ ° ¤ Now we choose V in such a way that ° R, Sh : Lγ → L° ≤ ε for all h ∈ V . Finally, combining all these estimates, for all y ∈ (L0 )α and h ∈ V we obtain °£ 0 0 ¤ ° °£ 0 0 ¤ ° ° T , Sh y ° ≤ ° R , Sh y ° + 2c+1 εkyk °£ ° ¤ ≤ ° R, Sh : Lβ → Lα ° · kyk + 2c+1 εkyk ° °£ ¤ ≤ ° R, Sh : Lγ → L° · kyk + 2c+1 εkyk ≤ εkyk + 2c+1 εkyk, which means that T 0 ∈ C. ¤ 5.6.5. The isomorphisms C(Lp∞ ) ' C(Lp0 ) and C(C∞ ) ' C(C0 ) Proposition. Let E be an arbitrary Banach space. ¡ ¢ ¡ ¢ (a) The algebras C Lp∞ (G, E) and ¡ C Lp0 (G, ¢ E) are ¡ naturally ¢ isometrically isomorphic. The algebras C C∞ (G, E) and C C0 (G, E) are naturally isometrically isomorphic, too. (b) An operator T ∈ C(Lp∞ ) is invertible if and only if the corresponding operator T ∈ C(Lp0 ) is invertible. An operator T ∈ C(C∞ ) is invertible if and only if the corresponding operator T ∈ C(C0 ) is invertible. Proof. (a) By 5.5.5 we have the isomorphism of the classes t on the corresponding spaces. It remains to observe that the membership in C is the property of the restriction of T to Lp0 or C0 , respectively. (b) follows from (a) and 5.6.3, cf. 5.2.7. ¤ 5.6.6. An equivalent representation of a convolution. The following assertion is the generalization of 4.1.6(c) to the case of a vector-valued f . Proposition. Let E be a Banach space. Let f ∈ L1 (G, E), g ∈ L1 (G, C). Then Z ¡ ¢ f ∗g = Sh f g(h) dλ(h). Proof. The proof repeats the reasoning of that of 4.1.6.

¤

5.6.7. The invariant subspaces Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤ q < ∞ or q = 0, but q 6= ∞. ¡ ¢ (a) The subspace Cq (G, E) is invariant under operators T ∈ C L∞q (G, E) . (b) Let E be finite-dimensional. Then the subspace L1q0 (G, E0 ) ⊆ L∞q (G, E)¢0 ¡ is invariant under the conjugate T 0 of an operator T ∈ C L∞q (G, E) . And the subspace L1q0 (G, E0 ) ⊆ Cq (G,¡E)0 (see 1.8.12) is invariant under ¢ the conjugate T 0 of an operator T ∈ C Cq (G, E) .

337

5.6. OPERATORS WITH CONTINUOUS COEFFICIENTS

¡ ¢ Proof. (a) Let T ∈ C L∞q (G, E) , and let x ∈ Cq ⊆ L∞q . We recall from 1.6.12 that the function h 7→ Sh x, h ∈ G, is continuous in the norm of L∞q (recall that x ∈ Cq and q 6= ∞). Since q 6= ∞ the one point set {x} is Ψ-bounded. Therefore from 5.6.2 it follows easily that the function h 7→ Sh T x, h ∈ G, is continuous in the norm of L∞q , too. For an arbitrary compact neighbourhood V of the zero of G we consider the function Z 1 yV = (Sh T x) 1V (h) dλ(h), λ(V ) where 1V is the characteristic function of the neighbourhood V . Since the function h 7→ Sh T x is continuous we have that yV converges to T x when V shrinks to zero. We show that for locally almost all t ∈ G, Z ¡ ¢ 1 yV (t) = T x (t − h) 1V (h) dλ(h). λ(V ) Indeed, if one wants to prove this identity for t in a compact set K ⊆ G, one may change the function T x outside the set K − V . Then one can apply 5.6.6. From the last representation and 4.4.11(d) it follows that yV ∈ Cq . Since Cq is a closed subspace of L∞q we obtain that T x ∈ Cq . (b) We restrict ourselves to the case of L∞q . The proof for the case of Cq is a trivial modification of that for the case of L∞q . Clearly, it suffices to consider the case E = C. We break the argument into five steps. (i) For any compact neighbourhood V of the zero of G we consider the operator Z ¡ ¢ 1 NV x (t) = x(t − h) 1V (h) dλ(h) λ(V ) ¡ ¢ on L∞q of the convolution with ν = 1V /λ(V ) λ. We prove that the conjugate NV0 of NV maps (L∞q )0 into L1q0 . By 4.4.11 NV maps L∞q into Cq . Hence we can q

N0

V represent NV0 as the composition (L∞q )0 − → (L∞q )0 /(Cq )⊥ ' (Cq )0 −−→ (L∞q )0 (see 1.2.9). Clearly, it suffices to verify that the conjugate NV0 : (Cq )0 → (L∞q )0 of NV : L∞q → Cq maps (Cq )0 into L1q0 . First we show that NV0 maps (C0 )0 ⊆ (Cq )0 into L1 ⊆ L1q0 . We claim that on (C0 )0 = M the operator NV0 acts as the convolution with the transposed measure ν T , cf. 4.4.5. Indeed, for any µ ∈ M and x ∈ C0 , by the definition of the convolution of measures (see 4.1.3) we have

hx, NV0 µi = hNV x, µi Z ¡ ¢ = NV x (s) dµ(s) Z ¡ ¢ = NV x (−s) dµT (s) Z = x(−s) d(µT ∗ ν)(s) Z = x(s) d(µ ∗ ν T )(s) = hx, µ ∗ ν T i.

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V. OPERATORS WITH VARYING COEFFICIENTS

Consequently by 4.1.4 NV0 maps M into L1 . From 1.8.12(a) we see that (C0 )0 = M is dense in (Cq )0 = Mq0 . Hence NV maps (Cq )0 into the closure L1q0 of L1 . (ii) We take an arbitrary ξ ∈ L1q0 and show that ζ = T 0 ξ belongs to L1q0 , too. We recall from 5.5.2 that T 0 ∈ t. Therefore to show it in the case q 0 = ∞, by 5.5.5, it suffices to prove that ζ ∈ L10 provided ξ ∈ L10 . By (i) the measures ζV = NV0 ζ belong to L1q0 (to L10 if q 0 = ∞). Hence to complete the proof it suffices to show that ζV converges to ζ in norm. (iii) We observe that the function h 7→ Sh ξ is continuous in norm. Hence by 5.6.2 (cf. the proof of (a)) the function h 7→ Sh ζ is continuous in norm, too. As an auxiliary result, we prove that for x ∈ Cq Z 1 NV x = (Sh x) 1V (h) dλ(h). λ(V ) Indeed, since the function h 7→ Sh x is continuous in the norm of Cq , this integral exists. Applying the functional δt ∈ (Cq )0 to this integral we obtain Z D 1 Z E 1 (Sh x) 1V (h) dλ(h), δt = hSh x, δt i 1V (h) dλ(h) λ(V ) λ(V ) Z 1 x(t − h) 1V (h) dλ(h) = λ(V ) ¡ ¢ = NV x (t). (iv) Next, we prove that 1 ζV = λ(V )

Z (S−h ζ) 1V (h) dλ(h).

(1)

(This integral makes sense because the function h 7→ Sh ζ is continuous in norm.) Indeed, for all x ∈ Cq , by (iii) we have hx, ζV i = hx, NV0 ζi = hNV x, ζi D 1 Z E = (Sh x) 1V (h) dλ(h), ζ λ(V ) Z 1 = hSh x, ζi 1V (h) dλ(h) λ(V ) Z 1 = hx, S−h ζi 1V (h) dλ(h) λ(V ) Z D E 1 = x, S−h ζ 1V (h) dλ(h) . λ(V ) By 1.8.2 the same identity holds for all x ∈ L∞q . (v) Now from the representation (1) and the continuity of the function h 7→ Sh ζ (see (iii)) it is evident that ζV converges to ζ in norm when V shrinks to zero. It remains to recall (ii). ¤

5.6. OPERATORS WITH CONTINUOUS COEFFICIENTS

339

5.6.8. The main example: S(Cq ) ⊆ C(Cq ) Proposition. Let L be either Lpq or Cq , 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) An operator D ∈ S(L) belongs to Cu (L) if and only if it can be represented in the form ∞ X ¡ ¢ Dx (t) = am (t)x(t − hm ), (2) m=1

¡ ¢ P∞ where am ∈ C G, B(E) are uniformly continuous and m=1 kam kC < ∞. (b) An operator D ∈ S(L) belongs to ¡C(L) if and ¢ only Pif∞it can be represented in the form (2), where am ∈ C G, B(E) and m=1 kam kC < ∞. In particular, S(Cq ) ⊆ C(Cq ). Proof. (a) First, we assume that D has the form (2). We observe that ¡

∞ X ¢ Sh DS−h x (t) = am (t − h)x(t − hm ). m=1

From this formula it is easy to see that the operator D belongs to Cu provided am are uniformly continuous. Conversely, we assume that D ∈ S(L) belongs to Cu (L) and show that the functions am in the representation We represent the ¡ ¢ P∞(2) are uniformly continuous. operator D in the form D = m=1 Am Shm , where Am x (t) = am (t)x(t), and am ∈ L∞ (G,PB(E)) if L = Lpq , and am ∈ C(G, B(E)) if L = Cq , see 5.2.1. Clearly, ∞ Sh DS−h = m=1 (Sh Am S−h )Shm . We note that Sh Am S−h is the operator of the multiplication by the function t 7→ am (t − h). From 5.2.4(b) and the definition of Cu it follows that the operator (Sh Am S−h )Shm depends on h continuously for all m. Consequently the operator Sh Am S−h depends on h continuously, too. We ¡ ¢ recall from 5.1.1 that the norm of the operator of multiplication Ax (t) = a(t)x(t) is equal to kakL∞ on any space L. Hence, by what has been proved, the L∞ -norm of the function t 7→ am (t) − am (t − h) tends to zero as h → 0. If am ∈ C this means that am are uniformly continuous. Now assume that L = Lpq and am ∈ L∞ . From the continuity of the function h 7→ Sh am in the L∞ -norm it follows easily that Am belongs to Cu (Lpq ) for all p and q; in particular, Am ∈ Cu (L∞q ). By 5.6.7 this implies that Cq ⊆ L∞q is invariant under Am . Consequently, by 5.1.6, at x ∈ Cq the operator Am can be represented in the form ¡ ¢ Am x (t) = a ˜m (t)x(t), where a ˜m : G → B(E) is strongly continuous. Clearly, the restriction Am : Cq → Cq belongs to Cu (Cq ). Repeating the above reasoning we obtain that a ˜m is uniformly continuous in the norm To complete the proof it suffices to show that ¡ of B(E). ¢ in the representation Am x (t) = am (t)x(t) on Lpq one may change am to a ˜m . This follows from 1.8.2(b) and 5.1.1(c). Indeed, Am coincides with the operator ¡ ¢ em x (t) = a A ˜m (t)x(t) on the hL∞q , L1q0 i-dense subspace Cq . On the other hand, em are hL∞q , L1q0 i-continuous. Hence Am x = A em x for all x ∈ L∞q . both Am and A It remains to observe that L∞q is dense in Lpq in norm.

340

V. OPERATORS WITH VARYING COEFFICIENTS

(b) is proved in a similar way. We only note that one should, instead of 5.2.4(b), use the estimate kAm Shm : Lα → Lk ≤ kD : Lα → Lk; its proof is a word for word repetition of that of 5.2.4. ¤ 5.6.9. The isomorphism C(L∞q ) ' C(Cq ) Theorem. Let 1 ≤ q ≤ ∞ or q = 0, and let E be finite-dimensional. ¡ ¢ ¡ ¢ ¡ ¢ (a) The algebras C L∞q (G, E) and C Cq (G, E) , and C L1q0 (G, E0 ) are naturally isometrically isomorphic. (b) An operator T ∈ C(L∞q ) is invertible if and only if the corresponding operator T ∈ C(Cq ) is invertible. Proof. (a) We make use of 5.6.4, 5.6.5, and 5.6.7. If q 6= 0, 1, ∞ the proof follows from the diagram C(L∞q ) → C(Cq ) → C(Cq0 ) → C(L1q0 ) → C(L∞q ). If q = 0 the proof follows from the diagram C(L∞0 ) → C(C0 ) → C(C00 ) → C(L11 ) → C(L011 ) ' C(L∞∞ ) → C(L∞0 ). If q = 1 the proof follows from the diagram C(L∞1 ) → C(C1 ) → C(C10 ) → C(L1∞ ) → C(L10 ) → C(L∞1 ). If q = ∞ the proof follows from the diagram C(L∞∞ ) → C(L∞0 ) → C(C0 ) → C(C00 ) → C(L11 ) → C(L011 ) ' C(L∞∞ ) and the isomorphism C(C0 ) ' C(C). (b) follows from (a) and 5.6.3. ¤ Example. We denote by At = At (L) the intersection of A(L) and t(L) (by the example in 5.1.11, A \ t may be non-empty). Clearly, shift invariant operators has uniformly continuous coefficients. Hence At coincides with the intersection of A and C. Suppose E = C. By 4.1.2 and 1.8.5 At (C0 ) = A(C0 ) consists of all operators of convolution with bounded measures. From the last theorem it follows that At (C), At (L0 ), At (L∞ ), and At (L1 ) consist of all operators of convolution with bounded measures, too. Moreover, these are full subalgebras of B since they are defined as intersections of full subalgebras. 5.6.10. The invertibility of D ∈ S(Cq ). We recall from 5.2.1 (see also 5.1.1) that S(Cq ) has been defined to be the P∞algebra of all operators D ∈ B(Cq ) of the form (2) with am ∈ C(G, B(E)) and m=1 kam k < ∞. Clearly, this class does not depend on q. Moreover, these operators act on Lpq and S(Cq ) ⊆ S(Lpq ). Corollary. Let 1 ≤ p ≤ ∞, ¡and 1 ≤ ¢q ≤ ∞ or q = 0. Let E¡ be finite¢ dimensional. Then the algebra S Cq (G, E) is full in the algebra B Cq (G, E) . Moreover, if an operator D ∈ S(Cq ) is invertible on one of the spaces Cq or Lpq then it is invertible on all other spaces. Proof. By 5.6.9(b) and 5.2.10 the invertibility of D ∈ S(Cq ) does not depend on the space it is considered on. Assume D ∈ S(Cq ) is invertible on either Cq or on Lpq . Then by what has been proved it is invertible on L∞q . By 5.2.8(b) its inverse D−1 on L∞q belongs to S(L∞q ). On the other hand, by 5.6.3 D−1 belongs to C(L∞q ). Then by 5.6.8 D−1 on L∞q is a difference operator with continuous coefficients, i.e., D−1 is an operator of the class S(Cq ). Clearly, it acts and is the inverse of D on all spaces. ¤

5.6. OPERATORS WITH CONTINUOUS COEFFICIENTS

341

5.6.11. The equivalence of the lower norms on L∞q and Cq Proposition. Let 1 ≤ p ≤ ∞, and 1 ≤¡ q < ∞ or ¢q = 0, but q 6= ∞, and let E be finite-dimensional. Then for all T ∈ C L∞q (G, E) |T : L∞q → L∞q |+ ≤ |T : Cq → Cq |+ ≤ 2c |T : L∞q → L∞q |+ . Proof. The first inequality follows immediately from definitions. We prove the second one. Namely, we show that for an arbitrary r > 0 from the estimate |T : L∞q → L∞q |+ < r it follows that |T : Cq → Cq |+ < 2c r. So we assume that |T : L∞q → L∞q |+ < r. We choose x ∈ L∞q , kxk > 1, such that kT xk < r, and for each compact neighbourhood V of the zero of G consider the functions Z 1 xV (t) = x(t − h)1V (h) dλ(h), λ(V ) Z ¡ ¢ 1 T x (t − h)1V (h) dλ(h). (T x)V (t) = λ(V ) By 4.4.11(d) these functions belong to Cq . We show that kT xV − (T x)V k vanishes as V shrinks to zero. We recall from 5.6.7(b) that L1q0 (G, E0 ) is invariant under the operator T 0 . For an arbitrary y ∈ L1q0 (G, E0 ) we have hT xV , yi = hxV , T 0 yi Z DZ ¡ ¢ E 1 = x(t − h)1V (h) dλ(h), T 0 y (t) dλ(t) λ(V ) Z ³Z ­ ´ ¡ ¢ ® 1 = 1V (h) x(t − h), T 0 y (t) dλ(t) dλ(h) λ(V ) Z 1 1V (h)hSh x, T 0 yi dλ(h) = λ(V ) Z 1 = 1V (h)hT Sh x, yi dλ(h) λ(V ) (We do some comments of the usage of Fubini’s theorem in the preceding calculations. Since the beginning and the end of the formula do not depend on the choice of equivalent functions in the classes x ∈ L∞q and T 0 y ∈ L1q0 , we may assume that x and T 0 y are defined everywhere and, moreover, x is equal to­zero outside ¡ 0a σ-finite ¢ ® set. Under these assumptions the integrand (t, h) 7→ 1V (h) x(t − h), T y (t) is equal to zero outside a σ-finite set. Finally, by 4.1.5 it is measurable.) In a similar way Z DZ E ­ ® ¡ ¢ 1 (T x)V , y = T x (t − h)1V (h) dλ(h), y(t) dλ(t) λ(V ) Z ³Z ­¡ ¢ ´ ® 1 = T x (t − h), y(t) dλ(t) dλ(h) 1V (h) (3) λ(V ) Z 1 1V (h)hSh T x, yi dλ(h). = λ(V )

342

V. OPERATORS WITH VARYING COEFFICIENTS

Combining these two identities, for any y ∈ L1q0 (G, E0 ) we obtain 1 hT xV − (T x)V , yi = λ(V )

Z 1V (h)hT Sh x − Sh T x, yi dλ(h).

Next, since the one point set {x} is Ψ-bounded, by 5.6.2 we obtain lim kT xV − (T x)V k

V →0

= lim sup{ |hT xV − (T x)V , yi| : y ∈ L1q0 , kyk ≤ 1 } V →0 ¯ n¯ 1 Z o ¯ ¯ 0 = lim sup ¯ 1V (h)hT Sh x − Sh T x, yi dλ(h)¯ : y ∈ L1q , kyk ≤ 1 V →0 λ(V ) n 1 Z o 0 ≤ lim sup 1V (h)kT Sh x − Sh T xk · kyk dλ(h) : y ∈ L1q , kyk ≤ 1 V →0 λ(V ) ≤ lim sup { kT Sh x − Sh T xk} V →0 h∈V

= 0. On the other hand, since kSh k ≤ 2c , from the representation (3) and the assumption kT xk < r it is easy to see that k(T x)V k < 2c r. Consequently for sufficiently small V we have kT xV k < 2c r. Finally, we show that kxV k > 1 for sufficiently small V . We choose y ∈ L1q0 , kyk = 1, such that hx, yi > r (if q 0 = ∞ we can choose such a y in L10 ). Clearly, we have (cf. (3)) hxV , yi = = = =

Z 1 λ(V ) Z 1 λ(V ) Z 1 λ(V ) Z 1 λ(V )

DZ

E

x(t − h)1V (h) dλ(h), y(t) dλ(t) ³Z ­ ´ ® 1V (h) x(t − h), y(t) dλ(t) dλ(h) 1V (h)hSh x, yi dλ(h) 1V (h)hx, S−h yi dλ(h).

We recall from 1.6.12 that the function h 7→ S−h y is continuous in norm. Hence from this representation it follows that for sufficiently small V Z 1 kxV k ≥ RehxV , yi = Re 1V (h)hx, S−h yi dλ(h) > 1. λ(V ) Thus we have shown that there exists V such that kT xV k/kxV k < 2c r, which proves that |T : Cq → Cq |+ < 2c r. ¤

CHAPTER VI

DIFFERENTIAL DIFFERENCE EQUATIONS In this chapter we return to differential equations. Mainly we shall be concerned with differential difference equations and closed equations, which explains the title. The results of chapter 5 will be extensively employed. To save space we speak mostly about invertibility and causal invertibility of operators. We recall from chapter 3 that the causal invertibility is usually equivalent to input-output and exponential stability, and that the ordinary (but not causal) invertibility usually means rough instability and exponential dichotomy. It is worth noticing that ordinary invertibility possesses one interpretation which is more direct. Let us consider an ordinary differential equation x(t) ˙ + a(t)x(t) = f (t), where t runs over the whole of R. The problem of finding bounded solutions x of this equation is called the bounded solution problem. The bounded solution problem can be interpreted as a kind of boundary value problem with boundary conditions of the kind lim supt→−∞ |x(t)| < ∞ and lim supt→+∞ |x(t)| < ∞. The natural assumptions for its well posedness is the boundedness and continuity of a and f . Clearly, under these assumptions the boundedness and continuity of x imply these properties for x. ˙ Furthermore, it is easy to see that the operator Lx = x+ax ˙ 1 acts continuously from C to C and the existence of a unique bounded solution x for all f ∈ C is equivalent to the invertibility of L. Thus the bounded solution problem can be easily reformulated in operator terms. By misuse of language we also regard any equation Lx = f with L acting in one of our usual pairs (Cq1 , Cq ),

(Cq , Cq ),

(Lpq , Lpq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq )

as a bounded solution problem.

6.1. The local Fredholm property The main examples of compact operators are various kinds of integral operators. Unfortunately, integral operators on R (and on other non-compact sets) are usually non-compact. Nevertheless, they possess some properties of compact operators. For example, the restrictions (U −1 )a/b (see 2.1.1 for the definition) of the operator ¡ −1 ¢ R +∞ U f (t) = 0 e−s f (t − s) ds to bounded segments [a, b] are compact. We call operators of this kind locally compact. In this section we generalize some facts of the theory of compact and Fredholm operators to the local case.

343

344

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Difference operators D and some others possess the following property: if D is not invertible then it is so degenerate that its perturbations by compact operators can not make it invertible. More strictly, for such a D the invertibility of D is a necessary condition for the invertibility of the perturbed operator D +K, where K is compact. This property remains valid if the perturbation K is locally compact. The latter fact is the main result of this section, see 6.1.11. The discussion of applications is too long. We postpone it to the next section. 6.1.1. Locally compact operators. Let X and Y be Banach spaces with directions, see 5.5.1 for the definition. We say that an operator K ∈ B(X, Y ) is: (a) ∆+-compact if the operator K : X α → Y is compact for all α ∈ ∆; (b) ∆−-compact if the operator K : X → Yα is compact for all α ∈ ∆; (c) locally compact if it is both ∆+-compact and ∆−-compact. Example. Let K ∈ B(lq ) be restored by its matrix. Clearly, K is ∆+-compact if and only if each column {Kij : i ∈ I} of its matrix {Kij } (see 1.6.4 for the definition of a matrix) induces a compact operator acting from Ej to lq . And K is ∆−-compact if and only if each row {Kij : j ∈ J} of its matrix induces a compact operator acting from lq to Ei . We recall from 5.5.7 that a +-direction and a −-direction on X are called consistent if for any α ∈ ∆ there exists β ∈ ∆ such that |1 : X α → Xβ |+ = 1

and

|1 : X β → Xα |− = 1.

For example, the natural and standard directions on lq , Lpq , and Cq are consistent. Proposition. (a) Equivalent (see 5.5.1 for the definition) directions induce the same classes of ∆±-compact and locally compact operators. (b) The set of all ∆+-compact (∆−-compact, locally compact) operators K ∈ B(X, Y ) is a closed subspace of B(X, Y ). If K ∈ B is ∆+-compact then so is AK for all A ∈ B. If K ∈ B is ∆−-compact then so is KA for all A ∈ B. If K ∈ B is locally compact then so are both AK and KA for all A ∈ B. (c) An operator K ∈ B(X, Y ) is ∆±-compact if and only if the conjugate operator K 0 is ∆∓-compact. (d) Assume that directions on X are consistent and K ∈ t(X), see 5.5.2 or 1.6.6 for the definition of t. Then the following four assumptions are equivalent: (i) the operator K is ∆+-compact; (ii) the operator K is ∆−-compact; (iii) the operator K is locally compact; (iv) the operator K : X α → Xβ is compact for all α, β ∈ ∆. (e) Let L be lq (I, Ei ), Lpq (G, E), or Cq (G, E), and let K ∈ t(L). Then the following four assumptions are equivalent: (i) the operator K is ∆+-compact; (ii) the operator K is ∆−-compact; (iii) the operator K is locally compact; (iv) all entries Kij of the matrix of K are compact operators. (Here for L = Cq we consider the matrix induced by the isomorphism Cq ' lq (I, C♦i ), see 1.6.10.)

6.1. THE LOCAL FREDHOLM PROPERTY

345

Proof. (a) follows immediately from the definitions. (b) follows from the definitions and algebraic properties of compact operators. (c) follows from the definition of the directions on conjugate spaces (see 5.5.1) and 1.1.11. ¡ ¢ (d) Clearly, (i) & (ii) ⇔ (iii) ⇒ (iv). First, we show that (iv) ⇒ (ii). Namely, we show that for an arbitrary ε > 0 and α ∈ ∆ the image of the unit ball of the space X under the action of the operator K : X → Yα has a conditionally compact 2ε-net. We take R ∈ tf (X, Y ) such that kK −Rk < ε. By the definition of tf we choose β ∈ ∆ such that RX \β ⊆ Y \α , which means that the operator R : Xβ → Yα is well defined. It is easy to see that the images of the open unit balls under the action of operators R : X → Yα and R : Xβ → Yα are the same set. Consequently the images of the closed unit balls coincide, too. Next, according to the definition of consistent directions we choose γ ∈ ∆ such that |1 : X γ → Xβ |− = 1, i.e., the closed unit ball of X γ covers the closed unit ball of Xβ . Therefore the images of the closed unit balls under the action of operators R : Xβ → Yα and R : X γ → Yα are the same set. Thus we see that the images of the closed unit balls under the action of operators R : X → Yα and R : X γ → Yα are the same set, too. Since kK − R : X γ → Yα k ≤ kK − Rk < ε we may take for an ε-net for the image of the closed unit ball under the action of the operator R : X γ → Yα the image of the closed unit ball under the action of the operator K : X γ → Yα which is conditionally compact by assumption. By what has been proved, it is also an ε-net for the image of the closed unit ball under the action of the operator R : X → Yα . Since kK − Rk < ε it is a 2ε-net for the image of the closed unit ball under the action of the operator K : X → Yα as well. Finally, we show that (iv) ⇒ (i). We observe that the conjugate of the operator K : X α → Yβ is the operator K 0 : (Y 0 )β → (X 0 )α , see 1.2.9. Therefore the validity of the assumption (iv) for K implies its validity for K 0 . By what has been proved it follows that K 0 is ∆−-compact (we recall from 5.5.7 that the directions conjugate of consistent directions are consistent themselves). It remains to refer to (c). (e) follows immediately from (d). ¤ 6.1.2. The ideal k. The previous proposition shows that the theory of ∆±compact operators of the class t(L) is simpler than the general one. On the other hand, the class t covers all our applications. Therefore below we restrict ourselves to the consideration of locally compact operators of the class t. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let G be a locally compact abelian group and E be a Banach space. Let I be a set and Ei , i ∈ I, be Banach spaces. Let L be lq (I, Ei ), Lpq (G, E), or Cq (G, E). We denote by k = k(L) the subspace of t(L) consisting of all locally compact operators. By 6.1.1 k consists of all operators K whose matrix entries Kij are compact. And we denote by kf = kf (L) the intersection k ∩ tf ; thus kf consists of all operators induced by matrices of the class tf with compact entries.

346

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Proposition. (a) kf is dense in k. (b) kf is a two-sided ideal in tf . (c) k is a closed two-sided ideal in t. ¡ ¢ Proof. (a) We make use of 5.5.3. Let K ∈ k lq (I, Ei ) . For given ε > 0 we take R ∈ tf such that kK − Rk < ε. Next, we construct a representation of X = Y = lq in the form X = lq (Λ × N, X αδk ) and Y = lq (Λ × N, Y αδk ) such that the corresponding matrix of R is two-diagonal. Clearly, kKδi,δj − Rδi,δj k < ε. e δi,δj } as follows. We set K e δi,δj = Kδi,δj if Rδi,δj 6= 0, We define the matrix {K e δi,δj = 0 otherwise, and we denote by K e the operator induced by {K e δi,δj }. and K e ∈ kf and kK e − Kk ≤ kK e − Rk + kR − Kk ≤ K e − R + ε ≤ 3ε. Obviously, K (b) follows from 1.1.11. (c) follows from (a) and (b), and 6.1.1. ¤ 6.1.3. Locally Fredholm operators. Below we assume that ±-directions on X are consistent. We recall from 5.5.7 that the directions on lq , Cq , and Lpq possess this property. We say that an operator T ∈ t(X) is locally Fredholm if its natural projection to the quotient algebra t/k = t(X)/k(X) is invertible, cf. the example in 1.4.7. From the technical point of view it is more convenient to deal with the algebra of operators on the quotient space X # = X V /X K defined below than with the quotient algebra t/k. Let V be an infinite set; usually we shall use for V the set of all neighbourhoods of the zero of a non-discrete locally compact abelian group G. We denote by X V the space l∞ (V, X), i.e., the space of all bounded families Z = { zV ∈ X : V ∈ V } with the supremum norm. Clearly, any operator T ∈ B(X) induces the operator T V {zV } = {T zV } acting on X V with kT V k = kT k. We say that a subset Z ⊆ X is conditionally locally compact if for any α ∈ ∆ the natural projection of Z to Xα is conditionally compact, see 1.1.1 for the definition of a conditionally compact set. We denote by X K the subspace of X V consisting of all families Z ∈ X V which are conditionally locally compact as subsets of X. Proposition. (a) X K is a closed subspace of X V . (b) For any T ∈ t(X) the subspace X K is invariant under T V . (c) An operator K ∈ t(X) is locally compact if and only if Im K V ⊆ X K . Proof. (a) is obvious. (b) For T ∈ tf the assertion is evident. To consider the case of T ∈ t we observe that if the closed subspace X K is invariant under the family { T V : T ∈ tf } then it is invariant under its closure { T V : T ∈ t }, too. (c) We make use of 6.1.1. By definition an operator K is ∆−-compact if and only if it takes the unit ball into a conditionally locally compact set. Hence if K

6.1. THE LOCAL FREDHOLM PROPERTY

347

is locally compact it takes bounded families Z into conditionally locally compact ones, i.e., Im K V ⊆ X K . Conversely, we assume the contrary: let an operator K ∈ t(X) be not ∆−compact, i.e., there exists α ∈ ∆ such that the operator K : X → Xα is not compact. We recall (see 1.1.1) that a metric space is compact if and only if any sequence of its elements contains a Cauchy subsequence. Therefore there exists a bounded countable subset Z ⊆ X such that the image of Z under K : X → Xα is not conditionally compact. Clearly, the set Z can be indexed, i.e., there exists an indexed family Ze ∈ X V that coincides with Z as set. Obviously, K V Ze ∈ / X K. ¤ 6.1.4. The operator T # . We denote by X # the quotient space X V /X K and assign to each operator T ∈ t(X) the quotient operator T # : X # → X # ; this construction makes sense by 6.1.3(b). According to 6.1.3(c) this assignment induces an injective embedding of t/k in B(X # ). Proposition. If an operator T ∈ t(X) is locally Fredholm then the operator T is invertible. #

Proof. It suffices to observe that the natural embedding of t/k in B(X # ) is a morphism of algebras and refer to 1.4.6. ¤ 6.1.5. The conjugate of a locally compact operator. We observe that by 5.5.7 the induced directions on X 0 are consistent, too. Thus the above definitions and constructions make sense for operators acting on X 0 as well. Proposition. Assume K ∈ B(X). Then K ∈ k(X) if and only if K 0 ∈ k(X 0 ). Proof. The proof follows from 6.1.1(c, d) and 5.5.2(b). ¤ 6.1.6. The conjugate of a locally Fredholm operator Corollary. If an operator T ∈ t(X) is locally Fredholm then the operator T 0 ∈ t(X 0 ) is locally Fredholm, too. Proof. What T is locally Fredholm means that there exist operators R ∈ t(X) and K, K1 ∈ k(X) such that T R = 1 + K and RT = 1 + K1 . Consequently R0 T 0 = 1 + K 0 and T 0 R0 = 1 + K10 , which implies that T 0 is locally Fredholm (see 1.1.11). ¤ 6.1.7. The lower norms of T # Theorem. Assume T ∈ t(X) is locally Fredholm. Then kT # k+ > 0 and k(T 0 )# k+ > 0. Proof. The proof follows from 6.1.4, 1.3.2, and 6.1.6. ¤ 6.1.8. The lower norm on a subspace. Let G be a locally compact abelian group, and let L0 be the space Cq0 = Cq (G, E)0 or L0pq = Lpq (G, E)0 , 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ or q = 0. We denote by F ⊆ L0 the subspace L1q0 (G, E0 ) or Lp0 q0 (G, E0 ), respectively. We may consider on F two pairs of ±-directions. The first of them is the usual (say, natural, see example in 5.5.1 for the definition) directions on L1q0 and Lp0 q0 . On the other hand, we have ±-directions on L0 induced by directions on

348

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Cq and Lpq (for convenience we again take the natural directions on Cq and Lpq ). Clearly, the spaces F α = F ∩ (L0 )α and F \α = F ∩ (L0 )\α arising in the latter way coincide with the spaces from the first pair of directions. Proposition. Let E be finite-dimensional. Assume that the subspace F is invariant under an operator T ∈ t(L0 ). We denote by T0 : F → F the corresponding restriction. (It follows immediately from the definition in 5.5.2 that T0 ∈ t(F ).) Then |(T0 )# |+ ≥ |T # |+ . Proof. We consider the natural embedding J : F → L0 and the natural projector P : L0 → F , see 1.8.1, 1.8.11, and 1.8.12. By assumption, P J = 1F , kJk = 1, and kP k = 1. We consider the operators J V : F V → (L0 )V and P V : (L0 )V → F V . From the construction of P (see the proofs of 1.8.12 and 1.8.11) it is easy to see that P V (L0 )K ⊆ F K . Also it is evident that J V F K ⊆ (L0 )K . Thus the quotient operators J # : F # → (L0 )# and P # : (L0 )# → F # are well defined. (It could be simply noted that J and P are defined by main diagonal matrices; therefore they belong to tl and hence J # and P # are well defined. But formally our general construction T # : X # → Y # does not embrace the case X 6= Y .) It is straightforward to verify that P # J # = (P J)# = 1# . On the other hand, by 1.2.2 kJ # k ≤ kJk = 1 and kP # k ≤ kP k = 1. By 1.3.7 these imply that |J # |+ = 1. What T0 is the restriction of T to F means that T J = JT0 , which implies T # J # = J # (T0 )# . From 1.3.7 we have |J # |+ ·|T # |+ ≤ |T # J # |+ = |J # (T0 )# |+ ≤ kJ # k · |(T0 )# |+ , which in turn imply the estimate |(T0 )# |+ ≥ |T # |+ . ¤ 6.1.9. Rapidly oscillating characters Proposition. For any non-discrete locally compact abelian group G there exists I = 2, 3 . . . such that for any ε > 0, and h1 , h2 , . . . , hK ∈ G, and a neighbourhood√V of the zero of G there exists a point h0 ∈ V ∩ (−V ) such that for any value % of I 1 there exists a character χ ∈ X(G) such that |χ(hm ) − 1| < ε for m = 1, . . . , K

and

|χ(h0 ) − %| < ε.

Proof. Let H be a finitely generated subgroup of G. We say that the order of h ∈ G \ H (with respect to H) is equal to n if n is the smallest natural number such that nh ∈ H (by nh we mean h + · · · + h); we set n = ∞ if nh ∈ / H for all natural n. For any finitely generated subgroup H of G and a neighbourhood U of the zero of G we consider the set N(H, U ) of all n ≥ 2 such that there exists an element h ∈ U \ H of the order n. We observe that the set U \ H is not empty; indeed, by 1.6.2 λ(U ) > 0, but λ(H) = 0 because H is countable. Hence N(H, U ) 6= ∅ for all H and U . The two cases are possible. First: assume N(H, U ) contains arbitrary large numbers for all H and U . Then we set I = 2, and consider the subgroup H of

6.1. THE LOCAL FREDHOLM PROPERTY

349

¡ ¢ G generated by the set {h1 , . . . , hK }, and choose h0 ∈ V ∩ (−V ) \ H with a sufficiently large order n (for instance, it suffices to take n such that 2π/n < ε/2). e the subgroup generated by the family {h0 , h1 , . . . , hK }. We define We denote by H e as follows. We set κ(h) = 1 for h ∈ H, and define κ(h0 ) to the character κ √ of H n be the value of 1 the most close to %, where n is the order of hk+1 ; if n = ∞ we set simply κ(h0 ) = %. If n is large enough we have |κ(h0 ) − %| < ε/2. Clearly, κ e By Kronecker’s theorem there exists a possesses a continuation to a character of H. continuous character χ of G such that |χ(hm ) − κ(hm )| < ε/2 for m = 0, 1, . . . , K. Clearly, this is a desired character χ. We turn to the second case: assume that the set N(H, U ) is finite for at least one pair of H and U . We observe that H1 ⊇ H2 and U1 ⊆ U2 imply N(H1 , U1 ) ⊆ N(H2 , U2 ). Hence for any H1 and H2 , and U1 and U2 we have N(H1 +H2 , U1 ∩U2 ) ⊆ N(H1 , U1 ) ∩ N(H2 , U2 ). Since all sets N(H, U ) are not empty and at least one of them is finite it follows that the intersection of N(H, U ) over all H and U is not empty. We define I to be an element of this intersection. We consider the subgroup H of G generated by the set {h1 , . . . , hK }. For given e generated V we take h0 ∈ V ∩ (−V ) of the order I and consider the subgroup H e as follows. We set by the family {h0 , h1 , . . . , hK }. We define a character κ of H κ(h) = 1 for h ∈ H, and κ(h0 ) = %. Clearly, κ can be continued to a character e It remains to refer to the Kronecker theorem. ¤ of H. 6.1.10. An estimate of |D# |+ . Let G be a non-discrete locally compact abelian group. Let E be a Banach space. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by L = L(G, E) one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E). We recall from 5.2.9 that we denote by Dz = Dz (L) the closure in the norm of B(L) of the set of all operators D of the form D=

K X

Am Shm

(1)

m=1

with Am ∈ Xz (L). Clearly, Dz ⊆ t. Theorem. Let V (see 6.1.3) be the family of all neighbourhoods of the zero of the group G. For any D ∈ Dz (L) one has |D# |+ ≤ 8c |D|+ , where c is the constant from the representation G = Rc × V, see 1.6.1. Proof. We break the argument into three steps. (i) First we observe that by 1.3.5, without loss of generality we may assume that the operator D has the form (1). Moreover, without loss of generality we PK may assume that D = m=1 kAm Shm k ≤ 1 (this will simplify notation). We take an arbitrary ε > 0 and choose z ∈ L, kzk = 1, such that kDzk < |D|+ +ε. We would like to assume that z is supported in a compact set K. Indeed, if q 6= ∞ it follows from the subspace of compactly supported functions being dense

350

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

in L. For the space L = Lp∞ it follows from the equality |T : Lp∞ → Lp∞ |+ = |T : Lp0 → Lp0 |+ , see 5.5.5. Finally, let L = C. We consider the isomorphism Φ : C → l∞ (I, C♦i ) from 1.6.10 and recall that kΦk, kΦ−1 k ≤ 2c . Therefore by 5.5.5 and 1.3.7 we have |T : C0 → C0 |+ ≤ 4c |T : C∞ → C∞ |+ . Thus in all the cases without loss of generality we may assume that z is compactly supported and kDzk < 4c |D|+ + ε. (Actually one can show that the constant 4c may be changed to 1.) For the ε taken above, and the shifts h1 , h2 , . . . , hK in (1), and each neighbourhood V ∈ V of the zero of G√we define a character χ = χV according to 6.1.9. We suppose that the value % of I 1 in 6.1.9 is chosen in such a way that Re % ≤ 0. Note that in this case the radius of the smallest circle centred at zero and containing 1, %, and %¯ (¯ % is the complex conjugate of %) is equal to 1. We define the family Z = {zV } ∈ LV by the rule zV (t) = χV (t)z(t). We observe (cf. formula (6) in 5.2.4) that Ψ−1 χ DΨχ = Therefore by 6.1.9

PK m=1

χ(−hm )Am Shm .

K ° −1 ° X °Ψχ DΨχ − D° ≤ |χ(−hm ) − 1| · kAm Shm k V V m=1 K X

0 imply that |D# |+ ≤ 8c |D|+ . Thus the proof will be complete. By the definition of the norm on a quotient space we choose Y = {yV } ∈ LK such that kZ + Y k ≤ k(Z + LK )k + ε. Without loss of generality we may assume that the supports of the functions yV are contained in the same compact set K where the supports of the functions zV are. Then by the definition of LK the family Y forms a conditionally compact subset of L. Therefore again without loss of generality we may assume that the family Y consists of a finite number of distinct points. Next, we prove the estimate (2) for the cases of concrete spaces L. (ii) Assume L = Lpq , p 6= ∞. We choose a neighbourhood U of the zero of G such that (we use the strong continuity of Sh , see 1.6.12) kS±h z − zk < ε

for all h ∈ U

(3)

and kS±h yV − yV k < ε

for all h ∈ U and V ∈ V.

By the choice of Y we have kzV + yV k ≤ kZ + Y k ≤ k(Z + LK )k + ε

for all V ∈ V.

Thus we arrive at the estimate (we use the inequality kSh k ≤ 2c , see 1.6.12) kS±h zV + yV k ≤ kS±h (zV + yV )k + kS±h yV − yV k ≤ 2c (k(Z + LK )k + ε) + ε = 2c k(Z + LK )k + ε(2c + 1)

for all h ∈ U .

The two last estimates show that the three functions zV , Sh zV , and S−h zV are contained in a ball of the radius 2c k(Z + LK )k + ε(2c + 1) for all V ∈ V and h ∈ U . We estimate the radius of this ball from below. Let U be the neighbourhood chosen above. We take χ = χU and h0 ∈ U ∩ (−U ) in accordance with 6.1.9. Then from (3) we have kS−h0 zU − χ(h0 )zU k = kS−h0 Ψχ z − χ(h0 )Ψχ zk = kΨ−1 χ S−h0 Ψχ z − χ(h0 )zk = kχ(h0 )S−h0 z − χ(h0 )zk = kS−h0 z − zk < ε. In a similar way we have kSh0 zU − χ(−h0 )zU k < ε. We recall from (i) that by the choice of % the complex numbers 1, χ(h0 ), and χ(−h0 ) can not be located in a circle with radius less than 1 − ε. The estimates kS±h0 zU − χ(∓h0 )zU k < ε and

352

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

the equality kzU k = kzk = 1 show that the functions zU , Sh zU , and S−h zU can not be located in a circle with radius less than 1 − 2ε. Comparing this result with the above estimate of this radius we obtain the inequality 1 − 2ε ≤ 2c k(Z + LK )k + ε(2c + 1), which yields (2). (iii) Assume L = L∞q or L = Cq . We recall that Cq is embedded isometrically in L∞q . Thus essentially it suffices to consider the case L = L∞q . (We note that in the case L = Cq the following argument can be simplified.) We denote by J the family of all i ∈ I such that the restriction zi of z to Qi is not the zero function. Since the support K of z is compact, by 1.6.3 the set J is finite. Say, J consists of N points. For each i ∈ J we take an essential value ei ∈ E of the function zi such that |ei | > kzi kL∞ (Qi ) − ε/2N . Such an ei really exists. Indeed, we consider the set E consisting of all t ∈ Qi such that |zi (t)| > kzi kL∞ (Qi ) − ε/2N . Clearly, it has a positive measure. It suffices to take for ei an essential value of zi on E. Next, for any i ∈ J we choose a non-null measurable set Bi ⊆ Qi such that: the values of zi on Bi are contained in the ε/2N -neighbourhood of ei ; the values on any function yV on Bi are contained in a ball of radius ε/2N ; the set Bi is separated from the boundary of Qi , i.e., the closure of Bi is contained in the interior of Qi . We choose a neighbourhood U of the zero of G such that the sets Bi ± h are contained in Qi and the sets Ai = (Bi − h) ∩ Bi ∩ (Bi + h) are non-null for all h ∈ U . By the choice of Bi and Ai we L∞ (Qi ) − ε/N , and ° have ¡ k1Ai zi k ≥ kzi k¢° ° k1Ai (S±h zi − zi )kL∞ (Qi ) ≤ ε/N , and 1Ai S±h (yV )i − (yV )i °L (Q ) ≤ ε/N for ∞ i all h ∈ U . We set A = ∪i∈J Ai . Clearly, k1A zkL∞q (G) ≥ kzkL∞q (G) − ε = 1 − ε, and k1A (S±h z − z)kL∞q (G) ≤ ε

for all h ∈ U ,

(4)

and k1A (S±h yV − yV )kL∞q (G) ≤ ε

for all h ∈ U .

By the choice of Y we have k1A (zV + yV )k ≤ kzV + yV k ≤ k(Z + LK )k + ε

for all V ∈ V.

Using this estimate and the previous one we obtain (we recall that kSh k ≤ 2c , see 1.6.12) k1A (S±h zV + yV )k ≤ kS±h (zV + yV )k + k1A (S±h yV − yV )k ≤ 2c (k(Z + LK )k + ε) + ε = 2c k(Z + LK )k + ε(2c + 1)

for all h ∈ U .

The two last estimates show that the three functions 1A zV , 1A Sh zV , and 1A S−h zV are contained in a ball of the radius 2c k(Z + LK )k + ε(2c + 1) (centred at yV ) for all V ∈ V and h ∈ U .

6.1. THE LOCAL FREDHOLM PROPERTY

353

We estimate the radius of this ball from below. Let U be the neighbourhood chosen above. We take χ = χU and h0 ∈ U ∩ (−U ) in accordance with 6.1.9. Then from (4) we have ° ¡ ¢° ° ¡ ¢° °1A S−h zU − χ(h0 )zU ° = °1A S−h Ψχ z − χ(h0 )Ψχ z ° 0 0 ° ¡ ¢° ° = °1A Ψ−1 χ S−h0 Ψχ z − χ(h0 )z ° ¡ ¢° = °1A χ(h0 )S−h0 z − χ(h0 )z ° = k1A (S−h0 z − z)k < ε. ¡ ¢ In a similar way we have k1A Sh0 zU − χ(−h0 )zU k < ε. We recall from (i) that by the choice of % the complex numbers 1, χ(h0 ), and χ(−h ¡ 0 ) can not be located ¢ in a circle with radius less than 1 − ε. The estimates k1A S±h0 zU − χ(∓h0 )zU k < ε and the inequality k1A zU k = k1A zk ≥ 1−ε show that the functions 1A zU , 1A Sh zU , and 1A S−h zU can not be located in a circle with radius less than 1 − 3ε. Comparing this result with the above estimate of this radius we obtain the inequality 1 − 3ε ≤ 2c k(Z + LK )k + ε(2c + 1), which yields (2). ¤ 6.1.11. Difference operators can not be locally Fredholm. Let G be a non-discrete locally compact abelian group. Let E be a finite-dimensional Banach space. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by L = L(G, E) one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E). Theorem. If an operator D ∈ D(L) (see 5.2.9 for the definition of D) is locally Fredholm then in fact it is invertible. Remark. (a) We recall from 5.2.2 that since E is finite-dimensional Dz = D. (b) Note that the following proof employes implicitly the property of the subalgebra t being full, see, e.g., 6.1.3(b). Proof. We break the argument into five steps. (i) First we suppose that D has the form (1). From 6.1.7 we have |D# |+ > 0

and

|(D0 )# |+ > 0.

(ii) Assume L = Lpq , p, q 6= ∞. Then L0 = Lp0 q0 and 0

D =

K X

S−hm A0m .

m=1

Immediately from 6.1.10 we have |D|+ > 0 and |D0 |+ > 0. Hence by 1.3.2 the operator D is invertible. (iii) Assume L = Lpq , and p = ∞ or q = ∞. From 6.1.10 we have |D|+ > 0.

354

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

We consider the subspace Lp0 q0 ⊆ L0pq , see 1.8.1. It is easy to see that Lp0 q0 is invariant under the operator D0 : L0pq → L0pq . We denote by Dc : Lp0 q0 → Lp0 q0 the restriction of the operator D0 : L0pq → L0pq to Lp0 q0 . By 6.1.8 |(Dc )# |+ > 0. Hence by 6.1.10 |Dc |+ > 0. We consider four cases. Assume p = ∞ and q 6= 0, 1, or p 6= 1 and q = ∞. Then we observe that the conjugate of Dc is the operator D. Consequently, by 1.3.4 and what has been proved, |D|− > 0, which together with |D|+ > 0 implies the invertibility of D. Assume p = ∞ and q = 1. Then we consider the restriction D0 : L10 → L10 of the operator Dc : L1∞ → L1∞ to L10 , see 5.5.5. By 5.5.5 |Dc |+ > 0 implies |D0 |+ > 0. It remains to observe that the conjugate of D0 is the operator D and refer to 1.3.4 and 1.3.2. Assume p = ∞ and q = 0. Then we consider the extension D∞ : L∞∞ → L∞∞ of the operator D : L∞0 → L∞0 , see 5.5.5. By 5.5.5 the proved inequality |D|+ > 0 implies |D∞ |+ > 0. We observe that (Dc )0 = D∞ . Therefore by 1.3.4 |Dc |+ > 0 implies |D∞ |− > 0. Hence by 1.3.2 the operator D∞ is invertible. Consequently by 5.5.6 the operator D is invertible, too. Assume p = 1 and q = ∞. Then we consider the restriction D0 : L10 → L10 of the operator D : L1∞ → L1∞ , see 5.5.5. By 5.5.5 the inequality |D|+ > 0 proved above implies |D0 |+ > 0. We observe that (D0 )0 = Dc . Hence by 1.3.4 |Dc |− > 0. We recall that |Dc |+ > 0. Therefore by 1.3.2 the operator Dc is invertible. Consequently the operator D0 is invertible, too. Finally, by 5.5.6 the operator D is invertible as well. (iv) Assume L = Cq . By 6.1.7 and 6.1.10 we have |D|+ > 0 and |(D0 )# |+ > 0. Let us consider the extension D∞ : L∞q → L∞q of D : Cq → Cq (see 5.6.8 and 5.6.9), and the restriction Dc : L1q0 → L1q0 of D0 : Cq0 → Cq0 (see 5.6.7). By 5.6.11 |D|+ > 0 implies |D∞ |+ > 0. On the other hand, from 6.1.8 it follows that |(Dc )# |+ ≥ |(D0 )# |+ > 0, which by 6.1.10 implies |Dc |+ > 0. Above we have seen that the inequalities |D∞ |+ > 0 and |Dc |+ > 0 imply the invertibility of D∞ . It remains to recall (see 5.6.10) that D∞ and D are invertible simultaneously. (v) Now we assume that D ∈ D is arbitrary. From 6.1.7 and 6.1.10 we have |D|+ ≥ 8−c |D# |+ > ε and |(D0 )# |+ > ε for some ε > 0. We choose an operator e of the form (1) such that kD − Dk e < ε/2. Then from 1.3.5 we have D e # |+ ≥ |D# |+ − kD# − D e #k |D e > ε − kD − Dk > ε/2 > 0, e 0 )# |+ ≥ |(D0 )# |+ − k(D0 )# − (D e 0 )# k |(D e 0k > ε − kD0 − D > ε/2 > 0.

6.2. THE SOLUBILITY FOR DERIVATIVES

355

e is invertible. By what has been proved, from these inequalities it follows that D e < ε/2, by 1.3.5 we have |D| e + > ε/2. From the estimates |D|+ > ε and kD − Dk −1 e Hence by 1.3.2 kD k < 2/ε. Applying 1.3.6 we obtain that D is invertible, too. ¤

6.2. The solubility for derivatives In this section we discuss applications of the theorem 6.1.11. The main idea is described in the following simple statement. 6.2.1. Functional equations. Let E be a finite-dimensional Banach space. Let G be a non-discrete locally compact abelian group. Finally, let L = L(G, E) be one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Theorem. Let D ∈ D(L) (see 5.2.9) and K ∈ k(L) (see 6.1.2). Assume for any f ∈ L the equation (D + K)x = f has a unique solution x ∈ L, i.e., the operator D + K is invertible. Then the operator D is obligatorily invertible. Proof. Indeed, the operator D + K is locally Fredholm (see 6.1.3). Therefore it remains to apply 6.1.11. ¤ 6.2.2. The operator U −1 belongs to k. The following auxiliary statement will allow us to apply 6.2.1 to differential difference equations. Proposition. Let E be finite-dimensional. (a) Let c ∈ N and n : [0, 1]c ×[0, 1]c → B(E) be a continuous function. Then the ¡ ¢ c Fredholm integral operator N ∈ B Lp ([0, 1] , E) defined by the formula ¡

¢ N x (t) =

Z n(t, s)x(s) ds [0,1]c

is compact for all p. (b) Let n : [0, 1] × [0, 1] → B(E) be a continuous function. Then the Volterra ¡ ¢ integral operator N ∈ B Lp ([0, 1], E) defined by the formula ¡ ¢ N x (t) =

Z

t

n(t, s)x(s) ds 0

is compact for all p. (c) The operator (see 2.3.4 and 2.3.9) ¡ −1 ¢ U f (t) =

Z

+∞

e−s f (t − s) ds

0

¡ ¢ ¡ ¢ belongs to k Lpq (R, E) and k Cq (R, E) for all p and q.

356

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Proof. (a) Clearly, kN : Lp → Lp k ≤ knkC . We observe that if n is a polynomial then N is a finite-dimensional operator, i.e., its image is finite-dimensional. Finite-dimensional operators are trivially compact. By the Stone–Weierstrass theorem the kernel n can be approximated by polynomials. Since the set of all compact operators is closed (see 1.1.11) it follows that N is compact. (b) Let αδ : [0, 1] × [0, 1] → [0, 1], δ > 0, be a continuous function such that αδ (t, s) = 0 for s ≥ t, and αδ (t, s) = 1 for s ≤ t − δ. We consider the operator Z t ¡ ¢ Nδ x (t) = αδ (t, s)n(t, s)x(s) ds 0

Z

1

αδ (t, s)n(t, s)x(s) ds.

= 0

From 1.5.12(c) it follows immediately that kN − Nδ : Lp → Lp k → 0 for all p. We observe that the operator Nδ can be considered as a Fredholm integral operator with a continuous kernel. Thus one may refer¡ to (a). ¢ (c) ¡From 5.3.9¢ it follows that¢ U ∈ M∞ Lpq (R, E) . Consequently we have ¡ U ∈ e Lpq (R, E) ⊆ t Lpq (R, E) . We recall from the proof of 2.3.4 the explicit formulae for the matrix entries (U −1 )ij ∈ B(Lp [0, 1]) of U : Lpq → Lpq . We have Z 1 ³¡ ¢ ´ −1 −(i−j) −t U f (t) = e e es f (s) ds for j < i, ij 0 Z t ³¡ ¢ ´ −1 −t U f (t) = e es f (s) ds for j = i, ij 0 ¡ −1 ¢ U =0 for j > i ij We observe that the operators (U −1 )ij are compact. Indeed, for j > i this fact is trivial, since (U −1 )ij is one-dimensional, namely, its image is the span of the function t 7→ e−t . The case j = i follows from (b). It remains to refer to 6.1.1(e). The case of Cq follows from the case of L∞q by the definition of a ∆−-compact operator, see 6.1.1. ¤ 6.2.3. Bounded solution problem. The equations ∞ X

am (t)x(t ˙ − hm ) +

m=1 ∞ X

∞ X

bm (t)x(t m=1 ∞ X

− hm ) = f (t),

d am (t)x(t − hm ) + bm (t)x(t − hm ) = f (t) dt m=1 m=1 are called differential difference equations. In this section we usually write them in the forms Dx˙ + Bx = f (1) and

d Dx + Bx = f, (2) dt respectively. These are equations of neutral type. Earlier we have considered similar equations in 2.5.9, 2.5.10, 2.6.3, 2.6.7, and 4.5.9.

6.2. THE SOLUBILITY FOR DERIVATIVES

357

Theorem. Let E be a finite-dimensional Banach space. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let L be either Lpq = Lpq (R, E) or Cq = Cq (R, E). Let D ∈ D(L) and B ∈ t(L), see 5.2.9 and 5.5.2 for the definitions of D and t, respectively. (a) Assume that for any f ∈ Lpq (respectively, f ∈ Cq ) equation (1) has a 1 unique solution x ∈ Wpq (x ∈ Cq1 ), in other words, the operator L = d d 1 D dt + B : Wpq → Lpq (L = D dt + B : Cq1 → Cq ) is invertible. Then the operator D : Lpq → Lpq (D : Cq → Cq ) is obligatorily invertible. −1 (b) Assume that for any f ∈ Wpq (respectively, f ∈ Cq−1 ) equation (2) has a unique solution x ∈ Lpq (x ∈ Cq ), in other words, the operator L = d d −1 −1 dt D + B : Lpq → Wpq (L = dt D + B : Cq → Cq ) is invertible. Then the operator D : Lpq → Lpq (D : Cq → Cq ) is obligatorily invertible. Proof. (a) By 2.3.4 the operator L is invertible if and only if the operator T = LU −1 : L → L is invertible. It is easy to see that T = D + K, where K = (B − D)U −1 . By 6.2.2 and 6.1.2 K ∈ k. It remains to refer to 6.2.1. (b) By 2.3.9 the invertibility of L is equivalent to the invertibility of T = U −1 L : L → L. Clearly, T = D + K, where K = U −1 (B − D). By 6.2.2 and 6.1.2 K ∈ k. It remains to refer to 6.2.1. ¤ Remark. (a) We do not state that if D is invertible then L is invertible, too. The theorem only states that the invertibility of D is a necessary condition for the invertibility of L. If D is invertible the equation of neutral type (1) is equivalent to the equation x˙ + D−1 Bx = D−1 f,

(3)

which, generally speaking, is simpler than (1). Nevertheless, to pass from (1) to (3) one should invert the operator D first. Therefore assertion (a) of the theorem means that (1) is uniquely soluble on R if and only if both the equation Dy = g

(4)

and equation (3) are uniquely soluble on R. In a similar way, if D is invertible then performing the change of variables Dx = z in (2) we arrive at the equation z˙ + BD−1 z = f,

(5)

which, generally speaking, is simpler than the initial equation (2). Thus assertion (b) of the theorem means that (3) is uniquely soluble on R if and only if both equation (4) and equation (5) are uniquely soluble on R. (b) We recall from 5.2.9 and 5.6.10 that the inverse of D ∈ D also belongs to D. In applications operators D and B usually have bounded memory, e.g., D ∈ Df (see 5.2.9) and B ∈ tf (see 5.5.2). As a rule, the inverse of an operator with bounded ¡ ¢ memory has unbounded memory. For instance, the inverse ¡ ¢of the operator Dx (t) = x(t)− 21 x(t−1), i.e., D = 1−S1 /2, is the operator D−1 x (t) = P∞ −k x(t−k). In other words, subalgebras of operators with bounded memory k=0 2 usually are not full. This is one of the main reasons for the investigation of full subalgebras of operators with infinite memory.

358

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

6.2.4. Periodic solution problem. Let 1 ≤ p ≤ ∞. We denote by Lp per = Lp per (R, E) the subspace of Lp∞ (R, E) consisting of all 1-periodic functions and by Cper = Cper (R, E) the subspace of C(R, E) consisting of all 1-periodic functions. In a similar way we define the spaces Wp1 per = Wp1 per (R, E), Wp−1per = Wp−1per (R, E), 1 1 −1 −1 Cper = Cper (R, E), and Cper = Cper (R, E). Clearly, all these spaces are Banach. ¡ ¢ We observe that the mapping Φz (t) = z(e2πit ) establishes isometric isomorphisms from Lp (U, E) and C(U, E) onto Lp per (R, E) and Cper (R, E), respectively. (We do not discuss the spaces C 1 (U, E) and so on for technical reasons.)¡ It ¢is easy to see that according to this isomorphism the operator of multiplication Az (u) = a(u)z(u) on and C(U, E) corresponds to the operator of multiplication ¡ ¢ Lp (U, E) −1 2πit ΦAΦ x (t) = a(e )x(t) by a 1-periodic t 7→ a(e2πit ) on Lp per (R, E) ¡ coefficient ¢ and Cper (R, E), respectively. The formula ΦAΦ−1 x (t) = a(e2πit )x(t) also defines the operator ΦAΦ−1 on Lp∞ (R, E) and C(R, E), respectively. Clearly, on both these spaces it has the same norm. In a similar way the shift operator Sh on Lp (U, E) and C(U, E) corresponds to the shift operator on Lp per (R, E) and Cper (R, E), respectively. This correspondence is isometric. Nevertheless, kSh : Lp∞ → Lp∞ k ≤ 2 (see 1.6.12) whereas kSh : Lp per → Lp per k = 1 and kS¡h : Cper → C¢per k = ¡1. ¢ We define the subalgebra Df Lp per (R, E) of B Lp per (R, E) to be the set PK of all operators D of the form D = m=1 Am Shm , where Am are operators of multiplication by 1-periodic coefficients, cf. 5.2.9. By the above, the ¢algebra ¡ ¢ ¡ Df Lp per (R, E) is isometrically isomorphic to¢ the algebra Df L¢p (U, E) . Note ¡ ¡ that the natural embedding ¡ of Df Lp¢per (R, E) in Df Lp∞ (R, E) ¡ is not isomet¢ ric. We define D = D L (R, E) to be the closure of D L (R, E) p per f p per ¡ ¢ ¡ ¢ in norm. Evidently, ¡ D Lp per¢(R, E) is isometrically isomorphic to D Lp (U, E) . We define D = D ¡ ¢ Cper (R, E) similarly. Arguing as¡ above, ¢one can easily show that D Cper (R, E) is isometrically isomorphic to D C(U, E) . Theorem. Let 1 ≤ p ≤ ∞, and let E be a finite-dimensional Banach space. Let L be either Lp per = Lp per (R, E) or Cper = Cper (R, E). Let D ∈ D(L) and B ∈ B(L). (a) Assume that for any f ∈ Lp per (respectively, f ∈ Cper ) equation (1) has a unique solution x ∈ Wp1 per (x ∈ C 1per ), in other words, the operator d d 1 L = D dt + B : Wp1 per → Lp per (L = D dt + B : Cper → Cper ) is invertible. Then the operator D : Lp per → Lp per (D : Cper → Cper ) is obligatorily invertible. −1 (b) Assume that for any f ∈ Wp−1per (respectively, f ∈ Cper ) equation (2) has a unique solution x ∈ Lp per (x ∈ Cper ), in other words, the operator d d −1 L = dt D + B : Lp per → Wp−1per (L = dt D + B : Cper → Cper ) is invertible. Then the operator D : Lp per → Lp per (D : Cper → Cper ) is obligatorily invertible. Proof. The proof is similar to that of 6.2.3. We discuss, e.g., case (a). We 1 consider the operator U x = x˙ + x, see 2.3.4. Clearly, U maps Wp1 per ⊆ Wp∞ into 1 1 Lp per ⊆ Lp∞ and Cper ⊆ C into Cper ⊆ C. Next, from the explicit formula for

6.2. THE SOLUBILITY FOR DERIVATIVES

359

1 U −1 (see 2.3.4 or 6.2.2) it is clear that U −1 maps Lp per ⊆ Lp∞ into Wp1 per ⊆ Wp∞ 1 and Cper ⊆ C into Cper ⊆ C 1 . Thus U is invertible in the pairs (Wp1 per , Lp per ) 1 and (Cper , Cper ). It is easy to see that in these pairs U −1 can be represented as

Z 2π 1 e−s f (t − s) ds 1 − e−1 0 Z t 1 = e−(t−s) f (s) ds. 1 − e−1 t−2π

¡ −1 ¢ U f (t) =

Thus U −1 is a Fredholm operator with a continuous kernel considered as acting from Lp per to Lp per and from Cper to Cper (strictly speaking, to state this one should represent Lp per and Cper as the spaces of functions on U or on [0, 1]). Therefore U −1 is compact as acting from Lp per to Lp per and from Cper to Cper , cf. the proof of 6.2.2. Clearly, the operator L is invertible if and only if the operator T = LU −1 is invertible. It is easy to see that T = D + K, where K = (B − D)U −1 . By 1.1.11 K ∈ K. We now represent L as L(U, E) and refer to 6.2.1. ¤ 6.2.5. Elliptic equations with shifts. Theorem 6.2.3 possesses a generalization to partial differential equations. In this subsection we describe only an idea of such a generalization because the problem is vast and beyond the general scope of this book. For simplicity we restrict ourselves to the consideration of equations with the Laplace operator ∆ in W22 (Rn , C). We fix n ∈ N. We denote by W22 = W22 (Rn , C) the set of all functions x ∈ ∂ α1 +···+αn L2 (Rn , C) such that ∂t α1 αn x belongs to L2 for all αk = 0, 1, 2; α1 +· · ·+αn ≤ 2. 1 ...∂t1 Here, by the derivative we mean the distribution derivative, cf. 2.3.5 and 6.4.3 below. The space W22 endowed with the norm X

kxk =

α1 +···+αn ≤2

° ∂ α1 +···+αn ° ° ° ° α1 αn x° ∂t1 . . . ∂t1 L2

is Banach. It is called the Sobolev space. See, e.g., [BerL, §6.2], [LiM, ch. 1, §1], or [Ste1 , ch. 5, §2] for the detailed definition and discussion of W22 . Clearly, the Laplace operator ∆=

∂2 ∂2 ∂2 + + · · · + ∂t21 ∂t22 ∂t2n

acts continuously from W22 to L2 . Theorem. Let L2 = L2 (Rn , C), and let D ∈ D(L2 ) and B ∈ t(L2 ). Assume that for any f ∈ L2 the equation −D∆x + Bx = f

360

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

has a unique solution x ∈ W22 , i.e., the operator L = −D∆ + B : W22 → L2 is invertible. Then the operator D : L2 → L2 is obligatorily invertible. Sketch of proof. We break the argument into three steps. (i) We consider the operator U = 1 − ∆ : W22 → L2 . First we recall that U is invertible. This is a well known fact of the theory the Sobolev spaces (see, e.g., [LiM, ch. 1, theorem 1.2]), its proof can be obtained easily from the observation that the operator F−1 U¡F, where¢ F : L2 → L2 is the Fourier transform, is the multiplication operator F−1 U Fy (ω) = (1 + ω 2 )y(ω). Next, we show that Z ¡ −1 ¢ U f (t) = En (s)f (t − s) ds, Rn

where 1 E1 (t) = − e−|t| , 2 1 E2 (t) = K0 (|t|), 2π 1 −|t| E3 (t) = e , 4π|t| 1 Kn/2−1 (|t|), En (t) = (2π)n/2 |t|n/2−1

n ≥ 4,

here Kν is the Hankel function with imaginary argument (see, e.g., [Tikh, Appendix 2, Part 1, §3, 3] for a definition and discussion) and |t| is the l2 -norm of t ∈ Rn . We p recall (see [Tikh, Appendix 2, Part 1, §3, 3]) that for t → +∞ one has Kν (t) ∼ π/2te−t ; and for t → 0 one has K0 (t) ∼ − ln t and Kν (t) ∼ t−ν for ν 6= 0. Thus the functions En belong to L1 ; see also [Ste1 , ch. 5, §3, proposition 2] for the direct proof of this fact. Hence by 4.4.11 the convolution with En defines at least the operator acting from L2 to L2 . We recall from [Vla3 , problem 11.10] that En is a solution (in the sense of distributions) of the equation U En = δ. It is known (see, e.g., [Vla1 , ch. 3, §11, 3]) that for any f ∈ C00 the distribution x = En ∗ f is a solution of the equation U x = f . Thus the operator f 7→ En ∗ f coincides with the operator U −1 on C00 . Since C00 is dense in L2 , these operators coincide on the whole of L2 . In particular, f ∈¢ L2 . it follows that En ∗ f ∈ W22 provided ¡ (ii) We show that U −1 ∈ k L2 (Rn , E) . We recall from 4.4.11 that the norm of the operator G : L2 → L2 of the convolution with a function g ∈ L1 is less than or equal to kgkL1 . By 6.2.2(a) the operator of the convolution with a function g ∈ C00 belongs to k. But by definition C00 is dense in L1 . (iii) We consider the operator T = LU −1 . Clearly, T and L are invertible simultaneously. Evidently, LU −1 = (−D∆ + B)U −1 ¡ ¢ = D(1 − ∆) + B − D U −1 = D + (B − D)U −1 .

361

6.2. THE SOLUBILITY FOR DERIVATIVES

It remains to observe that by 6.1.2 (B − D)U −1 ∈ k and refer to 6.2.1.

¤

6.2.6. The representation of D ∈ D as a series. Let G be an arbitrary locally compact abelian group and E be an arbitrary Banach space. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let L be either Lpq = Lpq (G, E) or Cq = Cq (G, E). Proposition. One can associate with each D ∈ D (respectively, D ∈ Dz or D ∈ DΨ ; see 5.2.9 for the definitions) the formal series X

Ah Sh ,

(8)

h∈G

with Ah ∈ X (Ah ∈ Xz or Ah ∈ XΨ , respectively), in such a way that the following properties hold. Series (8) contains at most a countable number of non-zero P summands and kAh Sh k ≤ kDk for all h ∈ G. The correspondence D 7→ h∈G Ah Sh is linear (clearly, the set of all formal sums can be considered naturally as a linear space). If D ∈ S (respectively, D P ∈ Sz or D ∈ SΨ ) the series converges to D in norm. The correspondence D 7→ h∈G Ah Sh with these properties is determined uniquely. Remark. We do not state that, in general, series (8) converges to D in norm. Similarly, one can associate with a periodic function its Fourier series, but the latter may be not convergent in a suitable sense. Proof. For definiteness we consider P the case of D. We recall from 5.2.4 that for any D ∈ S of the form D = h∈G Ah S Ph we have kAh Sh k ≤ kDk for all h ∈ G. Thus the correspondence F ¢: D 7→ h∈G Ah Sh can be considered as a ¡ linear operator from S to l∞ G, B(L) . Moreover, if one endows S with the norm of B(L) one has kFk ≤ 1. Thus F possesses a unique extension to the closure D of S preserving the property kAh Sh k ≤ kDk for h ∈ G. ¤ 6.2.7. Input–output stability and causal invertibility for L ∈ t. In this subsection we return to the discussion of 3.3.6 and 3.3.8. We could not do it earlier because we had not known some special properties of the class t. Let G = R and E be an arbitrary Banach space. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let (Xq , Yq ) be one of the following pairs: (Cq , Cq ),

(Lpq , Lpq ),

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 ). (Lpq , Wpq

Theorem. Let L ∈ t(Xq , Yq ) be a causal operator, see 1.6.6, 5.5.2, and 5.5.11 for the definition of t. Let the equation Lx = f

(9)

be locally soluble, see 3.3.1 for the definition. Let τ ∈ R be a fixed point and H be (−∞, +∞), (−∞, τ ], or [τ, +∞). Equation (9) is input–output stable on H in the pair (Xq , Yq ) if and only if the operator T : Xq → Yq is causally invertible on H.

362

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Example. (a) Let L be either Lpq (R, E) or Cq (R, E), and let D, B ∈ t(L). d d Then the operators D dt + B and dt D + B belong to t, too. This fact has been mentioned actually in 5.5.11. (b) We also recall that D(L) ⊆ d(L) ⊆ t(L), see 5.2.11. Proof. The case q 6= 0, ∞ has been considered in 3.3.6. The case q = 0 has been considered in 3.3.8. The case q = ∞ and H = [τ, +∞) has been considered in 3.3.6. Thus it remains to consider the case q = ∞, and H = (−∞, +∞) or H = (−∞, τ ]. Assume H = (−∞, +∞). First we consider the case X = Y . By 5.5.5 and 5.5.6 the operator L acts simultaneously in (X0 , Y0 ) and (X∞ , Y∞ ) and is simultaneously invertible. We show that the causal invertibility holds simultaneously, too. Indeed, we recall from 1.6.6 that an operator of the class t is restored by its matrix. Consequently if L−1 exists in one pair and is causal then the matrix of L−1 induces the operator in another pair; evidently, this operator is causal and coincides with the inverse of L in that pair. It remains to observe that by 3.3.8 the causal invertibility in (X0 , Y0 ) is equivalent to input–output stability. Now let H = (−∞, +∞), but X 6= Y . To repeat the previous reasoning we need the following fact: the operator L acts and is causally invertible both in (X0 , Y0 ) and (X∞ , Y∞ ). It can be obtained easily from the consideration of the operator T = LU −1 or T = U −1 L. Assume H = (−∞, τ ]. Assume, e.g., that X 6= Y . We consider the operator Lτ = (1 − Pτ )L(1 − Pτ ) + Pτ U Pτ , where Pτ is defined as in 3.1.1 and U x = x˙ + x (if X = Y one should change U to 1). Clearly, the operators L and Lτ coincide on (−∞, τ ]. Hence on (−∞, τ ] they are simultaneously causally invertible and the corresponding equations are simultaneously input–output stable both in (X0 , Y0 ) and (X∞ , Y∞ ). On the other hand, on [τ, +∞) the operator Lτ = U (Lτ = 1) is obviously causally invertible and the equation Lτ x = f is input–output stable either in (X0 , Y0 ) and (X∞ , Y∞ ). These observations allow one to reduce the case H = (−∞, τ ] to the case H = (−∞, +∞). ¤ 6.2.8. Stability. We return to the case G = R. The space E is assumed to be finite-dimensional. Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0, and let L be either Lpq = Lpq (R, E) or Cq = Cq (R, E). Let D ∈ D(L) and B ∈ t(L) be causal operators. Let τ ∈ R and H be (−∞, +∞), (−∞, τ ], or [τ, +∞). (a) Assume equation (1) is locally soluble and input–output stable on H in 1 the pair (Wpq , Lpq ) (in the pair (Cq1 , Cq ), respectively). Then the operator D : Lpq → Lpq (respectively, D : Cq → Cq ) is obligatorily causally invertible on H. (b) Assume equation (2) is locally soluble and input–output stable on H in −1 the pair (Lpq , Wpq ) (in the pair (Cq , Cq−1 ), respectively). Then the operator D : Lpq → Lpq (respectively, D : Cq → Cq ) is obligatorily causally invertible on H. Proof. We consider the four cases separately.

363

6.2. THE SOLUBILITY FOR DERIVATIVES

1 (i) We begin with the case of the pair (Wpq , Lpq ). First, we show that without loss of generality we may assume that H = (−∞, +∞). Indeed, assume, e.g., H = [τ, +∞). We consider the projector Pτ on Lpq defined by the rule (see 3.1.1)

¡

¢ Pτ x (t) =

½

x(t)

for t ≥ τ ,

0

for t < τ .

d d Let L = D dt +B and U = dt +1. We consider the operator Lτ = Pτ L+(1−Pτ )U . d It is easy to see that Lτ = Dτ dt + Bτ , where Dτ = Pτ D + (1 − Pτ )1 and Bτ = Pτ B + (1 − Pτ )1. By assumption the equation Lτ x = f is locally soluble and input–output stable on [τ, +∞). It is also trivially locally soluble and input– output stable on (−∞, τ ]. Therefore by 3.3.5 it is locally soluble and input–output stable on (−∞, +∞). We shall show that this implies the causal invertibility of the operator Dτ on (−∞, +∞). Then in turn it will imply that D is causally invertible on H. Thus the case H = [τ, +∞) is reduced to the case H = (−∞, +∞). So we assume that H = (−∞, +∞). From 6.2.7 it follows that the stability of the equation (1) implies the causal invertibility of L. Therefore by 6.2.3 the operator D is invertible. From 2.5.9 and 2.5.10 there follows the causal invertibility of D on all finite segments [a, b]. By 2.1.5, to prove that D is causally invertible on (−∞, +∞) it suffices to prove that D is (ordinary) invertible, say, on [τ, +∞). We again consider the operators Lτ and Dτ . Repeating the above reasoning we obtain that Dτ is invertible on (−∞, +∞). But Dτ is evidently invertible on (−∞, τ ]. Hence by 2.1.4 it is invertible on [τ, +∞), too. But on [τ, +∞) the operator Dτ coincides with D. (ii) We turn to the case of the pair (Cq1 , Cq ). First we show that statements 2.6.2 and 2.6.3 remain valid for D ∈ D, i.e., equation (1) is locally soluble (by 2.5.9 it is equivalent to the causal invertibility of the operator D on all finite segments [a, b]) if and only if the coefficient a0 (t) (which corresponds to the operator D by 6.2.6) is invertible at all points t ∈ R. Clearly (see 2.1.7), it suffices to prove that D is causally invertible on [n, n + 1], n ∈ Z, if and only if the coefficient a0 (t) is invertible at all points t ∈ [n, n + 1]. Assume Dk ∈ Df converges operator D P to D in norm. We associate with the k the formal series D ∼ A0 + h6=0 Ah Sh (see 6.2.6) and represent D in the form PM Dk = Ak0 + m=1 Akm Shkm , where Akm are multiplication operators and hkm 6= 0 (we do not state that Dk are causal). By 6.2.6 Ak0 converges to A0 in norm. Hence PM Dk − Ak0 = m=1 Akm Shkm converges to D − A0 in norm, too. Consequently we may claim that the sequence (Dk − Ak0 )n/n+1 a fortiori converges to (D − A0 )n/n+1 in norm for any n ∈ Z. We observe that the space (Cq )n/n+1 is isomorphic to the space Cn/n+1 , see the second example in 2.1.1. So we shall consider (Cq )n/n+1 with the norm of the space Cn/n+1 . We show that

°³ X ´ ° Akm Shkm − (D − A0 ) ° hk m >0

n/n+1

° °³ X ´ ° ° Akm Shkm − (D − A0 ) °≤° hk m 6=0

n/n+1

° ° °.

364

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Indeed, for an arbitrary ε > 0 we choose x ∈ Cn/n+1 , kxk ≤ 1, and t ∈ (n, n + 1] such that ° °³³ X ´ ´ ° °³ X ´ ° ° ° ° k k Am Shkm −(D−A0 ) x (t)° ≥ ° Am Shkm −(D−A0 ) ° °−ε. hk m >0

n/n+1

hk m >0

n/n+1

By Urysohn’s theorem we change x to a function x ˜ ∈ Cn/n+1 , k˜ xk ≤ kxk ≤ 1, k such that x ˜(s) = x(s) if s ≤ t, and x ˜(s) = 0 if s ≥ − max{ hm : hkm < 0 }. Then we have °³³ X ´ ° ´ ° ° Akm Shkm − (D − A0 ) x ˜ (t)° ° hk m 6=0

n/n+1

°³³ X ´ ° Akm Shkm − (D − A0 ) =° hk m >0

n/n+1

which implies the inequality required. Now it is easy to see that °³ X ° °³ X ´ ´ ° ° ° Akm Shkm − (D − A0 ) Akm Shkm − (D − A0 ) °≤° ° hk m >0

n/n+1

hk m 6=0

´ ° ° x ˜ (t)°,

n/n+1

° ° °

° °¡ ¢ = ° (Dk − Ak0 ) − (D − A0 ) n/n+1 ° → 0.

P Clearly (cf. the proof of 2.6.3), the operator hkm >0 Akm Shkm is causal and belongs to the radical of the algebra B+ (Cn/n+1 ). Since the radical is closed the operator (D − A0 )n/n+1 belongs to the radical, too. By 1.4.8 it follows that the causal spectrum of Dn/n+1 is equal to the causal spectrum of the operator (A0 )n/n+1 , which by 2.6.1 implies that D is causally invertible on [n, n + 1] if and only if the coefficient a0 (t) is invertible at all points t ∈ [n, n + 1]. Next, we show that the general case is reduced to the case H = (−∞, +∞). Indeed, assume, e.g., H = [τ, +∞). We define the operator Πτ as follows  for t ≥ τ ,   (x)(t) ¡ ¢ Πτ x (t) = (1 + t − τ )(x)(t) for τ − 1 ≤ t ≤ τ ,   0 for t ≤ τ − 1. d ¯ ¡Let L¢ = D dt + B. We consider the operator ¡ ¢Lτ = Πτ L + (1 − Πτ )A0 U , where A¯0 x (t) = a0 (t)x(t) for x ≥ τ − 1, and A¯0 x (t) = a0 (τ − 1)x(t) for x ≤ τ − 1. d It is easy to see that Lτ = Dτ dt + Bτ with Dτ = Πτ D + (1 − Πτ )A¯0 and Bτ = ¯ Πτ B + (1 − Πτ )A0 . From the explicit formula

¡ ¢ for t ≥ τ ,   Dx (t) ¡ ¢ ¡ ¢ Dτ x (t) = (1 + t − τ ) Dx (t) + (τ − t)a0 (t)x(t) for τ − 1 ≤ t ≤ τ ,   a0 (τ − 1)x(t) for t ≤ τ − 1

6.2. THE SOLUBILITY FOR DERIVATIVES

365

it follows that Dτ is a difference operator. Indeed, assume Dk ∈ Df converges to D in norm. From the preceding formula it is clear that the operators Dτk = Πτ Dk + (1 − Πτ )A¯0 belong to Df . Evidently, Dτk converges to Dτ . Thus Dτ ∈ D. By assumption the equation Lτ x = f is locally soluble and input–output stable on [τ, +∞). It is easily locally soluble and input–output stable on (−∞, τ − 1], too. Finally, we observe that the coefficient a ¯0 of the operator Dτ is the function ½ a ¯0 (t) =

a0 (t)

for t ≥ τ − 1,

a0 (τ − 1)

for t ≤ τ − 1.

Therefore the equation Lτ x = f is locally soluble on [τ − 1, τ ]. Consequently by 3.3.5(a,b) it is locally soluble and input–output stable on (−∞, +∞). We shall show that this implies that the operator Dτ is causally invertible on (−∞, +∞). Then in turn it will imply that D is causally invertible on H. Thus the case H = [τ, +∞) is reduced to the case H = (−∞, +∞). So we assume that H = (−∞, +∞). Arguing as in (i) we show that L is causally invertible on (−∞, +∞). Then from 2.5.9 and 2.5.10 it follows the causal invertibility of D on all finite segments [a, b]. Hence by 2.1.5 to prove that D is causally invertible on (−∞, +∞) it suffices to prove that D is (ordinary) invertible, say, on [τ, +∞). We again consider the operators Lτ and Dτ . Applying the last result to the equation Lτ x = f we obtain that Dτ is invertible on (−∞, +∞). But, evidently, Dτ is invertible on (−∞, τ − 1]. Hence by 2.1.4 it is invertible on [τ − 1, +∞), too. Since Dτ is invertible on finite segments, by 2.1.4 it follows that Dτ is invertible on [τ, +∞). It remains to observe that on [τ, +∞) the operator Dτ coincides with the operator D. −1 (iii, iv) The cases of the pairs (Lpq , Wpq ) and (Cq , Cq−1 ) are handled similarly d with one exception. One should, for the operator L = dt D + B, consider the ¯ operator Lτ = LPτ + U (1 − Pτ ) or Lτ = LΠτ + U A0 (1 − Πτ ), respectively. ¤ Remark. We modify the remark from 6.2.3 to the case considered. First, we observe that the causal invertibility of D is equivalent to the stability of the equation Dy = g. Indeed, the cases q 6= 0, ∞ follow from 3.3.6. The case q = 0 follows from 3.3.8. Finally, from 3.3.8 and 5.5.6 it also follows the case q = ∞. (a) We do not claim that if D is causally invertible the equation Lx = f is then stable. The theorem only states that the causal invertibility of D is a necessary condition for the stability of Lx = f . If the operator D is causally invertible then the equation of neutral type (1) is equivalent to the equation (3), which, generally speaking, is simpler than (1). We stress that the operator D−1 B is causal; thus (3) is an equation of retarded type. Nevertheless, to pass from (1) to (3) one should invert the operator D first. Thus assertion (a) of the theorem means that (1) is stable if and only if both (4) and (3) are stable. In a similar way, if D is causally invertible then performing the change of variables Dx = z in (2) we arrive at the equation (5), which, generally speaking,

366

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

is simpler than the initial equation (2). Thus assertion (b) of the theorem means that (3) is stable if and only if both (4) and (5) are stable. (b) We recall that equations (3) and (5) may have unbounded memory even though equations (1) and (2) have bounded memory.

6.3. The independence from the choice of a norm Intuitively it seems natural that applied properties of an equation Lx = f must not depend on the pair of spaces in which the operator L is considered. Some mathematical results of this kind are presented is this section. For simplicity, throughout this section by input–output stability we mean input–output stability on (−∞, +∞). We recall that for L ∈ e the input–output stability is equivalent to exponential stability, and the invertibility is equivalent to exponential dichotomy. Corollaries of this kind are trivial and therefore usually omitted. Throughout this section we assume that G = R. As usual, we denote by L = L(R, E) one of the spaces Lpq = Lpq (R, E) or Cq = Cq (R, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We discuss the functional differential equations

and

Dx˙ + Bx = f

(1)

d Dx + Bx = f dt

(2)

d with the corresponding operators L = D dt + B and L =

d dt D

+ B.

6.3.1. The case of invertibility with D, B ∈ s. The following statement is an analogue of 5.2.7 for differential operators, cf. 5.5.11. Theorem. Let E be an arbitrary Banach space. Let D, B ∈ s(L), see 1.6.8 and 5.2.1 for the definition of s. d 1 + B. Assume the operator L : Wpq → Lpq (L : Cq1 → Cq ) is (a) Let L = D dt invertible for some q. Then it is invertible for all q. d −1 (b) Let L = dt D + B. Assume the operator L : Lpq → Wpq (L : Cq → Cq−1 ) is invertible for some q. Then it is invertible for all q. Proof. (a) We consider the operator T = LU −1 : L → L. Since U, U −1 ∈ df (see 5.5.11) we have T ∈ s(L). A reference to 5.2.7 completes the proof. (b) is handled in a similar way. ¤ Example. Let ξ ∈ M(R) be the measure from 4.2.12. We consider the operator Tξ x = ξ ∗ x of the convolution with ξ. We recall from the example in 5.3.1 that the operator i − Tξ is invertible on L2 , but not invertible on L1 , L∞ , and C. Hence by 5.2.7 the operator i − Tξ is not invertible on L1q , L∞q , and Cq for all q as well. Clearly, the same phenomenon holds for the differential operator L = (i − Tξ )U = U (i − Tξ ).

6.3. THE INDEPENDENCE FROM THE CHOICE OF A NORM

367

6.3.2. The case of stability with D, B ∈ s Corollary (cf. 3.4.8). Let E be an arbitrary Banach space. Let D, B ∈ s(L) be causal operators. (a) Assume equation (1) is locally soluble and input–output stable in the pair 1 (Wpq , Lpq ) (in the pair (Cq1 , Cq ), respectively) for some q. Then it is locally 1 soluble and input–output stable in the pair (Wpq , Lpq ) (in the pair (Cq1 , Cq ), respectively) for all q. (b) Assume equation (2) is locally soluble and input–output stable in the pair −1 (Lpq , Wpq ) (in the pair (Cq , Cq−1 ), respectively) for some q. Then it is −1 locally soluble and input–output stable in the pair (Lpq , Wpq ) (in the pair −1 (Cq , Cq ), respectively) for all q. Proof. By 6.3.1 the invertibility of L does not depend on q. Moreover, since an operator of the class s ⊆ t is restored by its matrix (see 1.6.6), if L−1 is causal for some q then it is also causal for all q. It remains to refer to 6.2.7. ¤ Example. Let ν be the measure from 4.3.11. We consider the causal operator Tν x = ν ∗ x. Arguing as in the example in 5.3.1 it is easy to show that the spectral radius ot Tν on L2 is less than or equal to 1/3. Hence the operator 1 − Tν is causally invertible on L2 . On the other hand (see 5.3.1), the operator 1 − Tν is not invertible on L1 , L∞ , and C. From 5.2.7 it follows that 1 − Tν is not invertible on L1q , L∞q , and Cq for all q as well. Clearly, the same phenomenon holds for the differential operator L = (1 − Tν )U = U (1 − Tν ). Let us interpret these facts in terms of stability. We recall that ν is concentrated on the segment [1, 5]. Therefore the equation Lx = f is uniformly soluble. Evidently, Tν ∈ df . Hence L ∈ e. By 3.5.9 the equation Lx = f is input– −1 1 , L2q ) and (L2q , W2q ), but is output and exponentially stable in the pairs (W2q −1 1 not input–output and exponentially stable in the pairs (W1q , L1q ) and (L1q , W1q ), 1 −1 1 −1 and (W∞q , L∞q ) and (L∞q , W∞q ), and (Cq , Cq ) and (Cq , Cq ). 6.3.3. The case of invertibility with D, B ∈ C. The results of this and the following subsections allow one to reduce the investigation of equations in Cq to the investigation of equations in L∞q and vice versa. Theorem. Let E be a finite-dimensional Banach space. Let D, B ∈ C(L∞q ) ' C(Cq ), see 5.6.2 and 5.6.9. d 1 (a) Let L = D dt + B. The operator L : W∞q → L∞q is invertible if and only 1 if the operator L : Cq → Cq is invertible. d −1 (b) Let L = dt D + B. The operator L : L∞q → W∞q is invertible if and only −1 if the operator L : Cq → Cq is invertible.

Proof. (a) We consider the operator T = LU −1 . We recall the identity T = D + (B − D)U −1 . Since U −1 ∈ A ∩ t ⊆ C (see 4.1.2) it is easy to see that T ∈ C. A reference to 5.6.9 completes the proof. (b) is handled in a similar way. ¤

368

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

6.3.4. The case of stability with D, B ∈ C Corollary. Let E be a finite-dimensional Banach space. Let D, B ∈ C(L∞q ) ' C(Cq ) be causal operators. (a) The local solubility and input–output stability of equation (1) in the pair 1 (W∞q , L∞q ) is equivalent to the local solubility and input–output stability of equation (1) in the pair (Cq1 , Cq ). (b) The local solubility and input–output stability of equation (2) in the pair −1 (L∞q , W∞q ) is equivalent to the local solubility and input–output stability of equation (2) in the pair (Cq , Cq−1 ). Proof. The proof is similar to that of 6.3.2. ¤ 6.3.5. The case of invertibility with D ∈ S and B ∈ V Theorem. Let E be a finite-dimensional Banach space. Let D ∈ S(L) and B ∈ V(L), see 5.2.1 and 5.3.1 for the definitions of S and V, respectively. d 1 (a) Assume the operator L = D dt + B : Wpq → Lpq is invertible for some p and q. Then it is invertible for all p and q. d −1 (b) Assume the operator L = dt D + B : Lpq → Wpq is invertible for some p and q. Then it is invertible for all p and q.

Proof. (a) We consider the operator T = LU −1 : Lpq → Lpq . We assume that T is invertible for some p and q. Then by 6.2.1 the operator D is invertible. Moreover, by 5.2.10 D is invertible on Lpq for all p and q, and the inverse D−1 does not depend on p and q (e.g., this fact can be interpreted as D−1 ∈ V). We turn to the operator R = D−1 T = 1 + D−1 (B − D)U −1 . Clearly, T and R are invertible simultaneously. Since U −1 ∈ N∞ (see 5.3.4) and g D−1 (B − D) ∈ V, by 5.3.2 we have D−1 (B − D)U −1 ∈ N∞ . Thus R ∈ N ∞ . It remains to refer to 5.3.5. (b) is handled in a similar way. ¤ 6.3.6. The case of stability with D ∈ S and B ∈ V Corollary. Let E be a finite-dimensional Banach space. Let D ∈ S(L) and B ∈ V(L) be causal operators. (a) Assume equation (1) is locally soluble and input–output stable in the pair 1 (Wpq , Lpq ) for some p and q. Then it is locally soluble and input–output 1 stable in the pair (Wpq , Lpq ) for all p and q. (b) Assume equation (2) is locally soluble and input–output stable in the pair −1 (Lpq , Wpq ) for some p and q. Then it is locally soluble and input–output −1 stable in the pair (Lpq , Wpq ) for all p and q. Proof. The proof is similar to that of 6.3.2. ¤

6.3. THE INDEPENDENCE FROM THE CHOICE OF A NORM

369

6.3.7. The case of invertibility with D, B ∈ S(Cq ) Corollary. Let E be a finite-dimensional Banach space. Let D, B ∈ S(Cq ), see 5.2.1 for the definition of S. d 1 (a) Let L = D dt + B. Assume the operator L : Wpq → Lpq is invertible for 1 some p and q, or the operator L : Cq → Cq is invertible for some q. Then 1 the operator L : Wpq → Lpq is invertible for all p and q, and the operator 1 L : Cq → Cq is invertible for all q. d −1 D + B. Assume the operator L : Lpq → Wpq is invertible for (b) Let L = dt −1 some p and q, or the operator L : Cq → Cq is invertible for some q. Then −1 is invertible for all p and q, and the operator the operator L : Lpq → Wpq L : Cq → Cq−1 is invertible for all q.

Proof. The proof follows from 6.3.3 and 6.3.5, see also 5.6.8.

¤

6.3.8. The case of stability with D, B ∈ S(Cq ) Corollary. Let E be a finite-dimensional Banach space. Let D, B ∈ S(Cq ) be causal operators. (a) Assume equation (1) is locally soluble and input–output stable in the pair 1 (Wpq , Lpq ) for some p and q, or locally soluble and input–output stable in the pair (Cq1 , Cq ) for some q. Then it is locally soluble and input–output 1 stable in the pair (Wpq , Lpq ) for all p and q, and locally soluble and input– output stable in the pair (Cq1 , Cq ) for all q. (b) Assume equation (2) is locally soluble and input–output stable in the pair −1 (Lpq , Wpq ) for some p and q, or locally soluble and input–output stable in the pair (Cq , Cq−1 ) for some q. Then it is locally soluble and input– −1 output stable in the pair (Lpq , Wpq ) for all p and q, and locally soluble and input–output stable in the pair (Cq , Cq−1 ) for all q. Proof. The proof follows from 6.3.4 and 6.3.6, see also 5.6.8.

¤

6.3.9. The case of invertibility with D ∈ S1 and B ∈ t. Let E be a finite1 1 dimensional Banach ¡ ¢ space. We denote by S = S (Lpq ) the set of all operators D ∈ B Lpq (R, E) of the form ∞ X ¡ ¢ Dx (t) = am (t)x(t − hm ), m=1

¡ ¢ P∞ 1 1 where am ∈ W∞ R, B(E) satisfy the assumption m=1 kam kW∞ < ∞. Since 1 W∞ ⊆ C, from 2.3.3 it follows that S1 (Lpq ) is an algebra. For D ∈ S1 (Lpq ) we set ∞ X ¡ ¢ ˙ Dx (t) = a˙ m (t)x(t − hm ). m=1

(3)

370

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

1 Thus D˙ ∈ S(Lpq ). From 2.3.3, for x ∈ Wpq we have

d d ˙ Dx = D x + Dx. dt dt

(4)

d 1 Hence if D ∈ S1 (Lpq ) and B ∈ B(Lpq ) the operator L = D dt + B : Wpq → Lpq d ˙ can be represented in the form L = dt D + D + B. This representation allows one −1 to consider L as acting from Lpq to Wpq as well. ¡ ¢ 1 1 We denote by S = S (Cq ) the set of all operators D ∈ B Cq (R, E) of the form ∞ X ¡ ¢ Dx (t) = am (t)x(t − hm ), m=1

¡ ¢ P∞ where am ∈ C 1 R, B(E) satisfy the assumption m=1 kam kC 1 < ∞. Obviously, S1 (Cq ) is an algebra. For D ∈ S1 (Cq ) we define the operator D˙ by the formula (3). Evidently, we have D˙ ∈ S(Cq ). Clearly, for x ∈ Cq1 the representation (4) holds. Therefore d if D ∈ S1 (Cq ) and B ∈ B(Cq ) the operator L = D dt + B : Cq1 → Cq can be d represented in the form L = dt D + D˙ + B, which allows one to consider it as acting from Cq to Cq−1 as well. Theorem. Let E be a finite-dimensional Banach space. d (a) Let D ∈ S1 (Lpq ) and B ∈ t(Lpq ), and L = D dt + B. For all p and q, if −1 1 the operator L : Lpq → Wpq is invertible then the operator L : Wpq → Lpq 1 is invertible, too. For p 6= ∞ and all q, if the operator L : Wpq → Lpq is −1 is invertible, too. invertible then the operator L : Lpq → Wpq d 1 (b) Let D ∈ S (Cq ) and B ∈ t(Cq ), and L = D dt +B. For all q, if the operator L : Cq → Cq−1 is invertible then the operator L : Cq1 → Cq is invertible, too. −1 is invertible. We take an f ∈ Lpq . Since Proof. (a) Assume L : Lpq → Wpq −1 Lpq ⊆ Wpq , by assumption the equation Lx = f has a unique solution x ∈ Lpq . 1 To complete the proof it suffices to show that actually x ∈ Wpq . d We rewrite the equation (D dt + B)x = f in the ¡ form (DU + B ¢ − D)x = f and −1 −1 then, employing 6.2.3, in the form x = U D f − (B − D)x . Clearly, in the 1 last representation the right side belongs to Wpq . Therefore the left side also does. 1 Conversely, assume L : Wpq → Lpq is invertible. We observe that by 6.3.1, without loss of generality we may assume that q 6= ∞. For more correctness d 1 → Lpq in the form L1 = D dt we represent the operator L : Wpq + BJ, and the d 1 −1 ˙ operator L : Lpq → Wpq in the form L2 = dt D + J(B − D), where J : Wpq → Lpq −1 and J : Lpq → Wpq are natural embeddings. We consider the conjugate operators. d 1 By 2.3.12 the conjugate of L1 : Wpq → Lpq is L01 = − dt D0 + J 0 B 0 : Lp0 q0 → Wp−1 0 q0 , −1 0 0 d 0 0 1 ˙ and the conjugate of L2 : Lpq → Wpq is L2 = −D dt + (B − D) J : Wp0 q0 → Lp0 q0 . By what has been proved, the invertibility of L01 implies that of L02 . (b) is handled in a similar way. ¤

6.3. THE INDEPENDENCE FROM THE CHOICE OF A NORM

371

6.3.10. The case of stability with D ∈ S1 and B ∈ t Corollary. Let E be a finite-dimensional Banach space. d (a) Let D ∈ S1 (Lpq ) and B ∈ t(Lpq ) be causal operators, and L = D dt + B. For all p and q, if the equation Lx = f

(5)

−1 is locally soluble and input–output stable in the pair (Lpq , Wpq ) then it is 1 locally soluble and input–output stable in the pair (Wpq , Lpq ). For p 6= ∞ and all q, if equation (5) is locally soluble and input–output stable in the 1 pair (Wpq , Lpq ) then it is locally soluble and input–output stable in the −1 pair (Lpq , Wpq ). d 1 (b) Let D ∈ S (Cq ) and B ∈ t(Cq ) be causal operators, and L = D dt + B. For all q, if equation (5) is locally soluble and input–output stable in the pair (Cq , Cq−1 ) then it is locally soluble and input–output stable in the pair (Cq1 , Cq ). −1 Proof. (a) We make use of 6.2.7. Assume L : Lpq → Wpq is causally invert−1 1 −1 −1 ible. Then, evidently, the restriction L : Lpq → Wpq of L : Wpq → Lpq is a causal operator, too. 1 Conversely, assume L : Wpq → Lpq is causally invertible. By 6.3.2, without loss 1 is dense of generality we may assume that q 6= ∞. Since p, q 6= ∞ we have that Wpq −1 in norm, too. Consequently the in Lpq in norm, and by 2.3.9 Lpq is dense in Wpq −1 1 extension L−1 : Wpq → Lpq of L−1 : Lpq → Wpq can be obtained by continuity. Evidently, such an extension gives a causal operator. (b) is handled in a similar way. ¤

6.3.11. The case of invertibility with D ∈ S1 (C) and B ∈ S(C) Corollary. Let E be a finite-dimensional Banach space. Let D ∈ S1 (C) and d B ∈ S(C), and L = D dt + B. If the operator L is invertible in one of the pairs (Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq )

(6)

(for at least one p and q) then the operator L is invertible in all these pairs (for all p and q). Proof. The proof follows from 6.3.7 and 6.3.9.

¤

6.3.12. The case of stability with D ∈ S1 (C) and B ∈ S(C) Corollary. Let E be a finite-dimensional Banach space. Let D ∈ S1 (C) and d + B. If equation (5) is locally soluble B ∈ S(C) be causal operators, and L = D dt and input–output stable in one of the pairs (6) (for at least one p and q) then equation (5) is locally soluble and input–output stable in all these pairs (for all p and q). Proof. The proof follows from 6.3.10 and 6.3.8.

¤

372

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

6.4. Green’s function The general solution of a differential equation can be usually represented by an integral operator. Its kernel is called Green’s function. Thus any result on Green’s function is an assertion on the representation of the inverse operator and vice versa. From this point of view the investigation of full subalgebras is closely connected with the investigation of Green’s functions. Throughout this section we assume that G = R and E is a finite-dimensional Banach space. 6.4.1. The integral representation for solutions. The following theorem shows that to a considerable extent the availability of the integral representation for L−1 depends on the spaces between which the operator L−1 acts. Theorem. (a) Let D, B ∈ s(L1q ), see 1.6.8 for the definition of s. Assume the operator d 1 L = D dt +B : W1q → L1q is invertible. Then the solution x of the equation Lx = f can be represented in the form Z +∞ x(t) = G(t, s)f (s) ds, (1) −∞

where G : R × R → B(E) is a measurable function satisfying the estimate kG(t, s)k ≤ β(t − s),

t, s ∈ R,

(2)

with a function β : R → [0, ∞) of the class L∞1 , cf. 5.3.4. (b) Let D, B ∈ e(L1q ), see 1.6.9 for the definition of e. Assume the operator d 1 → L1q is invertible. Then the solution x of the equation L = D dt +B : W1q Lx = f can be represented in the form (1), where G : R × R → B(E) is a measurable function satisfying the estimate kG(t, s)k ≤ N e−η|t−s| ,

t, s ∈ R,

(3)

for some N and η > 0, cf. 5.3.9. d −1 (c) Let D, B ∈ s(L∞q ). Assume the operator L = dt D + B : L∞q → W∞q is invertible. Then the solution x of the equation Lx = f for f ∈ Lpq (not −1 for f ∈ Wpq !) can be represented in the form (1), where G : R × R → B(E) is a measurable function satisfying the estimate (2) with a function β : R → [0, ∞) of the class L∞1 . d −1 (d) Let D, B ∈ e(L∞q ). Assume the operator L = dt D + B : L∞q → W∞q is invertible. Then the solution x of the equation Lx = f for f ∈ Lpq (not for −1 f ∈ Wpq !) can be represented in the form (1), where G : R × R → B(E) is a measurable function satisfying the estimate (3) for some N and η > 0. 1 Proof. (a) By 5.2.6 and 5.5.11 L−1 ∈ s(L1q , W1q ). We show that it follows −1 that L ∈ N∞ (see 5.3.2 for the definition of N∞ ), which by 5.3.4 will complete the proof.

373

6.4. GREEN’S FUNCTION

We represent L−1 as U −1 (U L−1 ). By 1.6.8 U L−1 ∈ s(L1q , L1q ). On the other hand U −1 ∈ s(L1q , L∞q ). Hence L−1 ∈ s(L1q , L∞q ), i.e., L−1 ∈ N∞ . −1 (d) By 3.5.2 and 5.5.11 L−1 ∈ e(W∞q , L∞q ). We show that it follows that −1 L ∈ M∞ (see 5.3.8 for the definition of M∞ ), which by 5.3.9 will complete the proof. We represent L−1 as (L−1 U )U −1 . By 1.6.9 L−1 U ∈ e(L∞q , L∞q ). On the other hand, U −1 ∈ e(L1q , L∞q ). Hence L−1 ∈ e(L1q , L∞q ), i.e., L−1 ∈ M∞ . (b) and (c) are proved in a similar way. ¤ 6.4.2. The integral representation for derivative Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. (a) Let D ∈ S(Lpq ) and B ∈ V(Lpq ), see 5.3.1 for the definition of V. Assume d 1 → Lpq is invertible. Then the derivative the operator L = D dt + B : Wpq x˙ of the solution x of the equation Lx = f can be represented in the form x(t) ˙ =

∞ X

Z

+∞

bm (t)f (t − gm ) +

H(t, s)f (s) ds,

(4)

−∞

m=1

P∞ where bm : R → B(E) are measurable and bounded, m=1 kbm kL∞ < ∞, gm ∈ R, and H : R × R → B(E) is a measurable function satisfying the estimate (2) with a function β : R → [0, ∞) of the class L∞1 . (b) Let D ∈ ER (Lpq ) and B ∈ W(Lpq ), see 5.2.12 and 5.3.7 for the definitions d 1 of ER and W. Assume the operator L = D dt → Lpq is invertible. +B : Wpq Then the derivative x˙ of the solution x of the equation Lx = f can be represented the form (4), where bm : R → B(E) are measurable and Pin ∞ bounded, m=1 e−ν|gm | kbm kL∞ < ∞, gm ∈ R, and H : R × R → B(E) is a measurable function satisfying the estimate (3) for some N and η > 0. d (c) Let D ∈ S(Lpq ) and B ∈ V(Lpq ). Assume that the operator L = dt D+ −1 B : Lpq → Wpq is invertible. Then the solution x of the equation Lx = f with f = u˙ + v, u, v ∈ Lpq , can be represented in the form x(t) =

∞ X m=1

Z

Z

+∞

bm (t)u(t − gm ) +

+∞

F (t, s)u(s) ds + −∞

G(t, s)v(s) ds,

(5)

−∞

P∞ where bm : R → B(E) are measurable and bounded, m=1 kbm kL∞ < ∞, gm ∈ R, and the functions F, G : R × R → B(E) are measurable and satisfy the estimate (2) with a function β : R → [0, ∞) of the class L∞1 . (d) Let D ∈ ER (Lpq ) and B ∈ W(Lpq ). Assume that the operator L = d −1 dt D + B : Lpq → Wpq is invertible. Then the solution x of the equation Lx = f with f = u+v, ˙ u, v ∈ Lpq , can be represented in the form (5), where P∞ bm : R → B(E) are measurable and bounded, m=1 e−ν|gm | kbm kL∞ < ∞, gm ∈ R, and the functions F, G : R × R → B(E) are measurable and satisfy the estimate (3) for some N and η > 0.

374

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Remark. (a) We observe (see 1.6.12) that S(Lpq ) ⊆ V(Lpq ) = s(L1q )∩s(L∞q ) and ER (Lpq ) ⊆ W(Lpq ) = e(L1q ) ∩ e(L∞q ). Hence under assumptions of this theorem the integral representations from 6.4.1 hold obviously. (b) Clearly, representations from 6.4.1(c, d) are special cases of 6.4.2(c, d) for u = 0. (c) We note that this theorem in distinction to 6.4.1 describes the properties of the representation of solutions (and their derivative) which actually ensure the action of L−1 in the corresponding pairs of spaces. Proof. (a) We represent the correspondence f 7→ x˙ as the operator d −1 L = U L−1 − L−1 dt ³ d ´−1 =U D +B − L−1 dt = U (DU + B − D)−1 − L−1 ¡ ¢−1 = (DU + B − D)U −1 − L−1 ¡ ¢−1 = D + (B − D)U −1 − L−1 ¡ ¢−1 = D(1 + D−1 (B − D)U −1 ) − L−1 ¡ ¢−1 −1 = 1 + D−1 (B − D)U −1 D − L−1 . We set N = D−1 (B − D)U −1 . By 5.2.8 D−1 ∈ S ⊆ V. Since U −1 ∈ N∞ , by 5.3.2 ¡ ¢−1 we obtain that N ∈ N∞ . Hence by 5.3.6 the operator 1 + D−1 (B − D)U −1 can be represented as 1 + N1 , where N1 ∈ N∞ . Thus d −1 L = (1 + N1 )D−1 − L−1 dt = D−1 + N1 D−1 − L−1 . By 6.4.1(a) L−1 ∈ N∞ . By 5.3.2 N1 D−1 ∈ N∞ , too. Thus we arrive at the representation (4). (d) The case u = 0 follows from 6.4.1 and the above remark. So it suffices to consider the case v = 0. We represent the correspondence u 7→ x as the operator L−1

d = L−1 U − L−1 dt ³d ´−1 = D+B U − L−1 dt = (U D + B − D)−1 U − L−1 ¡ ¢−1 = U −1 (U D + B − D) − L−1 ¡ ¢−1 = D + U −1 (B − D) − L−1 ¡ ¢−1 − L−1 = (1 + U −1 (B − D)D−1 )D ¡ ¢−1 − L−1 . = D−1 (1 + U −1 (B − D)D−1 )

6.4. GREEN’S FUNCTION

375

We set M = U −1 (B − D)D−1 . By 5.2.12 D−1 ∈ ER ⊆ W. Since U −1 ∈ M∞ , by ¡ ¢−1 5.3.8 we obtain M ∈ M∞ . Hence by 5.3.10 the operator 1 + D−1 (B − D)U −1 can be represented as 1 + M1 , where M1 ∈ M∞ . Thus L−1

d = D−1 (1 + M1 ) − L−1 dt = D−1 + D−1 M1 − L−1 .

By 6.4.1(d) L−1 ∈ M∞ . By 5.3.8 D−1 M1 ∈ M∞ , too. Thus we arrive at the representation (5). (b) and (c) are proved in a similar way. ¤ Remark. It is easy to see from the proof that bm and gm in (4) and (5) are coefficients and shifts of the operator D−1 . Namely, ∞ X ¡ −1 ¢ D f (t) = bm (t)f (t − gm ). m=1

6.4.3. Distributions of two variables. Let B be a finite-dimensional Banach space. For the main example of B we keep in mind the space B(E). Let ψ : R2 → C be a function. The closure of the set { t : ψ(t) 6= 0 } is called the support of ψ and denoted by supp ψ. We denote by D = D(R2 , C) the linear space of all infinitely many times differentiable functions ψ : R2 → C with a compact support. We say that a sequence ψk ∈ D converges to a function ψ ∈ D if: (a) there exists a compact set K ⊆ R2 such that supp ψk ⊆ K for all k; α+β α+β (b) the sequence ∂t∂α ∂sβ ψk converges uniformly to ∂t∂α ∂sβ ψ for all pairs α, β = 0, 1, 2, . . . . We note that the operator of differentiation ψ 7→

∂ α+β ψ ∂tα ∂sβ

acts continuously on D, α+β i.e., if ψn ∈ D converges to ψ ∈ D then converges to ∂t∂α ∂sβ ψ. Let f : D → B be a linear (vector-valued) functional. We denote the value of the functional f on ψ by hψ, f i. We say that the functional f is continuous if hψk , f i converges to hψ, f i whenever ψk converges to ψ. Any continuous linear functional f : D → B is called a distribution on R2 (with values in B). We denote the space of all distributions f by D0 = D0 (R2 , B). Since B is finite-dimensional the space D0 (R2 , B) is naturally isomorphic to B ⊗ D0 (R2 , C), cf. the example in 1.7.2. In a similar way, the space D = D(R2 , B0 ) of all infinitely many times differentiable functions ψ : R2 → B0 with a compact support is naturally isomorphic to B0 ⊗ D(R2 , C). Thus one can define naturally hψ, f i for ψ ∈ D(R2 , B0 ) and f ∈ D0 (R2 , B). ∂f Let f ∈ D0 . The functionals ∂f ∂t and ∂s defined by the rules ∂ α+β ψ ∂tα ∂sβ n

D ∂f E D ∂ψ E ψ, =− ,f ∂t ∂t

D and

D ∂ψ E ∂f E ψ, =− ,f ∂t ∂t

are called the partial derivative of the functional f . It is straightforward to verify ∂f 0 that ∂f ∂t and ∂s are actually linear and continuous; thus they belong to D .

376

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

We denote by L1 loc = L1 loc (R2 , B), 1 ≤ p ≤ ∞, the linear space of all measurable functions x : R2 → B whose restrictions to any compact set K ⊆ R2 belong to L1 (K, B). Let x ∈ L1 loc . It is easy to see that the functional fx defined by the rule Z hψ, fx i = ψ(t, s) x(t, s) dt ds R2

belongs to D0 . The distribution fx induced by a function x ∈ L1 loc is called regular. It is easy to see that fx = fy , x, y ∈ L1 loc , implies that x = y a.e.. ∂ ∂ We call the derivatives ∂t fx , ∂s fx ∈ D0 of the functional fx ∈ D0 the partial distribution derivatives of x ∈ L1 loc . Proposition ([Vla2 , ch. 1, §3, 2]). The space D(R, C) ⊗ D(R, C) is dense in D(R2 , C), i.e., for any ψ ∈ D(R2 , C) there exist functions ukm vkm ∈ D(R, C), Pn, m k = 1, . . . , nm , m = 1, 2 . . . , such that the sequence ψm (t, s) = k=1 ukm (t)vkm (s) converges to ψ. 6.4.4. The partial derivatives

∂G ∂t

and

∂G ∂s

Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let D ∈ S(Lpq ) and B ∈ V(Lpq ). d 1 → Lpq is invertible, and the (a) Assume the operator L = D dt + B : Wpq kernels G and H, and the coefficients bm , and the shifts gm are taken from the representations (1) and (4). Then ∞ X ¡ ¢ ∂G (t, s) = bm (t)δ s − (t − gm ) + H(t, s). ∂t m=1 d −1 is invertible, and the (b) Assume the operator L = dt D + B : Lpq → Wpq kernels G and F , and the coefficients bm , and the shifts gm are taken from the representation (5). Then ∞ X ¡ ¢ ∂G (t, s) = − bm (t)δ s − (t − gm ) − F (t, s). ∂s m=1 ∂G Here ∂G D(R, C) is the ∂s and ∂t are the distribution derivatives, and δ ∈ ¡ ¢ distribution hψ, δi = ψ(0), and by the distribution (t, s) 7→ bm (t)δ s − (t − gm ) we mean R +∞ the functional ψ 7→ −∞ bm (t)ψ(t, t − gm ) dt.

Remark. Let h, g ∈ D0 (R, C). We define h ⊗ g ∈ D0 (R2 , C) by the rule D E hψ, h ⊗ gi = hψ(t, s), g(s)i, h(t) ,

ψ ∈ D(R2 , C).

More strictly, for any t ∈ R we consider the function ψt (s) = ψ(t, s). Clearly, ψt ∈ D(R, C). Really, by hψ(t, s), g(s)i we mean the function ψ1 (t) = hψt , gi, and by hψ, h ⊗ gi we mean hψ1 , hi. It can be shown (see, e.g., [Vla2 , ch. 1, §3, 2])

6.4. GREEN’S FUNCTION

377

that for any ψ ∈ D(R2 , C) the function ψ1 (t) = hψt , gi belongs to D(R, C). Thus hψ1 , hi makes sense. It can also be shown that h ⊗ g is a continuous functional on D(R2 , C). For example, let δ ∈ D(R, C) be the distribution hψ, δi = ψ(0) and 1 be the regular distribution induced by the function 1(s) = 1, s ∈ R. Then by definition R +∞ hψ, δ ⊗ 1i = −∞ ψ(0, s) ds. The functional δ ⊗ 1 is usually denoted by δ(t). Let R : R2 → R2 be an invertible linear mapping and f be a distribution. We define the distribution f ◦R with the linear argument R by the rule hψ(t, s), f ◦Ri = hψ◦R−1 ), f i/ det R. It is easy to see that the distribution δ(s−t) can be interpreted as δ(t) with a relevant linear argument. Proof. (a) Let f ∈ D(R, E) and y ∈ D(R, E0 ). We denote by y⊗f the function (t, s) 7→ y(t) ⊗ f (s); it takes its values in E0 ⊗ E. Since E is finite-dimensional one can identify naturally E0 ⊗ E with the¡ conjugate¢ of B(E). Thus we may consider the function y ⊗ f as an element of D R2 , B(E)0 . By the definition of the distribution derivative we have D ∂G ∂t

E , y ⊗ f = −hG, y˙ ⊗ f i.

Since G ∈ L1 loc we have Z

+∞ DZ +∞

−hG, y˙ ⊗ f i = − −∞

E G(t, s)f (s) ds, y(t) ˙ dt.

−∞

R +∞ 1 By 6.4.1(a) the function t 7→ −∞ G(t, s)f (s) ds is the solution x ∈ Wpq ⊆ L1 loc of the equation Lx = f . Hence we can rewrite the right side as = −hx, yi, ˙ which in turn, by the definition of the distribution derivative, is equal to hx, ˙ yi. On the other hand, by 6.4.2(a) we have Z hx, ˙ yi =

∞ +∞ D X

−∞

Z

+∞

bm (t)f (t − gm ) +

E H(t, s)f (s) ds, y(t) dt

−∞

m=1

= hG1 , y ⊗ f i, where G1 (t, s) =

P∞

m=1 bm (t)δ

¡

¢ s − (t − gm ) + H(t, s). Thus, finally, we arrive at

D ∂G ∂t

E , y ⊗ f = hG1 , y ⊗ f i.

Since¡E is finite-dimensional, from 6.4.3 it follows that D(R, E0 ) ⊗ D(R, E) is dense ¢ in D R2 , B(E)0 . Thus the last equality implies that ∂G ∂t = G1 in the sense of distribution. (b) Let u ∈ D(R, E) and y ∈ D(R, E0 ). We denote by y ⊗ u the function 0 (t, s) 7→ y(t) ⊗ u(s) with values ¡ 2 in E0 ¢ ⊗ E. As in (a), we may consider the function y ⊗ u as an element of D R , B(E) .

378

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

By the definition of the distribution derivative we have D ∂G E , y ⊗ u = −hG, y ⊗ ui. ˙ ∂s Since G ∈ L1 loc and u˙ ∈ D we have Z +∞ DZ −hG, y ⊗ ui ˙ =− −∞

E

+∞

G(t, s)u(s) ˙ ds, y(t) dt. −∞

R +∞ ˙ ds is the solution of the equation By 6.4.1(c) the function t 7→ −∞ G(t, s)u(s) Lx = u. ˙ On the other hand, by 6.4.2(c) this solution is given by (5). Thus the right side of the last equality can be represented equivalently as Z +∞ D X Z +∞ ∞ E − bm (t)u(t − gm ) + F (t, s)u(s) ds, y(t) dt = hG2 , y ⊗ ui, −∞

−∞

m=1

where G2 (t, s) = − at

P∞

m=1 bm (t)δ

D ∂G ∂s

Arguing as in (a), we see that

¡ ¢ s − (t − gm ) − F (t, s). Thus, finally, we arrive

E , y ⊗ u = hG2 , y ⊗ ui.

∂G ∂s

= G2 in the sense of distribution.

¤

6.4.5. The conjugate equation Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let D ∈ S(Lpq ) and B ∈ V(Lpq ). d 1 (a) Assume the operator L = D dt +B : Wpq → Lpq is invertible and the kernels G and H, and the coefficients bm , and the shifts gm are taken from the d representations (1) and (4). Then the operator L0 = − dt D0 + B 0 : Lpq → −1 Wpq is invertible (for all p and q), and the solution y of the equation 0 L y = h, where h = u˙ + v, u, v ∈ Lpq , can be represented in the form

y(s) =

∞ X

Z 0

Z

+∞

bm (s) u(s + gm ) +

0

H(t, s) u(t) dt + −∞

m=1

+∞

G(t, s)0 v(t) dt.

−∞

d −1 D + B : Lpq → Wpq is invertible and the (b) Assume the operator L = dt kernels G and F , and the coefficients bm , and the shifts gm are taken from d 1 the representation (5). Then the operator L0 = −D0 dt + B 0 : Wpq → Lpq is invertible (for all p and q), and the solution y of the equation L0 y = h and its derivative y˙ can be represented in the form

Z

+∞

y(t) =

G(t, s)0 h(s) ds,

−∞

y(s) ˙ =

∞ X m=1

0

Z

+∞

bm (s) h(s + gm ) + −∞

F (t, s)0 h(t) dt.

379

6.4. GREEN’S FUNCTION

Proof. (a) First we recall from 6.3.5 that the invertibility of the operator d 1 L = Lpq = D dt + B : Wpq → Lpq does not depend on p and q. Moreover, all 1 operators Lpq coincide on the smallest space W∞1 . Therefore in turn the inverse −1 operators (Lpq ) coincide on L∞1 . Similar reasonings are valid for L0 (note that D0 ∈ S and B 0 ∈ V by 5.2.1 and 5.3.1, respectively). Hence without loss of generality we may assume that p = q = 2. Let G∗ be Green’s function of the equation L0 y = h. In order to prove that G∗ (s, t) = G(t, s)0 for almost all (s, t) ∈ R2 , it suffices to show that G∗ and G coincide as distributions. In its turn, to prove the latter, by 6.4.3 (cf. the previous proof) it suffices to show that for all f ∈ D(R, E) and v ∈ D(R, E0 ), hG∗ , v ⊗ f i = hG0 , v ⊗ f i, where v ⊗ f is the function (t, s) 7→ v(s) ⊗ f (t). Since G∗ , G0 ∈ L1 loc this equality means that Z

Z

+∞ D

+∞

f (t), −∞

Z E G (s, t)v(s) ds dt =

−∞

Z

+∞ D

∗

+∞

f (t),

−∞

E G(t, s)0 v(s) ds dt

−∞

or, equivalently, Z

Z

+∞ D

+∞

f (t), −∞

Z

E

∗

+∞ DZ +∞

G (s, t)v(s) ds dt = −∞

−∞

E G(t, s)f (t) dt, v(s) ds.

−∞

Next, by the definitions of G∗ and G, the last identity can be rewritten as Z

+∞

¡

0 −1

hf (t), (L )

¢ v (t)i dt =

−∞

Z

+∞

¡ ¢ h L−1 f (s), v(s)i ds

−∞

or, shortly, hf, (L0 )−1 vi(L2 ,L2 ) = hL−1 f, vi(L2 ,L2 ) . Finally, by 2.3.12(c) we have hf, (L0 )−1 vi(W −1 ,W 1 ) = hf, (L0 )−1 vi(L2 ,L2 ) 2

2

= hL−1 f, vi(L2 ,L2 ) = hL−1 f, vi(W −1 ,W 1 ) . 2

2

But here the left side coincide with the right one because, by 2.3.12, L0 is the conjugate of L with respect to the duality (W2−1 , W21 ). ∗ ∂G0 ∂G∗ The equality G∗ (s, t) = G(t, s)0 implies the equality ∂G ∂s = ∂t , where ∂s is the partial derivative of G∗ with respect to the second argument. It remains to apply 6.4.4. (b) is handled in a similar way. ¤

380

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

6.4.6. The fundamental solution. Let a ∈ R and α ∈ E. We recall from −1 2.3.10 that by α ⊗ δa = δa,α ∈ Wpq we denote the distribution hψ, α ⊗ δa i = ψ(a)α. Theorem. Let 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Let D ∈ S(Lpq ) and B ∈ V(Lpq ). d 1 (a) Assume the operator L = D dt +B : Wpq → Lpq is invertible. Then Green’s function G of the equation Lx = f can be changed almost everywhere in such a way that for all t ∈ R and β ∈ E0 the function s 7→ G(t, s)0 β is defined a.e. and is the solution of the conjugate equation L0 y = β ⊗ δt . d −1 (b) Assume the operator L = dt D+B : Lpq → Wpq is invertible. Then Green’s function G of the equation Lx = f can be changed almost everywhere in such a way that for all s ∈ R and α ∈ E the function t 7→ G(t, s)α is defined a.e. and is the solution of the equation Lx = α ⊗ δs . Proof. (b) Clearly, the validity of the assertions do not depend on p and q. So we set p = q = 1. Let xs,α be the solution of the equation Lx = α ⊗ δs . We define the function ¯ ¯ s)α = xs,α (t). Thus for any s ∈ R the G : R2 → B(E) from the identity G(t, ¯ s) belongs to L1 and, in particular, is defined almost everywhere. function G(·, Clearly, the function us,α ∈ L1 (see 2.3.10 for the definition of us,α ) depends on the parameter s continuously in norm. Hence the function α ⊗ δs ∈ W1−1 depends on the parameter s continuously in norm as well. In turn this implies that the ¯ s) depend on the parameter s continuously in the functions xs,α ∈ L1 and G(·, ¯ ∈ L1 loc . We show that norm of L1 , too. By Fubini’s theorem it follows that G ¯ ¯ = G almost everywhere G = G in the sense of distributions. This will imply that G and the proof will be complete. Let f ∈ D(R, E) and y ∈ D(R, E0 ). We denote by y ⊗ f the function (t, s) 7→ 0 y(t) ⊗ f (s) with values in ⊗ E. ¢We recall that we may consider the function ¡ E 2 0 y⊗ f as an element of D R , B(E) . Since the span of functions y ⊗ f is dense in ¡ 2 ¢ 0 ¯ = G it suffices to show that D R , B(E) , in order to prove the equality G Z +∞ DZ +∞ Z +∞ DZ +∞ E E ¯ G(t, s)f (s) ds, y(t) dt = G(t, s)f (s) ds, y(t) dt. −∞

−∞

−∞

−∞

By 6.4.1 the right side is equal to hx, yi, where x is the solution of the equation Lx = f . So it remains to show that the left side is equal to hx, yi, too. Let [a, b] be a segment which contains supp f . We consider a partition τ = { a = s0 < s1 < · · · < sn = b } of the segment [a, b], and take arbitrary points ξk ∈ [sk−1 , sk ]; and P we set¡ ∆sk = sk −¢sk−1 . It is straightforward to verify that the n distributions fτ = k=1 f (ξk )(∆sk ) ⊗ δξk tend to f is the norm of W1−1 when the diameter max ∆sk of the partition τ tends zero. ¡ to ¢ Therefore Pn ¯by the definition −1 −1 ¯ of G and the continuity of L the functions L fτ (t) = k=1 G(t, ξk )f (ξk )∆sk converge to the solution x. Consequently Z +∞ DX n E ¯ G(t, ξk )f (ξk )∆sk , y(t) dt → hx, yi. −∞

k=1

381

6.4. GREEN’S FUNCTION

On the other hand, Z Z +∞ DX n n D E X ¯ ξk )f (ξk )∆sk , y(t) dt = G(t, f (ξk )∆sk , −∞

k=1

+∞

E ¯ ξk )0 y(t) dt . G(t,

−∞

k=1

¯ s) is continuous We consider the right side. Since the correspondence s 7→ G(·, R +∞ ¯ s)0 y(t) dt is in the norm of L1 and y ∈ D ⊆ L∞ , the function s 7→ −∞ G(t, continuous. Hence the right side of the last equality can be interpreted as the integral sum for the Riemann integral Z +∞ D Z +∞ E ¯ s)0 y(t) dt ds. f (s), G(t, −∞

Thus Z +∞ DX n −∞

−∞

Z E ¯ G(t, ξk )f (ξk )∆sk , y(t) dt →

k=1

Z

Z

+∞ D

+∞

f (s),

−∞

+∞ DZ

−∞ +∞

= −∞

E 0 ¯ G(t, s) y(t) dt ds

E ¯ s)f (s) ds, y(t) dt, G(t,

−∞

and we arrive at the required equality Z +∞ DZ +∞ E ¯ s)f (s) ds, y(t) dt = hx, yi. G(t, −∞

−∞

(a) follows from 6.4.5 and (b). ¤ 6.4.7. The case of D ∈ S1 (C) and B ∈ S(C). We return to the discussion of the situation considered earlier in 6.3.11. d + B. Assume the Corollary. Let D ∈ S1 (C) and B ∈ S(C), and L = D dt d 1 operator L = D dt + B : Wpq → Lpq is invertible in one of the pairs

(Cq1 , Cq ),

1 (Wpq , Lpq ),

(Cq , Cq−1 ),

−1 (Lpq , Wpq ).

Then Green’s function G of the equation Lx = f can be changed almost everywhere in such a way that for all t ∈ R and β ∈ E0 the function s 7→ G(t, s)0 β is defined a.e. and is the solution of the conjugate equation L0 x = β ⊗ δt , and for all s ∈ R and α ∈ E the function t 7→ G(t, s)α is defined a.e. and is the solution of the equation Lx = α ⊗ δs . Proof. Let Ga and Gb be the functions equivalent to G form 6.4.6(a) and 6.4.6(b), respectively. By 6.4.6(b) for all s ∈ R and α ∈ E the function t 7→ Gb (t, s)α is defined a.e. and is the solution of the equation Lx = α ⊗ δs . By 6.4.6(a) the function Ga possesses the conjugate property. Clearly, Ga − Gb is a null function. Therefore by Fubini’s theorem, for almost all s ∈ R the function t 7→ |Ga (t, s) − Gb (t, s)| is null, in particular, it is defined for almost all t. We denote by E the set of all such s. It follows that for all s ∈ E and α ∈ E the function t 7→ Ga (t, s)α is also defined a.e. and is the solution of the ¯ s) to be Ga (t, s) for s ∈ E and we define equation Lx = α ⊗ δs . We define G(t, ¯ s) to be Gb (t, s) for s ∈ ¯ obtain the property of Gb from 6.4.6(b). G(t, / E. Then G ¯ At the same time G does not lose the property of Ga described in 6.4.6(a). Thus ¯ is a desirable function equivalent to G. ¤ the function G

382

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

6.5. Almost periodic operators If a solution of a non-linear equation is bounded it is most likely that the linearization of the equation along this solution is an equation with almost periodic coefficients. In this section we show that almost periodic equations possess a property similar to the Fredholm alternative for equations on compact set, e.g., for periodic equations. From the formal point of view the theory of almost periodic operators is dual with respect to the Fourier transform to the theory of difference operators. 6.5.1. Almost periodic operators. Let G be a locally compact abelian group, and let X be a Banach space. A mapping h 7→ Sh from G to B(X) is called a representation of the group G on X if Sg+h = Sg Sh

for all g, h ∈ G.

The representation is usually designated by {Sh } or, simply, by Sh (cf. the usual notation for sequence). We say that a representation Sh is bounded if there exists M such that kSh k ≤ M for all h ∈ G, cf. 4.1.2. We note that if Sh is bounded and one change the norm on X to kxk∗ = sup{ kSh xk : h ∈ G }, the operators Sh become isometric, i.e., kSh k = 1 for all h ∈ G. Assume X and Y are Banach spaces, and we have bounded representations Sh , h ∈ G, both on X and Y (it is convenient to use the same notation Sh for both representations). We say that an operator T ∈ B(X, Y ) is almost periodic with respect to the representations Sh if the function h 7→ Sh T S−h ,

h ∈ G,

taking its values in B(X, Y ) is almost periodic in the sense of definition 5.1.9. We denote by BS = BS (X, Y ) the set of all operators T ∈ B(X, Y ) almost periodic with respect to the representations Sh . We say that an operator T ∈ B(X, Y ) is S-invariant if T = Sh T S−h

for all h ∈ G.

We denote by AS = AS (X, Y ) the set of all S-invariant operators T ∈ B(X, Y ). In addition, we suppose that we also have a representation Ψχ, χ ∈ X(G), of the dual group, and the Weyl identity holds: hh, χiSh Ψχ = Ψχ Sh ,

h ∈ G, χ ∈ X(G).

(1)

Remark. The set Γ = U × G × X with the operation (u1 , h1 , χ1 )(u2 , h2 , χ2 ) = (u1 u2 + hh2 , χ1 i, h1 + h2 , χ1 χ2 ) is called the Heisenberg group of the group G (see, e.g., [How] or [Tayl2 , ch. 1] for more about it). Note that this group is non-commutative. Identity (1) means that the representations Sh and Ψχ can be extended naturally to the representation of Γ.

6.5. ALMOST PERIODIC OPERATORS

383

Example. Clearly, the usual representations ¡ ¢ Sh x (t) = x(t − h), ¡ ¢ Ψχ x (t) = χ(t)x(t),

h ∈ G, χ ∈ X(G),

on the spaces Lpq (G, E) and Cq (G, E) possess all the enumerated properties. Evidently, in the considered case S-invariant operators are shift invariant ones, see 4.1.2; and Ψ-invariant operators are oscillation invariant ones, see 5.1.10. If Sh is interpreted in the usual sense, we say simply that an operator is almost periodic without mentioning of the representation and employ the notation BAP instead of BS . In particular, the usual interpretation of Sh and Ψχ will be, as a rule, assumed in the references to the following theorem. For the dual interpretation, see 6.5.3 and 6.5.4 below. Theorem. For any T ∈ B(X, Y ) the following assumptions are equivalent. (a) The operator T is almost periodic with respect to the representations Sh . (b) The function h 7→ Sh T S−h , h ∈ G, possesses an extension to a continuous in norm function defined on Gb , see 5.1.7 for the definition of Gb . (c) The function h 7→ Sh T S−h , h ∈ G, is continuous in norm, and the set H(T ) = { Sh T S−h : h ∈ G } is conditionally compact in B(X, Y ). (d) The operator T can be approximated in norm by operators of the form PM m=1 Ψχm Tm , where χm ∈ X and Tm ∈ AS (X, Y ). (e) The operator T can be approximated in norm by operators of the form PM m=1 Tm Ψχm , where χm ∈ X and Tm ∈ AS (X, Y ). Suppose, in addition, that T belongs to a closed subspace R of B(X, Y ) such that R ∈ R implies Sh RS−h ∈ R, h ∈ G. Then the operators Ψχm Tm and Tm Ψχm in (d) and (e) can be chosen in R. Proof. Without loss of generality we may assume that kSh k = 1 for all h ∈ G. (a) ⇔ (b) is a special case of 5.1.9. (b) ⇒ (c) is evident. (c) ⇒ (b). This proof is similar to the corresponding part of the proof of 5.1.9. Clearly, for any h ∈ G the correspondence R 7→ Sh RS−h from B(X, Y ) into itself is an isometry and maps the set H = H(T ) onto itself. Therefore it maps the closure H of H onto itself, too. We stress that H is a compact metric space. Let us denote by A the compact group of all invertible isometric mappings A : H → H, with the metric %(A, B) = max{ kA(R) − B(R)k : R ∈ H }, see 5.1.8. For any h ∈ G we denote by Ah ∈ A the mapping from H to H which arises as the restriction of the mapping R 7→ Sh RS−h to H. We observe that the mapping R 7→ Ah (R) depends on h continuously for all R ∈ H. Indeed, for R ∈ H this assertion follows from assumption (c); to consider the case of an arbitrary R ∈ H it suffices to recall that Ah is an isometry. Next, arguing as in 5.1.9, it is easy to show that the mapping h 7→ Ah is continuous with respect to the metric % on H. We denote by K the closure of { Ah : h ∈ G } in A. Obviously, K is a compact abelian group and the mapping h 7→ Ah is a morphism of topological groups from

384

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

G to K. By 5.1.7 it possesses an extension to the morphism ge 7→ Age from Gb to K. We recall that Ah (R) = Sh RS−h . Therefore the function ge 7→ Age(T ), ge ∈ Gb , is the required extension of h 7→ Sh T S−h . (d) ⇒ (a) follows immediately from the identity (1). (d) ⇔ (e). From (1) it is easy to see that Tm ∈ AS implies Ψχ Tm Ψχ−1 ∈ AS for all χ ∈ X, cf. 5.2.1(a). Thus representations (d) and (e) are equivalent. (b) ⇒ (d). Let R denote the closure of the span of H = { Sh T S−h : h ∈ G } in B(X, Y ). Then the function g 7→ Sg T S−g , g ∈ G, can be considered as an almost periodic function with values in the Banach space R. By 5.1.9 it can be approximated by trigonometric polynomials with values in R, i.e., for any ε > 0 there exist χ1 , χ2 , . . . , χm ∈ X and R1 , R2 , . . . , Rm ∈ R such that M °X ° ° ° hg, χm iRm − Sg T S−g ° ≤ ε °

for all g ∈ G.

m=1

Since kSg k = 1 this estimate can be rewritten as M °X ° ° ° hg, χm iS−g Rm Sg − T ° ≤ ε °

for all g ∈ G

m=1

or, equivalently, M °X ° ° ° hg, χm iΨχm Ψ−1 S R S − T ° °≤ε χm −g m g

for all g ∈ G.

m=1

Then using (1) we can transform it as follows M °X ° ° ° Ψχm S−g Ψ−1 R S − T ° °≤ε χm m g

for all g ∈ G.

(2)

m=1

Next, we observe that since Rm ∈ R, without loss of generality we may assume PIm αmi Shmi T S−hmi or, simply, the that the operators Rm has the form Rm = i=1 form Rm = αm Shm T S−hm , where αm ∈ C and hm ∈ G. Then it is easy to see directly from the definition of an almost periodic operator that the operators −1 Ψ−1 χm Rm = αm Ψχm Shm T S−hm are almost periodic. Hence identity (2) can be extended to g ∈ Gb . Finally, we integrate (2) over Gb . After integration we obtain M °X ° ° ° Ψ T − T ° ° ≤ ε, χm m m=1

where

Z Tm = Gb

S−g Ψ−1 χm Rm Sg dg.

6.5. ALMOST PERIODIC OPERATORS

385

From the shift invariance of the Haar measure it is obvious that Sh Tm S−h = Tm , h ∈ G, i.e., Tm ∈ AS . (d, e) We observe that by (1) the definition of Tm can be rewritten as Z −1 Tm = Ψχm hg, χm iS−g Rm Sg dg, Gb

which shows that Ψχm Tm ∈ R ⊆ R. The consideration of the case (e) is similar and based on a simple modification of the reasoning (b) ⇒ (d). ¤ Remark. From the proof it is evident that the equivalence of (a), (b), and (c) holds without the assumption (1). 6.5.2. The subalgebra BAP is full Theorem. The set BAP (X, Y ) is a closed subspace of B(X, Y ). The product of almost periodic operators is almost periodic, too. The inverse of an almost periodic operator is almost periodic, too. In particular, the set BAP (X) is a closed full subalgebra of the algebra B(X). Proof. The proof follows easily from 6.5.1(b). ¤ 6.5.3. More about the algebra DΨ . Let G be a locally compact abelian group and E be a Banach space. Let L = L(G, E) be either Lpq (G, E) or Cq (G, E). Finally, let X = X(G) be the dual group. We recall from 5.2.9 that we denote by DΨ = DΨ (L) the closure in norm of the set DΨ f of all operators D ∈ B(L) of the form D=

M X

Am Shm ,

m=1

PM or, equivalently (see 5.2.1), of the form D = m=1 Shm Am , where Am ∈ XΨ and hm ∈ G. Clearly, DΨ (L) can be also defined as the closure of SΨ (L). We recall from 5.2.9 that DΨ (L) is a full subalgebra of B(L). The following corollary shows that DΨ can be defined to be the set of all almost periodic operators with respect to the representation Ψχ . Corollary. For any D ∈ B(L) the following assumptions are equivalent. (a) The function χ 7→ Ψχ DΨ−1 χ , χ ∈ X, is almost periodic in the sense of 5.1.9. (b) The function χ 7→ Ψχ DΨ−1 χ , χ ∈ X, possesses an extension to a continuous function defined on Xb . (c) The function χ 7→ Ψχ DΨ−1 χ , χ ∈ X, is continuous, and the set H(D) = −1 { Ψχ DΨχ : χ ∈ X } is conditionally compact in B(L). (d) The operator D can be approximated in norm by operators of the form PM m=1 Shm Am , where hm ∈ G and Am ∈ XΨ (L). (e) The operator D belongs to DΨ (L), i.e., it can be approximated in norm PM by operators of the form m=1 Am Shm , where hm ∈ G and Am ∈ XΨ (L).

386

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Suppose, in addition, that D belongs to a closed subspace R of B(L) such that R ∈ R implies Ψχ RΨ−1 χ , χ ∈ X. Then the operators Shm Am and Am Shm in (d) and (e) can be chosen in R. Proof. It suffices to observe that (a) means that D is almost periodic with respect to the representation Ψχ and refer to 6.5.1. ¤ 6.5.4. More about the algebra dψ . Let G be a locally compact abelian group and L = L(G, E) be one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We recall from 5.2.9¡ that fixing¢ d¯ ∈ d for each d ∈ D, and representing the space Lpq as lq (I) = lq I, Lp (Q, E) and the space Cq as lq (I) = lq (I, C♦ ) 5.1.12 for details) one can define the algebra dψ (L) as a special case of DΨ . Corollary. For any D ∈ B(L) the following assumptions are equivalent. −1 , κ ∈ X(I), is almost periodic in the sense of (a) The function κ 7→ ψκ Dψκ 5.1.9. −1 (b) The function κ 7→ ψκ Dψκ , κ ∈ X(I), possesses an extension to a continuous function defined on Xb (I). −1 , κ ∈ X(I), is continuous, and the set H(D) = (c) The function κ 7→ ψκ Dψκ −1 { ψκ Dψκ : κ ∈ X(I) }¡ is conditionally compact in B(L). ¢ (d) The operator D on lq I, Lp (Q, E) (respectively, on lq (I, C♦ )) can be apPM proximated in norm by operators of the form m=1 sjm Am , where jm ∈ I and Am ∈ xψ (L). ¡ ¢ (e) The operator D belongs to dψ (L), i.e., on lq I, Lp (Q, E) (respectively, on lq (I, C♦ )) it can be approximated in norm by operators of the form PM m=1 Am sjm , where jm ∈ I and Am ∈ xψ (L).

Suppose, in addition, that D belongs to a closed subspace R of B(L) such that −1 R ∈ R implies ψκ Rψκ , κ ∈ X(I). Then the operators sjm Am and Am sjm in (d) and (e) can be chosen in R. Proof. This is a special case of 6.5.3.

¤

6.5.5. The equality tAP = dAP . Let G be a locally compact abelian group and L = L(G, E) be one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), where 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by tAP (L) (respectively, by dAP (L)) the intersection of t(L) (respectively, d(L)) and BAP (L). Proposition. Let G be a locally compact abelian group, and let E be a Banach space, and let q be either 1, ∞, or 0. Then tAP (L) = dAP (L). Proof. We make use of 6.5.1. Assume T ∈ tAP . Clearly, Sh T S−h ∈ t, h ∈ G, too. Therefore by 6.5.1 the operator T can be approximated by a sum PM m=1 Ψχm Tm with Tm ∈ A(L) and Ψχm Tm ∈ t. Since Ψχm ∈ t we have Tm ∈ t and consequently Tm ∈ t∩A(L). It is straightforward to verify that A(L) ⊆ A(lq ), i.e., the shift¡ invariance¢ under ¡Sh : L → L, ¢ h ∈ G, implies the shift invariance under sj : lq I, Lp (Q, E) → lq I, Lp (Q, E) or sj : lq (I, C♦ ) → lq (I, C♦ ), j ∈ I, respectively. Thus Tm ∈ t ∩ A(lq ).

6.5. ALMOST PERIODIC OPERATORS

387

We approximate Tm by operators Rm ∈ tf . From the explicit formula for the norm of the operator on l1 and l∞ induced by a matrix (see 1.6.5) it is clear that without loss of generality we may assume that non-zero elements (Rm )ij of the matrix of Rm coincide with the corresponding elements (Tm )ij of the matrix of Tm . Moreover, since Tm ∈ A(lq ) we may assume that the matrix {(Rm )ij } has a finite number of non-zero diagonals and (Rm )ij = (Rm )i+h,j+h for all i, j, h ∈ I. Thus Rm ∈ df . Consequently T ∈ d. ¤ Remark. We show that in general t ∩ A * d. In particular, this implies that tAP 6= dAP , i.e., the above proposition does not hold. In the space l2 = l2 (Z, C) we consider the operator T induced by the matrix {aij } with the entries aij = 1/(i − j) if i 6= j, and aij = 0 if i = j. It can be shown that this matrix really induces a bounded operator. This is the operator of the convolution with the sequence ξn = 1/n if n 6= 0, and ξ0 = 0. We observe that the ˆ = (π − t)/2 for 0 ≤ t < 2π provided Fourier transform of {ξn } is the function ξ(t) we represent the dual group X(Z) as [0, 2π). If T were in d, it could be approximated by operators of the class df . Moreover, since both T and Ψχ T Ψ−1 χ belong to A, by the end of 6.5.1 T could be approximated by operators of the class df ∩ A. Clearly, any operator R ∈ df ∩ A is an operator of the convolution with a sequence with a finite (compact) support. The Fourier transform of such a sequence is a continuous function. But the function ξˆ is discontinuous on X(Z). Therefore it can not be approximated by a continuous function. Since the Fourier transform F : l2 (Z, C) → L2 (U, C) is an isometry (see, e.g., [Bou4 , ch. 2, §1, 3]) the operator T can not be approximated by operators of the class df ∩ A. Thus T ∈ / d. We show that T ∈ t. We fix ε > 0. For each k ∈ Z we choose nk ∈ N ¡P ¢ 2 1/2 such that 1/n < 2−|k| ε. Let Bk , k ∈ Z, be induced by the matrix |n|>nk {bij } = {bkij } defined as follows: bij = aij if i = k and ¯|i − j| > nk , and bij = ¯ 0 P ¯ ¯ otherwise. Clearly, (Bk x)i = 0 if i 6= k, and |(Bk x)k | = |k−j|>nk xj /(k − j) ≤ ¡P ¢ 2 1/2 kxkl2 ≤ 2−|k| εkxk. Thus {bij } actually induces an operator, |n|>nk 1/n and kBk k ≤ 2−|k| ε. In a similar way let Ck , k ∈ Z, be induced by the matrix {cij } = {ckij } defined as follows: cij = aij if j = k and |i − j| > nk , and cij = 0 otherwise. Clearly, (Ck x)i = 0 if |i − k| ≤ nk , and (Ck x)i = xk /(i − k) ¡P ¢ 2 1/2 otherwise. Evidently, kCk xk ≤ kxkl2 ≤ 2−|k| εkxk. Thus Ck ac|n|>nk 1/n tually an operator, and kCk k ≤ 2°−|k| ε. It is easy to ° see that Pinduces P∞ P∞ the operator ∞ T − −∞ (Bk + Ck ) belongs to tf . Since ° −∞ (Bk + Ck )° < 2 −∞ 2−|k| ε = 6ε and ε > 0 is arbitrary we obtain T ∈ t. 6.5.6. The ideal h. Let G be a locally compact abelian group and L = L(G, E) be one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by h = h(L) the subalgebra of d(L) consisting of all operators K whose matrix entries Kij are compact; and we denote by hf = hf (L) the intersection h ∩ df . Cf. the definition of k in 6.1.2.

388

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

We denote by hAP = hAP (L) the set of all almost periodic operators K ∈ h, i.e., the intersection of h(L) and BAP (L), and by hAP f = hAP f (L) the intersection of hf (L) and BAP (L). Remark. By the remark in 6.5.5, in general, k ∩ A * h (and, moreover, kAP 6= hAP ), see 6.1.2 for the definition of k. Proposition. (a) hf (L) is an ideal in df (L). (b) h(L) is an ideal in d(L). (c) hAP f (L) is an ideal in dAP f (L). (d) hAP (L) is an ideal in dAP (L). (e) hf (L) is dense in h(L). (f) Assume the representation G = Rc × V is fixed, and K ∈ hAP (L). Then e ∈ hf (L) such that K e is almost pefor any ε > 0 there exists an operator K c riodic with respect to the representation Sh , h ∈ Z × V (note Zc , not Rc !), e − Kk < ε. and kK Proof. (a), (b), (c), and (d) follow from 1.1.11 and 6.5.2. (e) We make use of the end of 6.5.4. We take for R the set h. For any κ ∈ X(I) we consider the operator (ψκ x)i = hi, κixi , i ∈ I, of the multiplication by κ. −1 ∈ h for all κ ∈ X(I) and K ∈ h. Therefore an operator Clearly, we have ψκ Kψκ PM K ∈ h can be approximated by operators m=1 Am sjm with Am ∈ xψ , jm ∈ I, and Am sjm ∈ h. Since sjm ∈ d, from (b) we have Am = (Am sjm )s−jm ∈ h. It remains to show that Am ∈ x. To this end we show that h ∩ xψ ⊆ x. Indeed, by 5.1.1(c) operators from h ⊆ d are hlq , lq0 i-continuous. But, by 5.1.11 and 5.1.4 hlq , lq0 i-continuous elements of xψ lie in x. (f) Assume K ∈ hAP ¡(L). In particular, this means K ∈ d(L) ⊆ dψ (L). We ¢ represent Lpq (G, E) as lq I, Lp (Q, E) and Cq (G, E) as lq (I, C♦ ) (recall that doing so we imply that we fix d¯ ∈ d for all d ∈ D, see 1.6.3, 1.6.10, and 6.5.4). By e ∈ dψ f , i.e., by 6.5.4(e) the operator K can be approximated by an operator K P M e = an operator of the form K m=1 Am sjm , where sjm : lq (I) → lq (I), jm ∈ I, and e possesses the desired Am ∈ xψ . We prove that one can choose Am sjm so that K properties. We denote by bAP the set of all operators T ∈ B(L) being almost periodic with respect to the representation Sh , h ∈ Zc × V (a part of the whole representation Sh , h ∈ G). Clearly, BAP ⊆ bAP ; thus K ∈ bAP . We apply the end of 6.5.4 taking for R the set h ∩ bAP . −1 Clearly, ψκ Kψκ ∈ h for all κ ∈ X(I). −1 Next, we show that ψκ Kψκ ∈ bAP for all κ ∈ X(I). To do this we shall employ the auxiliary identity (we postpone its verification till the end of the proof) hh, κiSh ψκ = ψκ Sh ,

h ∈ Zc × V, κ ∈ X(Zc × D)

(3)

−1 (note the difference: V, but D). From this identity we obtain Sh (ψκ Kψκ )S−h = −1 c c ψκ (Sh KS−h )ψκ for all h ∈ Z × V and κ ∈ X(Z × D). Therefore for any κ ∈

6.5. ALMOST PERIODIC OPERATORS

389

−1 X(Zc × D) the function h 7→ Sh (ψκ Kψκ )S−h , h ∈ Zc × V, is almost periodic since −1 the function h 7→ Sh KS−h , h ∈ Zc × V, is almost periodic. Thus ψκ Kψκ ∈ bAP . From the last two paragraphs it follow that the assumption from 6.5.4 for R = h ∩ bAP is satisfied. Now, by 6.5.4 we obtain that in the approximation e = PM Am sj we can take Am sj ∈ h ∩ bAP and Am ∈ xψ . K m m m=1 In the proof of (e) we have seen that the assumptions Am sjm ∈ h and Am ∈ xψ imply Am ∈ x. Consequently Am sjm ∈ df , which together with Am sjm ∈ h imply PM Am sjm ∈ hf . Thus we have Am sjm ∈ hf ∩ bAP . Hence m=1 Am sjm ∈ hf ∩ bAP as well, which completes the proof. We proceed to the proof of (3). We consider the more difficult case L = Cq . Note that more accurately by ψκ : Cq → Cq in (3) we mean Υ−1 ψκ Υ, where Υ : Cq → lq (I, C♦ ) is the isomorphism induced by the choice d¯ ∈ d. According to 4.2.1 we identify X(Zc × D) with X(Zc ) × X(D), and X(Zc × V) with X(Zc ) × X(V). We also recall that X(D) = X(V/K) can be considered as the subgroup K⊥ ⊆ X(V). Clearly, from (1) we have

hv, ηiS(0,v) Ψ(0,η) = Ψ(0,η) S(0,v) ,

v ∈ V, η ∈ X(V),

hk, uis(k,0) ψ(u,0) = ψ(u,0) s(k,0) ,

k ∈ Zc , u ∈ X(Zc ),

where (s(k,0) x)i = xi−(k,0) , i ∈ I, is the operator of the shift by (k, 0) on lq (I). Simple straightforward calculations (cf. 1.6.10) show that Ψ(0,η) = Υ−1 ψ(0,η) Υ for all η ∈ X(D) ⊆ X(V), and S(k,0) = Υ−1 s(k,0) Υ for all k ∈ Zc . We also note that for all u ∈ X(Zc ), η ∈ X(V), and v ∈ V (Υ−1 ψ(u,0) Υ)Ψ(0,η) = Ψ(0,η) (Υ−1 ψ(u,0) Υ), (Υ−1 ψ(u,0) Υ)S(0,v) = S(0,v) (Υ−1 ψ(u,0) Υ). ¯ (u,η) = (Υ−1 ψ(u,0) Υ)Ψ(0,η) . From what For u ∈ X(Zc ) and η ∈ X(V) we set Ψ has been proved, for all k ∈ Zc , u ∈ X(Zc ), v ∈ V, and η ∈ X(V) we have ¯ (u,η) = hk, ui · hv, ηiS(0,v) S(k,0) Ψ ¯ (0,η) Ψ ¯ (u,0) h(k, v), (u, η)iS(k,v) Ψ ¯ (u,0) = hk, ui · hv, ηiS(0,v) S(k,0) Ψ(0,η) Ψ ¯ (u,0) = hk, ui · hv, ηiS(0,v) Ψ(0,η) S(k,0) Ψ = hk, uiΨ(0,η) S(0,v) (Υ−1 s(k,0) Υ)(Υ−1 ψ(u,0) Υ) = Ψ(0,η) S(0,v) Υ−1 ψ(u,0) s(k,0) Υ ¯ (u,0) S(k,0) = Ψ(0,η) S(0,v) Ψ ¯ (u,0) S(0,v) S(k,0) = Ψ(0,η) Ψ ¯ (u,η) S(k,v) , =Ψ which contains (3) as a special case. The case L = Lpq is handled similarly. ¤

390

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

¡ ¢ 6.5.7. The case of hAP lq (Zn ) . Here we prove the main result of this section for the simplest group Zn . The basic idea appears just in this case. Moreover, further generalizations can be obtained by means of references to this result. Theorem. Let ¢E be a Banach space, and let 1 ≤ q ≤ ∞ or q = 0. Assume ¡ n K ∈ hAP lq (Z , E) . (a) If |1 + K|+ > 0 then the operator 1 + K is invertible. (b) If |1 + K|− > 0 then the operator 1 + K is invertible. Proof. Let us set ε = |1 + K|+ or ε = |1 + K|− , respectively. By 6.5.6(f) with e ∈ hAP f such that kK e − Kk < ε/2. c = 0 and V = Zn , we choose an operator K e + > ε/2 or |1 + K| e − > ε/2, respectively. If we show that 1 + K e By 1.3.5 |1 + K| is invertible then by 1.3.6 this will imply that 1 + K ¡is invertible, ¢ too. Thus it suffices to consider the case of the operator K ∈ hAP f lq (Zn , E) . (a) We break the argument into eight steps. PM (i) We represent the operator K in the form K = m=1 Am Shm . For i = (i1 , . . . , in ) ∈ Zn we set |i| = kikl∞ = max{|i1 |, . . . , |in |}. Let γ be the maximum PM of |hm |, where hm are taken from the representation K = m=1 Am Shm . We n n consider the subgroup γZ of Z consisting of all elements α = (γα1 , . . . , γαn ), α1 , . . . , αn ∈ Z, with the ‘norm’ |α| = kαkl∞ = max{|α1 |, . . . , |αn |}. Let Q = {0, 1, . . . , γ − 1}n (cf. 1.6.3) and Q¡α = γα + Q, α¢ ∈ Zn . It is easy to n see lq (Zn , E) ¡ that ¢ is naturally isomorphic to lq γZ , lq (Qα , E) or, equivalently, to n lq γZ , lq (Q, E) . It is straightforward to verify that Kαβ may be non-zero only if |α − β| ≤ 1. Thus the matrix of K on lq (γZn ) has at most 3n non-zero diagonals. Since K is almost periodic with respect to the representation Sh , h ∈ Zn , the operator K is a fortiori almost periodic with respect to the representation Sh¢, ¡ n n h ∈ γZ , i.e., K is almost periodic being considered on the space lq γZ , lq (Q, E) . Thus without loss of generality we may assume that a diagonal of the matrix of K may be non-zero only if it is next to the main one. (ii) We set T = 1 + K. For any η > 0 we consider the operator (cf. 3.4.1 and the proof of 5.5.5) (Φη x)i = (1 + η)−|i| xi , i ∈ Zn . Clearly, Φη maps l∞ into lq . We show that for all x ∈ l∞ (not only for x ∈ lq ) kT Φη x − Φη T xklq ≤ η T · kΦη xklq . Indeed, for any i ∈ Zn we have (T Φη x − Φη T x)i =

X |i−j|≤1

=

X

|i−j|≤1

=

X

|i−j|≤1

¡ ¢ Tij (1 + η)−|j| − (1 + η)−|i| xj ¡ ¢ Tij (1 + η)|i|−|j| − 1 (1 + η)−|j| xj ¡ ¢ Tij (1 + η)|i|−|j| − 1 (Φη x)j .

(4)

6.5. ALMOST PERIODIC OPERATORS

391

It remains to observe that |(1 + η)|i|−|j| − 1| ≤ (1 + η)|i−j| − 1 ≤ η when |i − j| ≤ 1. (iii) We show that |T : lq → lq |+ > 0 implies |T : l∞ → l∞ |+ > 0. Hence by virtue of 5.2.7 it suffices to restrict ourselves to the case q = ∞. Assume |T : l∞ → l∞ |+ < N , i.e., there exists x ∈ l∞ such that kxk = 1 and kT xk < N . Let us fix such an x. For the sake of definiteness, we assume that |x0 | > 1/2, where the index 0 is the zero of Zn . For any η > 0 we define Φη as in (ii), and consider y = Φη x. Clearly, y ∈ lq and kyk > 1/2. From (4) we have kT yklq = kT Φη xklq ≤ kΦη T xklq + η T · kΦη xklq = kΦη T xklq + η T · kyklq . We observe that kΦη T xklq ≤ k%klq · kT xkl∞ ≤ N k%klq , where %i = (1 + η)−|i| , i ∈ Zn . Therefore from the above estimate we obtain (we recall that kyklq > 1/2) |T : lq → lq |+ ≤ kT yklq /kyklq ≤ N k%klq /kyklq + η T ≤ 2N k%klq + η T . Taking the infimum over all N such that |T : l∞ → l∞ |+ < N we obtain |T : lq → lq |+ ≤ 2|T : l∞ → l∞ |+ · k%klq + η T or, equivalently, |T : l∞ → l∞ |+ ≥ (|T : lq → lq |+ − η T )/(2 · k%klq ). From this estimate it is clear that if η > 0 is small enough the inequality |T : lq → lq |+ > 0 implies |T : l∞ → l∞ |+ > 0. (iv) We fix an arbitrary ε > 0. We claim that since T = 1+K is almost periodic there exist arbitrary large natural numbers γ1 , . . . , γn such that kT − S±¯γi T S∓¯γi k < ε,

i = 1, . . . , n,

(5)

where γ¯1 = (γ1 , 0, 0, . . . , 0), . . . , γ¯2 = (0, γ2 , 0, . . . , 0), . . . , γ¯n = (0, 0, . . . , 0, γn ). Indeed, we take a sequence ik ∈ N such that ik+1 − ik → +∞. Let ¯ik = (ik , 0, 0, . . . , 0). By 6.5.1(c) the sequence k 7→ S¯ik T S−¯ik has a Cauchy subsequence. To simplify notation we shall assume that this subsequence coincides with the initial sequence. Then the sequence S¯ik+1 T S−¯ik+1 − S¯ik T S−¯ik tends to zero or, equivalently, the sequence S¯ik+1 −¯ik T S−¯ik+1 +¯ik tends to T . This proves the existence of γ1 . The existence of γ2 , . . . , γn is proved in a similar way. (v) Let γ1 , . . . , γn be as in (iv). We set γ¯ = (γ1 , . . . , γn ) and denote by γ¯ Zn the subgroup of Zn consisting of all elements of the form ξ¯ = (ξ1 γ1 , . . . , ξn γn ), where ¯ = kξk ¯ l = max{|ξ1 |, . . . , |ξn |}. From (5), ξ1 , . . . , ξn ∈ Z. For ξ¯ ∈ γ¯ Zn we set |ξ| ∞ for matrix entries of T we have the estimate kTαβ − Tα±¯γi ,β±¯γi k < ε,

i = 1, . . . , n,

392

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

which in turn implies kTαβ − Tα+ξ,β+ ¯ ξ¯k < 2nε,

¯ ≤ 2. ξ¯ ∈ γ¯ Zn , |ξ|

(6)

Let us consider the matrix {Rαβ } and the corresponding operator R defined as follows. We set Rαβ = Tαβ , −γi < αi ≤ γi . (7) And for all other α we define Rαβ from the 2¯ γ Zn -periodicity condition: Rαβ = Rα+ξ,β+ ¯ ξ¯,

ξ¯ ∈ 2¯ γ Zn .

(8)

(We stress that R depends on ε from (5) and γ1 , . . . , γn .) Note that Rαβ can be non-zero only if |α − β| ≤ 1. It is also clear that R ≤ T . We denote by l2¯γ the subspace of l∞ consisting of all 2¯ γ -periodic families x, n ¯ i.e., satisfying S2ξ¯x = x for all ξ ∈ γ¯ Z . Clearly, R acts on l2¯γ . (vi) We show that if ε > 0 is small enough and γ1 , . . . , γn are large enough then |R : l2¯γ → l2¯γ |+ >

1 |T : l∞ → l∞ |+ . 2

We define the family % = { %α ∈ [0, 1] : α ∈ Zn } as follows. We set %α = 1 for |α|i ≤ γi , %α = 0 for |α|i ≥ 2γi , and %α = max{2 − |α1 |/γ1 , . . . , 2 − |αn |/γn } for all other α. It is easy to see that |%α −%β | ≤ η|α−β|, where η = max{1/γ1 , . . . , 1/γn }. Let x ∈ l2¯γ , kxk = 1. We consider the family y = %x. Since x is 2¯ γ -periodic we have kykl∞ = 1. It is easy to show that (cf. (ii)) kRyk = kR%xk ≤ k%Rxk + η R · kxk

(9)

≤ kRxk + η T . We compare Ry and T y. Since Rαβ = 0 and Tαβ = 0 for |α − β| > 1, and yα = 0 for |αi | ≥ 2γi we have (Ry)α = 0 and (T y)α = 0 for |αi | > 2γi . Next, from (6) and (7) it follows that kRαβ − Tα+ξ,β+ ¯ ξ¯k < 2nε,

¯ ≤ 2; −γi < αi ≤ γi . ξ¯ ∈ γ¯ Zn , |ξ|

Applying (8) we obtain ¯ ≤ 2; −γi < αi ≤ γi . ξ¯ ∈ γ¯ Zn , |ξ|

kRα+ξ,β+ ¯ ¯ ξ¯ − Tα+ξ,β+ ξ¯k < 2nε, Thus kRα,β − Tα,β k < 2nε,

−3γi < αi ≤ 3γi .

Consequently, kRy − T yk ≤ kR − T k · kyk ≤ R − T · kyk ≤ 3n 2nεkyk,

6.5. ALMOST PERIODIC OPERATORS

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which implies (we recall that kyk = 1) |T : l∞ → l∞ |+ ≤ kT yk ≤ kRyk + kRy − T yk ≤ kRyk + 3n 2nεkyk. Finally, substituting in here the estimate (9) we obtain |T : l∞ → l∞ |+ ≤ kT yk ≤ kRyk + 3n 2nεkyk ≤ kRxk + η T + 3n 2nεkyk. Taking the infimum over all x ∈ l2¯γ , kxk = 1 (we recall that kykl∞ = 1), we arrive at the conclusion that |T : l∞ → l∞ |+ ≤ |R : l2¯γ → l2¯γ |+ + η T + 3n 2nε. Now, if η and ε are sufficiently small we obtain the desired estimate. (vii) From the definitions of R and T it is clear that R : l2¯γ → l2¯γ is the sum of the unit operator and a compact operator. Therefore from (vi) and 1.3.12 it follows that the operator R on l2¯γ is invertible with (see 1.3.2) kR−1 k ≤ 2/|T : l2¯γ → l2¯γ |+ . (viii) We show that the image of T : l∞ → l∞ coincides with l∞ . This will finish the proof. We take an arbitrary f ∈ l∞ . Then we take an arbitrary sequence εk → +0 and consider the corresponding sequences γ1k , . . . , γnk from (iv). We denote the corresponding operators R from (v) by Rk . We consider the family f k ∈ l2¯γ k which coincides with f for γik < αi ≤ γik . Clearly, kf k k ≤ kf k. We set xk = (Rk )−1 f k . By (vii) we have the estimate kxk k ≤ 2kf k/|T : l2¯γ → l2¯γ |+ . On the other hand, by (7) we have (T xk )α = (Rk xk )α ,

−γik < αi ≤ γik .

By assumption Rk xk = f k . But fαk = fα for −γik < αi ≤ γik . Therefore (T xk )α = fα ,

−γik < αi ≤ γik .

Thus we see that the sequence T xk converges to f in pointwise sense. We recall that T = 1 + K, where K ∈ hf . Since the sequence xk is bounded we can choose a subsequence z m = xkm such that Kz m converges pointwise. Then from the representation z m = T z m − Kz m it follows that the sequence z m also converges pointwise. Let x be the pointwise limit of z m . It is easy to see that x + Kx = f . Thus f ∈ Im T . (b) If q = ∞ we change it to q = 0. According to 5.5.5 and 5.5.6 this involves no loss of generality. Clearly, the conjugate operator K 0 belongs to hAP f , too. On the other hand, by 1.3.4 |1 + K 0 |+ = |1 + K|− > 0. By what has been proved, the operator 1 + K 0 is invertible. Hence 1 + K is invertible, too.

394

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

6.5.8. The case of hAP (L). Let G be a locally compact abelian group and L = L(G, E) be one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Theorem. Let K ∈ hAP (L). (a) If |1 + K|+ > 0 then the operator 1 + K is invertible. (b) If |1 + K|− > 0 then the operator 1 + K is invertible. Proof. We take a representation of G in the form G = Rc × V (see 1.6.1). We break the argument into eight steps. (i) Let us set ε = |1 + K|+ or ε = |1 + K|− , respectively. By 6.5.6(f) we e ∈ hf such that kK e − Kk < ε/2 and K e is almost periodic choose an operator K c e + > ε/2 or with respect to the representation Sh , h ∈ Z × V. By 1.3.5 |1 + K| e − > ε/2, respectively. If we show that under this assumption 1 + K e is |1 + K| invertible then by 1.3.6 this will imply that 1 + K is invertible, too. Hence it suffices to consider the case of the operator K ∈ hf (L) which is almost periodic with respect to the representation Sh , h ∈ Zc × V. PM (ii) According to (i) we represent K in the form K = m=1 Am sjm , where jm ∈ I and Am ∈ x. Next, we represent each jm in the form jm = (km , dm ) ∈ Zc ×D and denote by DJ the subgroup of D generated by the set { d1 , d2 , . . . , dM }. We denote by VJ ⊆ V the pre-image of DJ ⊆ D under the natural projection from V to D = V/K. Then we consider the groups GJ = Rc × VJ and J = Zc × DJ . Next, e j = (k + (0, 1]c ) × n and for each j = (k, n) ∈ Zc × DJ = J we consider the set Q e j , E), cf. 1.6.3; and in a similar way we consider the space C♦j , the space Lp (Q cf. 1.6.10. We note that since the groups Rc , Zc and K associated with G and e j , E) and Lp (Q e i , E), and GJ are the same, we may not distinguish the spaces Lp (Q C♦j and C♦i , j ∈ J, i ∈ I, with j = i, respectively. We set W = V/VJ ' D/DJ ' G/GJ . For any function x : G → E we consider the family of all its restrictions xw : w → E, w ∈ W. In each w ∈ W we fix a point w ¯ ∈ w. Then we transform each xw to the function x ¯w (t, v) = xw (t, v − v + VJ ) defined on VJ ; here v + VJ is the fixed element in the class v + VJ ∈ W. It is easy to see that the Ξ : x 7→ {¯ xw } defines the ¡ correspondence ¢ ¡ isomorphisms ¢ from Lpq (G, E) onto lq W, Lpq (GJ , E) and from Cq (G, E) onto lq W, Cq (GJ , E) . (iii) By the definitions of DJ and VJ the operator K on lq (W) is induced by a main diagonal matrix. We denote by Kww , w ∈ W, its main diagonal entries. Obviously, if 1 + K is invertible then the operators 1 + Kww are invertible, too. Conversely, assume all main diagonal entries 1 + Kww are invertible. Clearly, |1 + Kww |+ ≥ |1 + K|+ or |1 + Kww |− ≥ |1 + K|− , respectively. On the other hand, by 1.3.2 k(1 + Kww )−1 k = 1/|1 + Kww |± . Therefore the operators 1 + Kww are uniformly invertible. Consequently the operator 1+K is invertible itself. Thus it suffices to prove that each of operators 1 + Kww is invertible. (iv) It is easy to see that the isomorphism Ξ takes the operators Sh of the shift by h ∈ Zc × VJ on Lpq (G, E) (respectively, on Cq (G, E)) to the operators on lq (W) induced by main diagonal matrices whose all main diagonal entries (Sh )ww equal to Sh : Lpq (GJ , E) → Lpq (GJ , E) (respectively, Sh : Cq (GJ , E) → Cq (GJ , E)).

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Therefore from the almost periodicity of K with respect to the representation Sh , h ∈ Zc × V, it follows the almost periodicity of all Kww with respect to the representation Sh , h ∈ Zc × VJ . Thus Kww satisfy the same assumption as K in (i) to within G changed to GJ . Taking into account (iii) we may forget about the initial group G and pass from the consideration of the operator K to the consideration of the operator Kww . To simplify notation we shall abbreviate Kww to K and omit the index J. To unload the reader’s mind, we enumerate the facts we shall use below. The operator K belongs to hf (L) and is almost periodic with respect to the represenPM tation Sh , h ∈ Zc × V; K is represented in the form K = m=1 Am sjm , where jm = (km , dm ) ∈ I = Zc × D and Am ∈ x; and the group D is generated by the set { d1 , d2 , . . . , dM }. (v) By assumption D is finitely generated. According to 1.6.1(c) we represent D in the form D = Zb × F, where b is a non-negative integer, and F is a finite group. We denote by K0 the pre-image of the group {0} × F ⊆ Zb × F = D under the natural projection from V to D = V/K. Clearly, K0 is both compact and open (as K is). Hence we can change K to K0 and turn to the group D0 = V/K0 . Then PM from the representation K = m=1 Am sjm we can pass to the representation PM 0 0 0 0 , where j K = m=1 Am sjm m = (km , dm + K ). (We note that such a change of the group D implies the change of the norm on L by equivalent.) Now the group D0 is exactly isomorphic to Zb . Thus without loss of generality we may assume that the group D is isomorphic to Zb . In particular, it follows that I = Zc × Zb . (vi) We show that V can be identified with Zb × K (we recall that we have only V/K ' Zb ). From (v) we have D ' Zb . Let d1 , . . . , db ∈ D be generators of D. ¯ in V We fix arbitrary elements d¯1 ∈ d1 , . . . , d¯b ∈ db . Clearly, the subgroup D generated by the set { d¯1 , d¯2 , . . . , d¯b } is isomorphic to D ' Zb . For each d ∈ D ¯ corresponding to d ∈ D with respect to this we denote by d¯ ∈ d the element of D ¯ ∩ K = {0} and D ¯ + K = V. Thus V ' D ¯ × K. isomorphism. It is easy to see that D ¡ ¢ e E) and Υ : Cq (G, E) → lq (I, C♦ ) be the (vii) Let Υ : Lpq (G, E) → lq I, Lp (Q, isomorphisms induced by the fixed above choice d¯ ∈ d. Below we shall show that −1 s(k,d) = ΥS(k,d) ¯Υ

for all (k, d) ∈ Zc × D = I.

¡ ¢ e E) From this identity it is easy to see that the operator K considered on lq I, Lp (Q, or on lq (I, C¡ ♦ ) is almost ¢ periodic with respect to the representation si , i ∈ I. Thus K ∈ hAP f lq (Zn , E) . Now a reference to 6.5.7 completes the proof. (viii) So we verify the identity s(k∗ ,d∗ ) = ΥS(k∗ ,d¯∗ ) Υ−1 , k∗ ∈ Zc , d∗ ∈ D. As¡ ¢ e E) sume L = Lpq . We observe that the isomorphism Υ : L (G, E) → l I, L ( Q, pq q p ¡ c ¢ c c or, in detailed form, Υ : L (R ×V, E) → l Z ×D, L ((0, 1] ×K, E) has the form pq q p ¡ ¢ ¡ ¢ Υx (k, r; d, ζ) = x τ (k, r; d, ζ) , where τ : Zc × (0, 1]c × D × K → Rc × V is defined ¯ by the rule τ (k, r; d, ζ) = (k + r, d¯+ ζ). Clearly, τ −1 (t, v) = ([t], t − [t]; v + K, v − d), where [t] = ([t1 ],¡[t2 ], . . . , [t¢c ]) and [ti¡] is the ¢integer part of ti ∈ R, and d = v + K. In a similar way S(k∗ ,d¯∗ ) x (t, v) = x σ(t, v) , where σ : Rc ×V → Rc ×V is defined by the rule σ(t; v) = (t − k∗ ; v − d¯∗ ). It is easy to see that ¡ ¢ ¡ ¢ ΥS(k∗ ,d¯∗ ) Υ−1 x (k, r; d, ζ) = x τ −1 ◦ σ ◦ τ (k, r; d, ζ) .

396

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

Straightforward calculations show that τ −1 ◦ σ ◦ τ (k, r; d, ζ) = (k − k∗ , r; d − d∗ , ζ) because d − d∗ = d¯ − d¯∗ . This means that ΥS(k∗ ,d¯∗ ) Υ−1 is the operator s(k∗ ,d∗ ) . The case of L = Cq is handled in a similar way, but an explicit formula for Υ is unwieldy (cf. 1.6.10). So we omit the relevant calculations. ¤ Remark. (a) It is interesting to note that the direct implementation of the proof of the theorem (instead of the reference to 6.5.7) for the case of the spaces Lp causes the usage of the spaces Lp∞ . (b) From the analysis of the proof one can see that the assumption on the almost periodicity of K can be weakened. Namely, it suffices that K being almost periodic with respect to the representation Sh , h ∈ Zc × V. For example, the operator A of the multiplication on Lpq (R, C) by a coefficient a ∈ L∞ is almost periodic with respect to the representation Sh , n ∈ Z, if a is 1-periodic; whereas the almost periodicity with respect to the whole representation Sh , h ∈ R, implies the continuity of a. 6.5.9. Almost periodic difference operators. Let L be either Lpq (G, E) or Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. We denote by DAP = DAP (L) the set of all almost periodic operators from D(L), i.e., the intersection of D(L) and BAP (L); and we denote by DAP f = DAP f (L) the set of all almost periodic operators belonging to Df (L), i.e., the intersection of Df (L) and BAP (L). Proposition. (a) DAP f (L) consists of all operators of the form M X ¡ ¢ Dx (t) = am (t)x(t − hm ),

(10)

m=1

where am : G → B(L) are almost periodic functions. (b) DAP f (L) is dense in DAP (L). Proof. (a) From 5.6.8(a) it follows that the coefficients am are continuous. We recall that the operator Sh DS−h has the form ¡

M X ¢ Sh DS−h x (t) = am (t − h)x(t − hm ). m=1

Thus ¡

M X ¢ ¡ ¢ (Sh DS−h − D)x (t) = am (t − h) − am (t) x(t − hm ). m=1

By assumption the set { Sh DS−h : h ∈ G } is conditionally compact. Therefore by 5.2.4(b) the set { Sh am : h ∈ G } is conditionally compact, too. Hence by 5.1.9 the coefficients am are almost periodic. The opposite assertion is evident. (b) Assume D ∈ DAP (Lpq ). We make use of the end of 6.5.3, cf. the proof of 6.5.6. We observe that the operators Ψχ DΨ−1 χ , χ ∈ X, belong to BAP . Indeed,

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−1 by (1) we have Sh (Ψχ DΨ−1 χ )S−h = Ψχ (Sh DS−h )Ψχ . Therefore the function h 7→ Sh (Ψχ DΨ−1 χ )S−h is almost periodic provided the function h 7→ Sh DS−h is almost periodic. On the other hand, evidently, Ψχ DΨ−1 χ ∈ D for all χ ∈ X. Thus we may set R = D ∩ BAP = DAP . By 6.5.3, since D is almost periodic with respect to the representation Ψχ , D PM can be approximated by operators of the form T m=1 Am Shm , where Am Shm ∈ DAP (Lpq ) and Am ∈ XΨ . Hence Am ∈ DAP XΨ . It remains to show that actually Am ∈ X. Indeed, by 5.1.1(c), operators from D are hLpq , Lp0 q0 i-continuous; and, by 5.1.11 and 5.1.6 hLpq , Lp0 q0 i-continuous elements of XΨ lie in X. e ∈ DAP f (Lp∞ ) of the extension of D to If D ∈ DAP (Cq ) an approximation D Lp∞ (see 5.6.8 and 5.6.9) is an approximation of D on Cq , too. ¤

6.5.10. The case of DAP . Let G be a locally compact abelian group and L = L(G, E) be one of the spaces Lpq = Lpq (G, E) or Cq = Cq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Theorem. Let E be a finite-dimensional Banach space and D ∈ DAP (L). (a) If |D|+ > 0 then the operator D is invertible. (b) If |D|− > 0 then the operator D is invertible. Proof. By 6.5.9, without loss of generality we may assume that D ∈ DAP f , i.e., D has the form (10). (a) We break the argument into seven steps. (i) We show that without loss of generality we may assume that L = Lpq with q 6= ∞. Indeed, by 5.6.11 we can pass from the assumption |D : Cq → Cq |+ > 0 to the assumption |D : L∞q → L∞q |+ > 0. In turn, by 5.5.5 we can pass to the case q 6= ∞. Now suppose that we have succeed to prove that D : L∞q → L∞q , q 6= ∞, is invertible. Then by 5.6.10 we would obtain the invertibility on the initial space. (ii) Let H denote the subgroup of G generated by the set { h1 , h2 , . . . , hM } (see (10)). We consider the quotient group W = G/H with the discrete topology. ¢ ¡ Te For any w ∈ W we denote by lpq = lpq (w, E) the space lq I, lp (w Q i , E) . In more detail, lpq (w, E) is defined as follows. For a family y = { yt ∈ E : t ∈ w } and Te i ∈ I we denote by kyklp (Qe i ) the lp -norm of the subfamily { yt ∈ E : t ∈ w Q i }. Then we define kyk = kyklpq to be the lq -norm of the family i 7→ kyklp (Qe i ) . We define lpq (w, E) to be the set of all families y = { yt ∈ E : t ∈ w } bounded by the norm k · klpq . Clearly, lpq is a Banach space. On each of the spaces lpq (w, E) we consider the operator ¡

Dw y

¢ t

=

M X

am (t)yt−hm ,

t ∈ w.

m=1

It is easy to see that for the operators (Am y)t = am (t)yt and (Shm y)t = yt−hm on lpq , one has kAm k ≤ kam kC and kShm k ≤ 2c . Hence the operator Dw actually acts on lpq and is bounded. (iii) We prove an auxiliary estimate, cf. the step (ii) of the proof of 6.5.7. We fix a point w ¯ ∈ w and set H0 = {w} ¯ and Hk+1 = Hk + H, k ∈ N, where

398

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

H = { 0, h1 , −h1 . . . , hM , −hM }, see (10). By the definitions of H and w the union of the sets Hk is equal to w. We say that the order of a point t ∈ w (with respect to H) is equal to k(t) if t ∈ Hk \ Hk−1 (in particular, k(t) = 0 if t ∈ H0 ). We take η > 0. We set %t = (1 + η)−k(t) , t ∈ w, and (Φη y)t = %t yt . By the definition of k(·) we have |%t−hm − %t | ≤ %t−hm¯ (1 + η)|k(t)−k(t−hm )| ≤ %t−hm η ¯for PM all m. Therefore |(Dw Φη y)t − (Φη Dw y)t | = ¯ m=1 (%t−hm − %t )am (t)yt−hm ¯ ≤ PM η m=1 |am (t)%t−hm yt−hm |. Hence for all y ∈ l∞ (not only for y ∈ lpq ), kDw Φη y − Φη Dw yklpq ≤ η Dw · kΦη yklpq , PM where Dw = m=1 kAm Shm : lpq → lpq k. (iv) We show that for all w ∈ W, |Dw : lpq → lpq |+ ≥ |D : Lpq → Lpq |+ . We take an arbitrary ε > 0 and choose y ∈ lpq (w, E) such that kyklpq = 1 and kDw yklpq < |Dw |+ + ε. First we observe that without loss of generality we may assume that the support of y is finite. Indeed, if p 6= ∞, this fact is evident because q 6= ∞ as well. We consider the case p = ∞. From (iii) we have kDw Φη yk ≤ kΦη Dw yk + η Dw ≤ |Dw |+ + ε + η Dw . On the other hand, Φη y converges to y in pointwise sense as η → 0. This implies that kΦη yklpq tends to kyklpq = 1. Hence we may change y to Φη y and then change Φη y to a family with a compact support. ¡ So ¢we assume that ys 6= 0 only for s ∈ { s1 , s2 , . . . , sK }. With this notation Dw y t = 0 for all t 6= sk + hm . Let t ∈ w be a point of the form sk + hm . Let sk1 + hm1 , sk2 + hm2 , . . . , skN + hmN be all distinct representations of t in the form sk + hm with different sk and hm . Note that for such a t we have N X ¡ ¢ Dw y t = amn (t)yskn . n=1

Let neighbourhood of the zero of G. Let us consider the set ¡ V be ca compact ¢ E = (−1, 0] × K ∩ V . Clearly, it is not null. We denote by ϑE its characteristic function 1E multiplied by a scalar factor such that kϑE kLp = 1. We consider the function K X x(t) = xE (t) = ysk ϑE (t − sk ). m=1

We assume that V is taken so small that the sets sk + hm + E ⊆ sk + hm + V , k = 1, 2, . . . , K and m = 1, 2, . . . , M (on which the functions t 7→ ϑE (t − sk − hm ) are non-zero) are pairwise disjoint (provided sk + hm 6= sk0 + hm0 ), and each set e i , i ∈ I, only. In this case, sk + hm + E is contained entirely in one of the sets Q evidently, we have kxkLpq = kyklpq = 1.

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Clearly, ¡

¢

Dx (t) =

M X m=1

am (t)

K X

ysk ϑE (t − sk − hm ).

k=1

¡ ¢ It is obvious that Dx (t) = 0 if t ∈ / sk + hm + E for all k and m. We consider t such that t ∈ sk∗ + hm∗ + E for some k∗ and m∗ . Let sk1 + hm1 , sk2 + hm2 , . . . , skN + hmN be all distinct representations of sk∗ + hm∗ in the form sk0 + hm0 with various sk0 and hm0 . Then ϑE (t − sk − hm ) 6= 0 only if (sk , hm ) is (sk1 , hm1 ), (sk2 , hm2 ), . . . , or (skN , hmN ). Hence for the considered t we have N X ¡ ¢ Dx (t) = amn (t)yskn ϑE (t − skn − hmn ) n=1

=

N ³X

´ amn (t)yskn ϑE (t − sk∗ − hm∗ ).

n=1

Since the coefficients am of D are continuous and V is small the number kDxkLpq is close to kDw yklpq . In particular, we may take V so small that kDxkLpq is less than kDw yklpq + ε. Then kDxkLpq < |Dw |+ + 2ε. Now since kxk = 1 and ε > 0 is arbitrary we can state that |D|+ ≤ |Dw |+ . (v) We show that |Dw : lpq → lpq |+ > 0 implies |Dw : l∞ → l∞ |+ > 0, cf. the step (iii) of the proof of 6.5.7. Assume |Dw : l∞ → l∞ |+ < N . We take y ∈ l∞ such that kyk = 1 and kDw yk < N . For the sake of definiteness, we assume that |yw¯ | > 1/2, where w ¯ is the point fixed on the step (iii). For any η > 0 we define Φη as in (iii) and consider z = Φη y. Clearly, z ∈ lpq and kzk > 1/2. From (iii) we have kDw zklpq ≤ kΦη Dw yklpq + η Dw · kΦη yklpq = kΦη Dw yklpq + η Dw · kzklpq . We observe that kΦη Dw yklpq ≤ k%klpq · kDw ykl∞ ≤ N k%klpq ≤ N k%kl1 . From these two estimates we obtain (we recall that kzklpq > 1/2) |Dw : lpq → lpq |+ ≤ kDw zklpq /kzklpq ≤ 2N k%kl1 + η Dw . Taking here the infimum over all N such that |Dw : l∞ → l∞ |+ < N we obtain |Dw : l∞ → l∞ |+ ≥ (|Dw : lpq → lpq |+ − η Dw )/(2 · k%kl1 ). From this estimate it is clear that if η > 0 is small enough the inequality |Dw : lpq → lpq |+ > 0 implies |Dw : l∞ → l∞ |+ > 0. Moreover, we can state that |Dw : l∞ → l∞ |+ > δ, where δ > 0 is independent of w ∈ W. (vi) We claim that all the operators Dw : l∞ → l∞ , w ∈ W, are invertible. For the proof we can not apply 6.5.7 since H may be not isomorphic to Zn . But

400

VI. DIFFERENTIAL DIFFERENCE EQUATIONS

we can apply 6.5.8, implying that G in 6.5.8 is discrete. From 1.3.2 we have −1 kDw : l∞ → l∞ k ≤ 1/δ. By 5.2.12 the operators Bw = (Dw )−1 have the form X ¡ ¢ Bw y t = bh (t)yt−h , t ∈ w, (11) h∈H

with

kbh (t)k ≤ N e−η|h|

for some N and η > 0. The analysis of the proof of 5.2.12 shows easily that N and η can be chosen independently of w. Identity (11) determines functions bh on the whole of G. We show that the functions bh : G → B(E) are continuous. Indeed, we observe that the operator ¡ ¢ Sτ x (t) = x(t − τ ), τ ∈ G, maps l∞ (w, E) onto l∞ (w + τ, E). Hence the operators M X ¡ ¢ Sτ Dw−τ S−τ y t = am (t − τ )yt−hm ,

t ∈ w,

m=1

act on l∞ (w, E) for all τ ∈ G. Clearly, these operators depend on τ continuously in norm. Therefore their inverses X ¡ ¢ Sτ Bw−τ S−τ y t = bh (t − τ )yt−h , t ∈ w, h∈H

depend on τ continuously, too. By 5.2.4(b) it follows the continuity of bh . (vii) Let us consider the operator X ¡ ¢ Bx (t) = bh (t)x(t − h), t ∈ G. h∈H

Clearly, it acts on Lpq . It is easy to see that BD = DB = 1. Thus D is invertible. (b) Consider the conjugate operator D0 : L0 → L0 . By 1.3.4 |D0 : L0 → L0 |+ > 0. Assume L = Lpq . It is straightforward to verify that the operator D0 on the subspace Lp0 q0 ⊆ (Lpq )0 (see 1.8.1) is defined by the formula (cf. (10)) M X ¡ 0 ¢ D z (t) = a0m (t)z(t + hm )

(12)

m=1

and, in particular, Lp0 q0 is invariant under D0 . Clearly, |D0 : Lp0 q0 → Lp0 q0 |+ > 0. By (a) D0 : Lp0 q0 → Lp0 q0 is invertible. By 5.6.10 D0 is invertible on L1 . Consequently D00 = D is invertible on L∞ and, again by 5.6.10, on Lpq . Assume L = Cq . Then L1q0 is embedded isometrically in Cq0 (see 1.8.12) and D0 : L1q0 → L1q0 is defined by the formula (12). A simple modification of the above argument shows that D is invertible. ¤ 6.5.11. The case of DAP + hAP . Let G be a locally compact abelian group and Lpq = Lpq (G, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Theorem. Let E be finite-dimensional. Assume that D ∈ DAP (Lpq ) and K ∈ hAP (Lpq ). (a) If |D + K|+ > 0 then the operator D + K is invertible. (b) If p 6= ∞ and |D + K|− > 0 then the operator D + K is invertible.

6.5. ALMOST PERIODIC OPERATORS

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Proof. (a) We take a representation of G in the form G = Rc × V (see 1.6.1). We break the further argument into seven steps. (i) Our main goal will be the proof of the invertibility of D. Indeed, suppose that the invertibility of D is established. Then we represent the operator D + K in the form D + K = D(1 + D−1 K). By 5.2.9(c) D−1 ∈ D, and by 6.5.2 D−1 ∈ BAP . Thus D−1 ∈ DAP . Clearly, DAP ⊆ dAP . Therefore by 6.5.6(d) D−1 K ∈ hAP . By 1.3.7, from |D + K|+ > 0 it follow that |1 + D−1 K|+ > 0. Thus by 6.5.8 the operator 1 + D−1 K is invertible. Consequently D + K is invertible, too. Hence to complete the proof it suffices to show that D is invertible. e ∈ DAP f such (ii) We set ε = |D + K|+ . By 6.5.9 we choose an operator D e − Dk < ε/4. Next, by 6.5.6 we choose an operator K e ∈ hf such that that kD e − Kk < ε/4 and K e is almost periodic with respect to the representation Sh , kK e + K| e + > ε/2. If we show that D e +K e is invertible then h ∈ Zc × V. By 1.3.5 |D by 1.3.6 this will imply that D + K is invertible, too. Hence ¡ ¢ it suffices to consider the case where D ∈ DAP f and K ∈ hf Lpq (Rc × V, E) is almost periodic with respect to the representation Sh , h ∈ Zc × V. Moreover, we shall use only the almost periodicity of D with respect to the representation Sh , h ∈ Zc × V. We denote by bAP the set of all operators T ∈ B(L) being almost periodic with respect to the representation Sh , h ∈ Zc × V. Clearly, D, K ∈ bAP . (iii) By (ii), D and K belong to df . By the definition of df we can represent PM PM them in the forms D = m=1 Dm sjm and K = m=1 Km sjm , where jm ∈ I and Dm , Km ∈ x (without loss of generality we assume that the set {sjm } is common). Arguing as in the proof of 6.5.8, without loss of generality we may assume that the subgroup of D generated by the set { d1 , d2 , . . . , dM } coincides with the whole c b group D; and V = Zb × K (in particular, G = Rc ¡× Zb × K and ¢ I = Z ×Z ; n we set n = c + b); we may represent Lpq (G, E) as lq Z , Lp (Q, E) in such a way that the operators D and K remain to be almost periodic with respect to the representation si , i ∈ Zn . (iv) Arguing as in the step (i) of the proof of 6.5.7, without loss of generality we may assume that Dαβ and Kαβ , α, β ∈ Zn , may be non-zero only if |α − β| ≤ 1; here |α| = kαkl¡∞ = max{|α1¢|, . . . , |αc |}. (We note that this causes the change of the norm on lq Zn , Lp (Q, E) to equivalent.) ¡ n ¢ Again, arguing as in the proof of 6.5.7 we pass from the space l Z , L (Q, E) q p ¡ ¢ to the space l∞ Zn , Lp (Q, E) with the estimate |D + K : l∞ → l∞ |+ > 0. Then as in the proof of 6.5.7, for large natural numbers ¡ n γ1 , . . . , γ¢n and γ¯ = (γ1 , . . . , γn ) we consider the subspace l2¯γ of the space l∞ Z ,¡Lp (Q, E) consisting¢ of all 2¯ γ -periodic families. We interpret l2¯γ as the space l∞ Zn /2¯ γ Zn , Lp (Q, E) on the compact (actually finite) group Zn /2¯ γ Zn . We construct the operators D2¯γ and K2¯γ by the operators D and K in the same way as the operator R has been constructed by T on the step (v) of the proof of 6.5.7. We recall that |D2¯γ + K2¯γ : l2¯γ → l2¯γ |+ >

1 |D + K : l∞ → l∞ |+ . 2

(v) We show that |(D2¯γ + K2¯γ )# |+ ≥ 21 |D2¯γ + K2¯γ |+ , where # is defined as in 6.1.4. We abbreviate D2¯γ + K2¯γ to R and l2¯γ to l.

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Assume |R# |+ < N . This means that there exists Z ∈ lV , kZ + lK kl# = 1, such that kRV Z + lK kl# < N , i.e., kRV Z + Y klV < N for some Y ∈ lK . Since the group Zn /2¯ γ Zn is compact the assumption Y ∈ lK means that the family Y = { yV ∈ X : V ∈ V } is conditionally compact. Hence it can be changed to a family consisting of a finite number of distinct points with kRV Z + Y klV < N preserved. Thus the set RV Z can be covered by a finite collection of balls of radius N centred at the points yV . Unfortunately, it is unknown whether the centres of these balls belong to Im R. Therefore we replace yV by some elements y˜V ∈ Im R belonging to the appropriate ball (we assume that the family {˜ yV } is V finite). Obviously, the balls of radius 2N centred at the points y˜V cover R Z. But then the balls of radius 2N/|R|+ centred at the points R−1 y˜V cover Z. Consequently kZ+lK kl# ≤ 2N/|R|+ . We recall that by assumption kZ+lK kl# = 1. Hence |R|+ ≤ 2N . By the way N has been chosen, we obtain |R|+ ≤ 2|R# |+ . # (vi) Clearly, the operator K2¯γ is compact. Hence by 6.1.3 kK2¯ γ k = 0, which im1 1 # # plies |(D2¯γ ) |+ = |(D2¯γ + K2¯γ ) |+ ≥ 2 |D2¯γ + K2¯γ |+ > 2 |D + K : l∞ → l∞ |+ > 0. From 6.1.10 it follows that (note that D2¯γ is a difference operator) |(D2¯γ )|+ ≥ 8−c |(D2¯γ )# |+

¡ ¢ (strictly speaking, formally the case of the space Lp∞ = l∞ Zn /2¯ γ Zn , Lp (Q, E) on the compact group Zn /2¯ γ Zn is not enclosed in theorem 6.1.10; nevertheless, the proof for this case is carried out in the same way). (vii) We show that |D|+ > 0. Indeed, by 5.5.5 the infimum in the definition |D|+ = { kDxk : kxk = 1 } can be taken over families x with a finite (compact) support. Assume x is such a family. Then xα 6= 0 only if −γi + 1 < αi ≤ γi − 1 provided that γi are sufficiently large. Let x ¯ ∈ l2¯γ be the 2¯ γ -periodic extension of n x from −γi < αi ≤ γi to the whole of Z . Then clearly, kDxk = kD2¯γ x ¯k. Thus −c # |D|+ = |(D2¯γ )|+ ≥ 8 |(D2¯γ ) |+ . Now from 6.5.10 it follows that D is invertible. ¡ ¢0 (b) By 5.5.5 we may assume that q 6= ∞. Then Lpq (G, E) = Lp0 q0 (G, E0 ). Clearly, D0 ∈ DAP and K 0 ∈ hAP . By 1.3.4 |D0 + K 0 |+ > 0. Therefore by (a) D0 + K 0 is invertible. Hence D + K is invertible, too. ¤ 6.5.12. The case of a differential-difference operator. Let E be a finitedimensional Banach space and G = R. Let Lpq = Lpq (R, E), 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞ or q = 0. Corollary. Let D ∈ DAP (Lpq ) and B ∈ dAP (Lpq ). d 1 (a) We consider the operator L = D dt + B : Wpq → Lpq . If |L|+ > 0 then L is invertible. If p 6= ∞ and |L|− > 0 then L is invertible. d −1 (b) We consider the operator L = dt . If |L|+ > 0 then L D + B : Lpq → Wpq is invertible. If p 6= ∞ and |L|− > 0 then L is invertible.

Proof. (a) By 2.3.4 the operator L is invertible if and only if the operator T = LU −1 : L → L is invertible, see 2.3.4 for the definition of U . As we know, T = D + K, where K = (B − D)U −1 . We observe that B − D ∈ dAP and U −1 ∈ h(Lpq ) ∩ A. Therefore K ∈ hAP (Lpq ). By 6.5.11 T is invertible. (b) is handled in a similar way. ¤

COMMENTS These comments are intended to help the reader in finding additional related information. They do not pretend to be a complete or strict description of the history of the subject. A special attention is paid to the local publications which is hard to find in the usual data bases. Chapter 1. The notion of | · |+ was first suggested in [GiT]. A detailed authors’ exposition of theorems 1.4.11 and 1.4.12 can be found in [Gel] and [Nai]. See also comments to ch. 4. The history of the spaces Lpq is rather old. For example, the space L2∞ was employed in [Wie1 ]. A survey on the spaces Lpq can be found in [FoS]. See, e.g., [DaK] and [MaS] for applications to differential equations. The classical works on topological tensor products are [Gro] and [Schn], see also [Scha]. A treatment of the theory of Lp -spaces, 1 < p < ∞, from the tensor products point of view can be found in [Lev1 ]. Variants of theorem 1.7.8 were proved, e.g., in [EmR], [Fro1,3 ], [Har1,2 ], and [Ku4 ]. Close results on spectral mapping theorems for scalar functions of the tensor product of two unbounded operators can be found in [Ich1,2 ], [Ree1 ], and [Ree2 , vol. 1, theorem 8.33; vol. 4, theorems 13.34 and 13.35]. The first variant of theorem 1.7.10 was proved in [Boc2 ]. See [All] and [B¨ot2 , ch. 1] for further developments. Our exposition is borrowed from [Ku17 ]; it is based on [All] and [Boc2 ]. For the discussion of the Shilov boundary under assumptions of theorem 1.7.10, see [Fro5 ] and [Sl2 ], cf. also 4.5.10. The treatment in 1.7.12 is based on [Cas] and [Ver]. For applications of measures on abstract groups, see, e.g., [Berg] and [Hey]. A detailed discussion of the Yosida–Hewitt theorem can be found in [Lev2 ]. Chapter 2. In the history of mathematics the first famous equation with a causal operator was apparently the Volterra integral equation. The theory of Volterra’s equation exerted a deep influence on many branches of mathematics. At present causal operators are investigated (almost independently) in functional differential equations, functional analysis (see, e.g., [GoK], [Kadi], and [Rin]) and control (see, e.g., [Deso], [FeS], and [Wil2 ]). It is of interest that the connection between causal invertibility, and evolutionary solubility and stability was formulated explicitly in [Wil1 ] rather later. Probably it has not been done before because there had not been the language of Banach algebras and the investigation of functional and functional differential equations was not so extensive.

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The properties of causal invertibility and causal spectrum on a segment and at a point were discussed in [Ku1,5 ]. Similar results were also obtained in [You]. There is a large literature on spectral properties of causal invertibility and related problems. The semi-norm G (see 2.2.6) was first described explicitly in [Z2 ] in connection with the investigation of integral operators on the spaces of measurable functions. The case of the space C was discussed in [Ku2 ]; the case of abstract functional spaces was considered in [Ku5 ]. Estimates of spectral radius of the kind 2.2.7 were proved by many authors, see, e.g., [Ber1 ], [Ku2,5 ], [DeS1 ], and [Z2,5 ]. Actually the proof of the existence and uniqueness of solution based on so called the step method (see, e.g., [ElN]) and the contractive mapping principle is the realization of the same idea, see, e.g., [Cru2 ], [Hale7 ], and [Wil2 ]. For the connection of the semi-norm G with the radical (theorem 2.2.9), see, e.g., [ErL], [FeS], [Ku5 ], and [Rin]. Properties of compact causal operators of the kind of theorem 2.2.10 are discussed, e.g., in [Ber2 ], [DeS2 ], [ErL], [FeS], [GoK], [Ku2,5 ], and [Z1,2,5 ]. One can consider the following generalization of the notion of causal operator. Let G be a locally compact abelian group and S be a subsemi-group of it. Let X and Y be linear spaces of functions defined on G. We say that a linear operator T : X → Y is causal (with respect to S) if for any t ∈ G and x ∈ X, x(s) = 0,

s ∈ t − S0 ,

⇒

¡

¢ T x (s) = 0,

s ∈ t − S0 ,

where S0 is the interior of S. The most curios example is G = R4 with the light cone as S; see, e.g., [Far] and [Vla2 ]. Some properties of causal invertibility similar to those of §2.1 and §2.2 are discussed in [Ku21 ]. For the results of the kind 2.2.7 and 2.2.10, see [Kal] and [Z6 ]. The causal invertibility of shift invariant operators for the case, where G = Rn and S is an arbitrary cone, is discussed in [Stu]. An extensive standard theory of the Sobolev spaces Wps can be found, e.g., in [BerL] or [LiM]. For an extensive theory of distributions, see, e.g., [Rud2 ], [Schz2 ], and [Vla1−3 ]; the features of infinite-dimensional E are considered in [Schz1 ]. Different authors use many various definitions of a solution of a functional differential equation. Usually the choice of the definition can be interpreted as the choice of a functional space for solutions or as the choice of the pair of spaces for the corresponding differential operator. The simplest (but not the most convenient) definition is based on the usage of the classical derivative and the pair (C 1 , C). It was usually used in early works, see, e.g., [Bel1,2 ], [ElN], and [Mys]. Later, to avoid problems with the treating of the compatibility condition it was suggested (see, e.g., [Dri1 ] and [Mel1,2 ]) to use the notion of the Lebesgue derivative and the pairs of the kind (Wp1 , Lp ). The definition of a solution of the class C was suggested in [Hale7 ], see also [Cru1−4 ], [Hale2,6 ], and [Hen2 ]. In [Ku11 ] it was observed that this approach corresponds to the employment of the pair (C, C −1 ). The space C −1 was also used for the investigation of functional differential equations in [Akh2 ] and [Ku13,19 ]. See [Z7,8 ] for other applications of the spaces of the kind C −1 .

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−1 Pairs of the kind (Wpq , Lpq ) are connected closely with the semi-group approach in the theory of functional differential equations, see, e.g., [Bad1,2 ], [Bern], [Bor], [Burn], [Col], [Die], [Del2,3 ], [Hin], [Ito], [Ka], [Kap2,3 ], [Sal], and [St2−5 ]. An extensive exposition of this approach with a lot of references can be found in [Gri]. Distribution solutions also arise naturally in the investigation of boundary value problem for functional differential equations of neutral type, see, e.g., [Kam1−3 ] and [Sku1,2 ]. As we have mentioned above, evolutionary solubility was first investigated in [Wil1 ], see also a detailed account in [Wil2 ]. These works deal with functional equations which describe feedback systems on the spaces of the kind Lp ; here the evolutionary solubility was defined as the causal invertibility on extended spaces, cf. 3.3.3. For the case of differential equations, evolutionary solubility was described in [Ku1 ]; theorems of the kind 2.5.9 and 2.6.4 were also proved here. Theorem 2.6.6 was proved in [Ku4 ]. Chapter 3. The first work on the topic of the chapter was [Per]. This article is devoted to the equivalence of exponential dichotomy of solutions of an ordinary differential equation and the invertibility of the corresponding operator. The equivalence of input–output stability and causal invertibility (see 3.2.5 and 3.3.6) was first obtained in [Wil1 ], see also [Wil2 ]. As we have mentioned, these works are concerned with functional equations which describe feedback systems on the spaces of the kind Lp . The most attention was paid to the case of [τ, +∞). The case of (−∞, +∞) was also considered, but it was interpreted in another way than in this book. For the case of feedback systems, further developments can be found in [FeS]. The case of differential equations was considered in [Ku3,6,7,11 ]. In fact, the history of the connection between stability and invertibility is essentially older. First, statements of this kind can be considered as special cases of results on dichotomy of solutions. Second, if an equation is locally soluble then by 2.1.5 the causal invertibility on [τ, +∞) is equivalent to the ordinary invertibility. The first statement on the equivalence of the ordinary invertibility on [τ, +∞) and input–output stability for functional differential equations was proved in [Hal]; here an equation of retarded type of the class df was considered. Similar results for equations of neutral type can be found, e.g., in [Az], [Cru3 ], [Ku7 ], and [Nos2 ]. Extended spaces was first considered in [San1,2 ] and [Zam]. They are often used in control theory, see, e.g., [Deso], [FeS], and [Wil2 ]. Extended space can be constructed for an arbitrary space X with a T -direction to be the projective limit of the spaces Xa/b as b → +∞ and to be the inductive limit as a → −∞; see [Del1 ] for details. The idea on the equivalence of the invertibility on [τ, +∞) on the spaces l0 and l∞ (see 3.2.6) is borrowed from [Pu1−3 ], see also [Az] and [Ch3 ]. Apparently, one may consider [Per] as the first publication connected with the subalgebra e being full (see 3.4.2 and 3.5.2). Many close results can be found in the literature on exponential dichotomy of solutions and Green’s function. For explicit formulations in terms of inverse matrices, see, e.g., [Bas1−4 ], [Leb3 ], and [Shu4 ]. An interesting application of the idea of theorem 3.6.2 (the preservation of

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instability under small perturbations) is found in [TaB], see also its exposition in [Deso, ch. 6, §12]. An exposition of the history of the theorem on dichotomy of solutions for ordinary differential equations can be found, e.g., in [DaK] and [MaS]. For functional differential equations, theorems on dichotomy were proved by many authors, see, e.g., [Akh2 ], [Bir], [Burd], [Cof], [Cor1 ], [Hale7 ], [Hen2 ], [Hin], [Ko1 ], [Ku3,17 ], [Pec1,2 ], [Sch¨ a], and [St2 ]. A short review of these works can be found in [Akh2 ]. 1 Variants of theorem 3.6.4 for the pairs (C, C −1 ), (C 1 , C), and (Wpq , Lpq ) were published in [Akh2 ], [Ku3 ], and [Ku17 ], respectively; our exposition follows these works. An application of theorem 3.6.4 to the investigation of stability and instability of non-linear equations is given in [Akh2 ]. Different effective criteria of invertibility, causal invertibility, and their applications to the investigation of functional differential equations can also be found, e.g., in [Ant1,2 ], [Bren], [Deso], [FeS], [Fro2 ], [Kara], [Karl1−3 ], [Kit1−4 ], [Kra1 ] [Ku16 ], [Kuz1−3 ], [Lat], [Muh5 ], [Mya], [Saz], and [Wil2 ]. In this book we have not discussed the second Lyapunov method. An exposition of this enormous topic and related references can be found, e.g., in [Burt], [Dri2 ], [Gri], [Hale2,6 ], [KhS], [Kim], [Kol2,3 ], and [Raz]. We mention specially the work [Mil] where it is observed that only in L2 -metric a quadratic Lyapunov function can be equivalent to the norm, which explains some difficulties in the theory of Lyapunov’s functions for functional differential equations. Chapter 4. The topic of the chapter takes its origin in [Wie2 ]. In [Wie2 ] it was proved that if a function f : T → C has an absolutely convergent Fourier series and 1/f nowhere vanishes then 1/f has an absolutely convergent Fourier series, too.¢ Speaking modern language this is the main application of the equality ¡ X Md (Z) = T, see 4.2.3 and 4.2.6. In [Cam] and [Pit] it was a similar ¡ established ¢ result associated with the property of R being dense in X Md (R) , cf. 4.2.6 and 4.2.10. Finally, in [Wie3 ] it was proved that a measure µ ∈ Md⊕ac (R) is invertible in M(R) if and only if the Fourier transform of the measure µ is bounded away from zero. Proofs in these works are based on direct ε–δ-estimates. For an essentially simplified proof of this kind, see [New]. For the history of the approach based on theorem 1.4.11 see [Gel] and [Nai]. For the stability of the difference integral equation Z ∞ X ah x(t − h) + g(s)x(t − s) dλ(s) = f (t) h∈R+

0

a first scalar result similar to 4.5.6 was obtained in [Dav]. For differential difference equations, first scalar results similar to 4.5.9 were obtained in [Hen2 ]. + + The explicit description of X(M+ (see 4.3.8) is presented in d⊕ac ) as Xb t X [Fro4 ] and [Pap]. A similar statement for a subsemi-group of a locally compact abelian group can be found in [Stu]. The first example of the kind 4.2.12 for G = R is found in [Wie3 ]. An example ˇ for G = R based on a different idea is given in [Sr]. For further development and references, see, e.g., [Rud1 ] and [Tay]. We recall that our exposition of 4.2.12 follows [Will1 ].

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From [Ig] it follows that for any non-discrete locally compact abelian group G and 1 ≤ p ≤ ∞, p 6= 2, there exists a bounded measure µ on G such that the operator of convolution Tµ x = µ ∗ x is invertible on L2 , but not invertible on Lp . The following effective estimate of p for which Tµ is invertible on Lp is given in [Shn2 ] (the proof in [Shn2 ] is based on [Shn1 ]). Assume µ ∈ M(G) and inf{ |ˆ µ(χ)| : χ ∈ X(G) } > 0. Then Tµ is invertible on Lp for all p such that ¯1 1¯ 2 inf{ |ˆ µ(χ)| : χ ∈ X(G) } ¯ ¯ . ¯ − ¯ < arctan p 2 π kTµ : Lp → Lp k For a generalization of theorem 4.3.11 to a subsemi-group of a locally compact abelian group, see [Davi]. Theorem 4.4.1 is well known, see, e.g., [Hew, theorem 20.12] or [Gri]. For a detailed theory of vector-valued measures, see, e.g., [Bou3 ¡, ch. 6, §2]¢ or [Edw, §8.19]. A complete description of operators of the class A L1 (G, E) in terms of convolutions with operator-valued measures is given in [Tew]. For the major part, our exposition in §4.5 is based on [Fro4 ]. The idea of the subspaces Lu (see 4.5.11) takes its origin in [Blo] and [Flo], see also [Ree2 , vol. 4, ch. 13, §16]. Variants of theorem 4.5.11 and corollary 4.5.12 for operators of the class e are proved in [Ch4,5 ] and [Pu4 ]. There is a large literature on partial functional differential equations (mostly shift invariant), see, e.g., [Dat], [Hale4 ], [Kam3 ], [LeT], [Pr¨ u], and [Vlas]. Their discussion is beyond the scope of this book. We also have no space to discuss the features of L2 -theory in this book. Its fundamentals can be found, e.g., in [Ste2 ]. Chapter 5. Our exposition in §5.2 is based on [Ku17 ]. The idea of the proof of theorem 5.2.5 consisting in the usage of the Bochner–Phillips theorem first appeared in [Sem1,2 ]. Later different variants of 5.2.5 were published in [Bas1−4 ], [Khe1−3 ], [Ku17,18 ], [La3 ], and [Leb1−3 ]. Specifically, in [Sem1,2 ] and [Khe1,3 ] it is proved that SH under¡ the assumption H ' Z is ¡ the subalgebra ¢ ¢ full B L2 (T, C) (see [Sem1,2 ]), B L2 (R, C) (see [Khe1 ]) and ¡ in the algebras ¢ B F lp (T, C) (see [Khe3 ], here F lp is the image of lp (Z) under the Fourier trans2 form); in [Khe2 ] it ¡is considered ¢ the case ¡H ' nZ . ¢ In [La3 ] it is proved that n the subalgebra ¡sAP ¢lp (Z , C) is full in B lp (Z , C) . In [Leb1,2 ] it is proved P∞ of the kind that operators Dx (t) = m=1 am (t)x(τm (t)), ¡ under assumption ¢ P∞ m=1 kam kL∞ < ∞ form a full subalgebra of B L2 (X, C) ; it is assumed that the group generated by τm is similar to Zn . Similar problems in crossed products are considered in [Leb3 ]. Articles [Bas1−4 ] are devoted to the subalgebras induced by infinite matrices satisfying various weighted estimates. Algebras S and d are used also in [Ra3 ]. Some results on full subalgebras of difference operators with applications to numerical analysis are given in [Bla1−3 ] and [Dem1−3 ]. Our exposition in §5.3 is based on [Ku17 ]. The class t is a modification of the class of c-continuous operators introduced in [Muh2 ] (the term comes from compact convergence, see [Bou1 , ch. 10]). The

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idea of that definition is as follows. Let (X, Y ) be a pair with directions. The set of c-continuous operators is defined to be the closure in norm of all operators T ∈ B(X, Y ) possessing the following property: for any α ∈ ∆ there exists β ∈ ∆ such that T X \β ⊆ Y \α holds, cf. 5.5.2. It is easy to see that c-continuous operators form a Banach algebra. We show that the subalgebra of c-continuous operators on L∞ (R, C) is not full. Assume the function τ : R → R maps linearly (2n − 1, 2n] onto (2n − 1, 2n + 1], and maps linearly (2n − 2, 2n − 1] onto (1/(n + 1), ¡ 1/n] ¢ for all¡ n ∈¢N. Finally, for t ≤ 0 we set τ (t) = t. We consider the operator T x (t) = x τ (t) . It is easy to verify that T acts¡ on L¢∞ ; the operator ¡ −1 ¢ T is c-continuous and invertible; but the −1 inverse operator T x (t) = x τ (t) is not c-continuous. The exposition in §5.6 is based on [Ku14 ], see also [Ku19 ] for another approach. Chapter 6. Section 6.1 is connected closely with the theory of measures of noncompactness. The idea of the functor # is a modification of that of [Sad], where for X K it was taken the subspace of all conditionally compact families Z ∈ X V ; see also the exposition of results of [Sad] in [Akh1 ]. An alternative approach to measures of noncompactness can be found, e.g., in [Gol1,2 ], [LeS], and [Wei2 ]. To obtain a more complete theory for non-reflexive spaces (such as C, L1 , and L∞ ) one should consider weakly compact, strictly singular, and strictly co-singular operators instead of compact operators; see, e.g., [Pie] and [Schr] for basic definitions and properties. A class of c-completely compact operators (similar to h ⊆ k) is considered in [Muh2 ] in connection with the investigation of almost periodic operators, see §6.5. Operators of the class h are investigated also in [Ra1,3 ]. Variants of theorems 6.1.10 and 6.1.11 are published in [Ku9,17 ]. Close results are also found in [La4,5 ]. There is a large literature on operators with the following property: if it is known that an operator possesses the Fredholm property then it is actually invertible. See, e.g., [Ant1,2 ], [B¨ot1 ], [Ch1,2 ], [Kara], [Karl2,3 ], [Kra2 ] [Khe1−3 ], [Kol2,3 ], [Leb1−4 ], [Mya], [Nos1 ], and [Wei1,2 ]. There are known many criteria for compactness of integral operators more general than proposition 6.2.2, see, e.g., [Eve], [Gri], and [Z4 ]. Theorems 6.2.1 and 6.2.3 are found in [Ku9 ]. The equation −D∆x + Bx = f from 6.2.5 is investigated in [Pot] where an analogue of 6.5.12 for this equation is proved. In literature the assumption of the causal invertibility of D (see theorem 6.2.8) is usually formulated as the equivalent assumption on the stability of the auxiliary equation Dz = g. Apparently, the significance of this assumption for the investigation of stability was first noted explicitly in [Hale1 ]. See [St1 ] for an extensive discussion of this assumption. An initial variant of theorem 6.2.8 was published in [Ku8 ]. Example from 6.3.2 is described explicitly in [Ku10 ]. Theorem 6.3.9 was published in [Ku13,17 ]. Variants of theorem 6.3.10 were obtained in [Ch6 ], [Ko2,3 ], and [Ku13,19 ]; some related counter-examples can be found in [Ku13 ]. Different results on the (in)dependence of invertibility from functional space

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can also be found in [Ig], [KeT], [Shn1,2 ], [Sta], [Tyu], [Z3 ], and [Zaf]. The literature on Green’s function is abundant. See, e.g., [Az], [Ban], [Bel2 ], [Cru1 ], [Dra], [Gri], [Hal], [Hale2,6,7 ], [Hen1,2 ], [Is], [Kam1,2 ], [Kol1,2 ], [KoT], [Mak], [Mys], and [Z8 ]. The main ideas of §6.4 were announced in [Ku12 ]. The subject of §6.5 takes its origin in [Fav] where the following condition, now called the Favard condition, has appeared: all operators from the closure of the set H(L) = { Sh LS−h¡: h ∈ G } ¢has the zero kernel. Usually (for example, for L = 1 + K, K ∈ hAP l∞ (Zn , E) ), this condition is equivalent to |L|+ > 0; but in general this is not the case. In [Fav] it was proved that Favard’s condition guarantees the almost periodicity of bounded solutions of the ordinary differential equation Lx = f with almost periodic coefficients and almost periodic f . The next important step was made in [Muh1 ], where it was pointed out that Favard’s condition imply the invertibility in the pair (C 1 , C). Later this result was extended to broader classes of operators, see, e.g., [Ko1 ], [Ku15 ], [La1−3 ], [Muh2,3 ], [Shu1−3 ], [Sl4−6 ], and [Pot]. Our exposition is based on [Ku15 ]. L2 -theory of almost periodic operators is richer. In [Cob] it is constructed a representation of the algebra of almost periodic operators as the factor of the type II∞ (see, e.g., [Breu], [MuN], and [Nai] for the theory of factors). For further developments of this idea, see [Shu1−3 ] and [La1,2 ]. The theory of almost periodic operators have many points of contact with the e theory¡ of so called ¢ limit operators. An operator T is called a limit operator for T ∈ B Lpq (G, E) if there exists a sequence hk → ∞ such that Shk T S−hk converges to Te is some topology. The typical result of this theory states that the invertibility of all limit operators Te is equivalent to the Fredholm property for the initial operator T . For this topic, see, e.g., [Deu1,2 ], [Ku20 ], [La4,5 ], [Muh4 ], [Ra2−4 ], and [Sht].

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BIBLIOGRAPHY

[Stu]

[TaB]

[Tay] [Tayl1 ] [Tayl2 ] [Tew]

[Tikh] [Tyu] [Ver]

[Vla1 ] [Vla2 ] [Vla3 ] [Vlas]

[Wei1 ] [Wei2 ]

[Wie1 ] [Wie2 ] [Wie3 ] [Wil1 ]

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INDEX

429

INDEX algebra 20 Banach 20 commutative 29 normed 20 with a unit 20 with an adjoint unit 21 C ∗ -algebra 31 almost everywhere 35 locally 37 atom 123, 217 causal invertibility 101 at a point 105 at infinity 105 on a segment 102 causal spectrum 106 at a point 107 at infinity 107 on a segment 106 centralizer 23 character group 222 character space of a group 222 of a semi-group 239 of an algebra with a unit 29 of an algebra without a unit 29 character of a group 221 of a semi-group 239 of an algebra with a unit 29 of an algebra without a unit 29 coefficients almost periodic 382 continuous 334, 367, 369, 381 periodic 283, 284, 358 uniformly continuous 334 commutator 333

compactification Bohr’s 289 one point 3 compatibility condition 130 convergence absolute 4 of distributions 117 unconditional 50 convolution of a measure and a function 210 of functions 214 of measures 211 cross-norm 62 compatible with multiplication 76 uniform 70 ε 63 π 63 ∗-uniform 66 density of a measure 89 derivative classical 112 complex 27 distribution 117, 376 Lebesgue’s 112 diagonal of a matrix 53 direction complete 110 consistent 330, 344 equivalent 323 natural 323 standard 323 T -direction 99 ±-direction 322 distribution 116, 375 equality 117 regular 117, 376

430

INDEX

equation conjugate 378 difference 157, 176 differential difference 356, 368, 373 functional 127, 355 functional differential 127, 366, 372 of neutral type 139, 140, 356 of retarded type 139, 140 regularization 133 with external differentiation 97 with internal differentiation 97 function almost periodic 290 equivalent 35 essentially integrable 39 finite-dimensional 33 Green’s 372 integrable 36 locally integrable 89 locally null 37 lower semi-continuous 34 measurable 37 null 35 universally measurable 37 group 45 dual 222 ideal 24, 74 image of an operator 5 index 20 instability exponential 196 input–output 196 rough 196 integral 34, 36, 87 complex 85 essential 39 essential upper 39 upper 35 invertibility 5, 21 at a point 106 at infinity 106 on a segment 102 isomorphism isometric 6 of algebras 24

topological 6 kernel of an operator 5 locally almost everywhere 37 matrix 50 main diagonal 53, 287 of an operator 50 measure 37 absolutely continuous 89 atom 217 bounded 34, 85 complex 85 complex conjugate 86 concentrated on 87, 90 conjugate 235 continuous 216 discrete 217 Haar’s 46 mutually singular 90 operator-valued 251 positive 34, 85 product 41 real 85 singular 92 transposed 260 upper 35 σ-compact 38 σ-finite 38 memory exponential 55, 172, 182, 313, 322 locally bounded 52, 324 locally fading 52, 324 summable 54, 296, 308, 309, 317 uniformly bounded 53, 302 uniformly fading 53, 302 zero 286 modulus of a measure 87 of an operator-valued measure 252 morphism of algebras 24 of groups 45, 289 norm 3 degenerate 3 equivalent 3 lower 14

INDEX

operator acts from X (in)to Y 5 acts on X 5 acts onto Y 5 almost periodic 382 anticausal 101, 122 bijective 5 c-continuous 408 causal 23, 57, 98, 99 compact 7, 355 conjugate 7, 378 convolution 210, 274 difference 143, 217, 264, 296, 302 difference integral 274, 276 differential difference 356, 402 Fredholm 19 index 20 injective 5 integral 43, 212, 264, 317, 322 limit 409 locally compact 344, 355 locally Fredholm 346 multiplication 141, 286 oscillation 285, 292, 331 oscillation invariant 292 periodic 283, 284, 358 pre-conjugate 7 quotient 8 restriction to a segment 99 shift 23, 60, 296 shift invariant 210 smoothing 309, 313 surjective 5 universal 308, 313 ∆±-compact 344 partition of unity 32, 193 pre-image 2, 5 radical 25, 110 resolvent 22, 28 semi-group 239, 404, 406 set conditionally compact 2 locally null 37 measurable 37 null 35

431

summable 37 universally measurable 37 σ-compact 38 σ-finite 38 Ψ-bounded 331 solubility evolutionary 137, 158 evolutionary at a point 138 in a pair 137 local 158, 163 uniform 176, 187 unique at a point 138 unique on a segment 129–133, 158 space conjugate 6 extended 158, 164 quotient 4 spectrum 22 at a point 107 at infinity 107 causal 106 on a segment 106 radius 22, 109 stability exponential 177, 192 input–output 159, 165, 361, 367 subalgebra 23 full 23 without a unit 23 support 32, 116, 375 theorem Banach–Steinhaus’s 5 Banach’s 5 Bochner–Phillips’s 75 Fubini’s 42 Gel0 fand–Naimark’s 31 Hanh–Banach’s 6 Kronecker’s 232 Lebesgue–Radon–Nikodym’s 90 Pontrjagin’s 230 Riesz–Thorin’s 43 Schatten’s 67 Stone–Weierstrass’s 33 Urysohn’s 32 Yosida–Hewitt’s 93

432 topology 1 compact convergence 222 Hausdorff 2 product 3 strong 5 uniform 5 weak 6 ∗-weak 6 hX, F i-topology 6 transform Fourier’s 225, 268 Gel0 fand’s 30, 76, 267 Laplace’s 241, 271

INDEX

NOTATION INDEX D 46 G 45 Gb 289 Gd 221 I 48 K 46 N = { 1, 2, . . . } e 48 Q, Q e i 48 Qi , Q T 45 U 45 V 46 X 221 X+ 239 Xb 221 X+ b 239 C 21, 32 Cloc 163 −1 1 Cloc , Cloc 163 Cq 49 Cq1 113 Cq−1 119 C0 21, 32 C00 34 + C00 34, 85 C♥i 59 C♦i 57 Ia 154, 186 Im A 5 Ja 154 Ker A 5 Lp 36 Lp loc 112, 163 Lpq 48 L0 39

L∞ 39 lq 40 l00 40 M + 34 Pa 59, 150 p0 , q 0 81 Qa 99, 154, 186 Ra 154, 186 R(A) 22 Sh 60 sj 297 U 115, 122, 125 U 0 125 Uϑ 181 Wp loc 114 Wp1 loc 163 Wp−1loc 163 1 Wpq 113 −1 Wpq 119 Xloc 163 X♥i 152 X♦i 333 D0 116, 375 F, Fp 35 Kp 35 K∞ 39 Lp 36 Lpq 48 L∞ 38 M 85, 251 M+ 220 Mac 92, 212 M+ ac 220 Mc 92, 216 M+ c 220 433

Md 217 M+ d 220 Md⊕ac 219 M+ d⊕ac 221 Mloc 85 Mq0 95 Msc 218 M+ sc 220 A 210, 273 A+ 276 B5 B+ 98, 106 BAP 383 bAP 388, 401 C 334 Cu 334 D 302 d 53, 302 Df 302 df 53, 302 Dz , Dz f 302 dz , dz f 302 DΨ 302, 385 dψ 302, 386 DΨ f 302 dψ f 302 E 304 e 55, 172, 333 e+ 177 EH 304 h 387 hf 387 k 345 M1 320 M∞ 313

N1 315 N∞ 309 O 28, 71 S 143, 296 S1 369 s 54, 297, 333 SH 303 Sz 296 sz 297 SΨ 296 sψ 297 t 52, 324, 333 tf 52, 324, 333 V 307 W 313 X 286 x 295 Xz 287 xz 295 XΨ 292 xψ 295 δa 217 δa,α 123, 380 λ 34, 46 σ(A) 22 σ + (A) 106 Ψχ 292, 299 ψκ 295, 388 1 5, 20 1E 35 | · |± 14 · 54, 298 t 231 ⊥ 12, 222, 304 # 347