Denseness, Bases and Frames in Banach Spaces and Applications 9783110493863, 9783110484885

Citation preview

Aref Jeribi Denseness, Bases and Frames in Banach Spaces and Applications

Also of Interest Implicit Fractional Differential and Integral Equations. Existence and Stability Saïd Abbas, Mouffak Benchohra, John R. Graef, Johnny Henderson, 2018 ISBN 978-3-11-055313-0, e-ISBN (PDF) 978-3-11-055381-9, e-ISBN (EPUB) 978-3-11-055318-5 Generalized Inverse Operators. And Fredholm Boundary-Value Problems Alexander Andreevych Boichuk, Anatolii M. Samoilenko, 2016 ISBN 978-3-11-037839-9, e-ISBN (PDF) 978-3-11-037844-3, e-ISBN (EPUB) 978-3-11-038735-3 Wavelet Analysis on the Sphere. Spheroidal Wavelets Sabrine Arfaoui, Imen Rezgui, Anouar Ben Mabrouk, 2017 ISBN 978-3-11-048109-9, e-ISBN (PDF) 978-3-11-048109-9, e-ISBN (EPUB) 978-3-11-048109-9

Regularization Methods in Banach Spaces Thomas Schuster, Barbara Kaltenbacher, Bernd Hofmann, Kamil S. Kazimierski, 2012 ISBN 978-3-11-025524-9, e-ISBN (PDF) 978-3-11-025572-0

Complex Analysis. A Functional Analytic Approach Friedrich Haslinger, 2017 ISBN 978-3-11-041723-4, e-ISBN (PDF) 978-3-11-041724-1, e-ISBN (EPUB) 978-3-11-042615-1

Aref Jeribi

Denseness, Bases and Frames in Banach Spaces and Applications |

Mathematics Subject Classification 2010 46A35, 46B15, 34L10, 35P10, 47F05, 58C40, 47J10, 58E07, 42C15, 47A70, 47B06, 05C50, 34L15, 11R45 Author Prof. Dr Aref Jeribi University of Sfax Faculty of Sciences/Department of Mathematics BP 1171 Road Soukra Km 3.5 3000 Sfax Tunisia [email protected]

ISBN 978-3-11-048488-5 e-ISBN (PDF) 978-3-11-049386-3 e-ISBN (EPUB) 978-3-11-049240-8 Library of Congress Control Number: 2018935074 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck Cover image: Thierry Foulon/PhotoAlto Agency RF Collections/getty images ∞ Printed on acid-free paper ○ Printed in Germany www.degruyter.com

|

To my mother Sania, my father Ali, my wife Fadoua, my children Adam and Rahma, my brothers Sofien and Mohamed Amin, my sister Elhem, and all members of my extended family

Preface This book gives a fairly comprehensive review of the important aspects of spectral theory, covering, in particular, the completeness of generalized eigenvectors, Riesz bases, semigroup theory, families of analytic operators, frames in Hilbert spaces, ν-convergence, and the Γ-hypercyclic set of linear operators. We present an interesting selection from the literature of topics and results, including a number of unfamiliar ones. This book is intended for students, researchers in the field of spectral theory of linear non-self-adjoint operators, and mathematicians interested in applications in mathematical physics. Pure analysts will also find some new problems to tackle. Readers interested in mathematical physics will find a new theory of practical importance. Most of the material can be understood by a reader with a relatively modest knowledge of functional analysis and operator theory, so I have included some chapters connected to this background material, which is needed to develop the results of this book, rendering the text almost entirely self-contained. There is inevitably an overlap with some existing books, since this is (and has been for some time) a very active area of research, but it must be emphasized that all the results of importance for physics or technology are obtained in this book by means of some new mathematical theory, result, or idea. Moreover, all these results are closely connected with problems of physics and technology of great interest in applications to a variety of problems from mechanics, fluid mechanics, and mathematical physics. Some of the problems which have remained unsolved for years are solved in this book for the first time. It must be emphasized that all results are of great importance, so this book is sufficiently different and it has many interesting extra features that make it recommendable. This well-written book covers an excellent list of important topics in applied functional analysis. The main topics include: – bases in Hilbert and Banach spaces; – semi-groups and diagonalization; – discrete operators and denseness of generalized eigenvectors; – summability of series in the principal vectors of non-self-adjoint operators; – frames in Hilbert spaces; – ν-convergence operators; – the Γ-hypercyclic set of linear operators; – analytic operators in Béla Szökefalvi-Nagy’s sense; – the perturbation method for sound radiation by a vibrating plate in a light fluid; – Reggeon field theory; and – applications to mathematical models. https://doi.org/10.1515/9783110493863-201

VIII | Preface We do hope that this book will be very useful for several researchers, since it represents not only a collection of previously heterogeneous material, but also an innovation through several extensions. Of course, it is impossible for a single book to completely cover such a huge field of research. In making personal choices for the inclusion of material, we tried to give useful complementary references in this research area, probably missing some relevant works. We would be very grateful to receive any comments from readers and researchers that provide us with information concerning missing references. We would like to thank Salma Charfi for the improvements she has made in the introduction of this book. We are indebted to her. Moreover, we should mention that the thesis work, performed under my direction by my former students and presently colleagues Bilel Krichen, Naouel Ben Ali, and Hanen Ellouz, and the obtained results have helped us in writing this book. Finally, we apologize in case we have forgotten to quote any author who has contributed, directly or indirectly, to this work. Sfax, November 2017

Aref Jeribi

Contents Preface | VII 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.8.1 1.8.2 1.8.3 1.8.4 1.8.5 1.8.6 1.9

Introduction | 1 Bases and semi-groups | 1 Discrete operator and denseness of the generalized eigenvectors | 2 ν-convergence operators | 3 Γ-hypercyclic set of linear operators | 4 Analytic operators in Béla Szökefalvi-Nagy’s sense | 5 Bases of the perturbed operator T (ε) | 6 Frames in Hilbert spaces | 7 Applications | 10 Perturbation method for sound radiation by a vibrating plate in a light fluid | 10 The shape memory alloys operator | 12 Heat exchanger equation with boundary feedback | 12 Expansion of solution for a hyperbolic system | 13 Frame of a one-dimensional wave control system | 14 Reggeon field theory | 15 Outline of contents | 15

2 2.1 2.2 2.3 2.3.1 2.3.2 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

Linear operators | 19 Closed and closable operators | 19 Basis | 20 Fredholm operators | 22 Bounded operators | 22 Unbounded operators | 24 Fredholm perturbations | 25 Compact, A-bounded, and A-compact operators | 25 Weakly compact operators | 26 Strictly singular operators | 26 Closable operator perturbation | 26 Adjoint operator | 27 Resolvent set and spectrum | 28 Generalized eigenvectors | 31 Operators on Hilbert spaces | 32 Discrete operator | 33 Ascent and descent operator | 34

X | Contents 2.15 2.16 2.17 2.17.1 2.17.2 2.17.3 2.17.4 2.18 2.18.1 2.18.2 2.19 2.20 2.21 2.22

Riesz operators | 35 Norm of the resolvent | 36 Convergence operators | 37 Convergence sequence | 37 Limit inferior and superior | 39 ν-convergence | 40 ν-continuity | 41 Normal, hyponormal, and pseudo-inverse operator | 43 Normal and hyponormal operator | 43 Pseudo-inverse operator | 43 Dunford–Pettis property | 44 Jeribi, Wolf, and Weyl essential spectra | 44 Hypercyclic operators | 47 Closed subspace | 48

3 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.5 3.5.1 3.5.2 3.6 3.6.1 3.6.2 3.6.3 3.7 3.8 3.9 3.10 3.11

Basic notations and results | 55 Bessel sequences in Hilbert spaces | 55 Orthogonal sequences in Hilbert spaces | 58 Projection | 62 Generalities | 62 Spectral projection | 63 Infinite sum of projections | 65 Commutation of projections | 66 p-subordinate operator | 68 Finite-order and sine-type functions | 70 Finite-order function | 70 A function of the sine type | 71 Singular values | 74 Singular values of a compact operator | 74 Numerical range | 75 Carleman class Cp | 76 Fractional operators | 77 Phragmén–Lindelöf theorems | 78 Fredholm determinant | 79 Denseness of generalized eigenvectors | 80 Inequalities | 81

4 4.1 4.2 4.3 4.4

Bases | 85 Hamel bases | 85 Schauder basis | 85 Coefficient functionals | 86 Natural projections associated with a basis | 86

Contents | XI

4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.12.1 4.12.2 4.12.3 4.12.4 4.12.5 4.12.6 4.12.7 4.12.8 4.12.9 4.12.10 4.13 4.14 4.15 5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2 5.2.3 6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5

A characterization of bases | 94 Biorthogonal systems | 99 Orthonormal bases | 99 The Gram matrix | 104 Hilbert–Schmidt operator | 105 The trace class | 105 Equivalent bases | 106 Riesz basis | 110 Definition | 110 Completeness results | 111 Dual basis associated to a Riesz basis | 113 Characterization of Riesz basis | 114 Collection of vectors near a Riesz basis | 118 Sequence quadratically close | 119 Sum direct of subspaces | 121 Concept of the angle between two closed linear subspaces | 123 Paley–Wiener criterion | 125 Pavlov result | 127 Basis of subspaces | 127 Unconditional bases | 128 Riesz basis of finite-dimensional A-invariant subspaces | 130 Semi-groups | 133 C0 -semi-group | 133 Definitions | 133 Hille–Yosida theorem | 133 Dissipative operator | 134 Eigenvalues | 134 Diagonalization of semi-groups | 135 Diagonalization of the semi-group e−tA | 135 1 2

Diagonalization of the semi-group e−tA | 137 Riesz basis formed by eigenvectors of A | 140 Discrete operator and denseness of the generalized eigenvectors | 145 Hilbert–Schmidt discrete operator | 145 Elementary properties | 145 Direct sum | 147 Laurent expansion for the resolvent | 148 Spectral theory of the inverse of an HS discrete operator | 149 Singular values of the resolvent | 152

XII | Contents 6.2 6.2.1 6.2.2

Denseness of the generalized eigenvectors of an HS discrete operator | 153 The subspaces M∞ and S∞ for a Fredholm operator | 153 Sufficient conditions S∞ = X and M∞ = {0} | 154

7 Frames in Hilbert spaces | 159 7.1 Frame in Hilbert space | 159 7.1.1 Frame | 159 7.1.2 Frames and operators | 163 7.1.3 The projection method | 167 7.1.4 Frame perturbations | 170 7.2 Frame of subspace or fusion frame | 171 7.2.1 Definitions | 172 7.2.2 A characterization of frame of subspaces | 172 7.2.3 Synthesis operator | 174 7.2.4 A characterization of Bessel fusion sequences | 175 7.2.5 A characterization of fusion frames | 177 7.2.6 Oblique projections and biorthogonality | 178 7.2.7 A characterization in terms of orthonormal fusion bases | 182 7.2.8 Duals of fusion frames | 183 7.2.9 Fusion frames and operators | 185 7.2.10 Non-orthogonal fusion frame | 188 8 8.1 8.2 8.3 8.4

Summability of series | 191 Series in principal vectors | 191 Jordan chain | 191 Summation of series by Abel’s method | 192 Coefficients of a series in principal vectors | 193

9 9.1 9.1.1 9.1.2 9.1.3 9.2 9.3 9.4 9.5

ν-convergence operators | 201 ν-convergence and spectral properties | 201 Semi-Fredholm sequence | 201 Convergence of the spectrum | 204 ν-convergence | 206 Jeribi, Wolf, and Weyl essential spectra | 208 Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators | 214 ν-continuity of Wolf and Weyl essential spectra | 220 2 × 2 operator matrix | 224

10 10.1

Γ-hypercyclic set of linear operators | 231 Γ-hypercyclic set of a bounded linear operator | 231

Contents | XIII

10.1.1 10.1.2 10.1.3 10.1.4 10.1.5 10.1.6 10.1.7

Properties of Γhyp (T ) | 231 Essentially quasi-nilpotent operators | 232 Operators with small Γhyp (T ) | 235 Operators with big Γhyp (T ) | 236 Aluthge transforms | 237 Operator equations ABA = A2 and BAB = B2 | 238 Upper triangular matrices | 239

11 Analytic operators in Béla Szökefalvi-Nagy’s sense | 241 11.1 Invariance of the closure | 241 11.1.1 Hypotheses | 241 11.1.2 Closeness | 241 11.2 Eigenvalues and eigenvectors | 242 11.2.1 Entire series of ε | 243 12 Bases of the perturbed operator T (ε) | 257 12.1 Completeness and Riesz basis of generalized eigenvectors | 257 12.2 On a Schauder basis in a separable Banach space | 257 12.3 Riesz basis property of families of non-harmonic exponentials | 259 12.3.1 The value of ε is variable | 259 12.3.2 The value of ε is fixed | 263 12.3.3 Values of ε are both fixed and variable | 267 12.4 Unconditional basis with parentheses | 271 13 Frame of the perturbed operator T (ε) | 275 13.1 Frames of eigenvectors of the perturbed operator T (ε) | 275 13.1.1 Frames of analytic eigenvectors in the sense of Kato | 275 13.1.2 Combination of eigenvectors | 277 13.1.3 Existence of a fixed complex number ε | 279 13.2 Non-orthogonal fusion frames | 280 13.2.1 Analytic projection | 280 13.2.2 Combination of projections | 285 13.2.3 Existence of a fixed complex number ε | 287 14 14.1 14.2 14.3 14.4 14.5 14.6

Perturbation method for sound radiation by a vibrating plate in a light fluid | 289 Problem of radiation of a vibrating structure in a light fluid | 289 Compactness results | 291 Spectral properties of the operator T0 | 293 On a Riesz basis in L2 (]−a, a[) | 297 Closeness operators | 297 Denseness and Riesz basis of eigenvectors | 299

XIV | Contents 14.7 14.8 14.9

Unconditional basis with parentheses | 299 Riesz basis property of families of non-harmonic exponentials | 300 Frames | 301

15 Applications to mathematical models | 305 15.1 Heat exchanger equation with a boundary feedback | 305 15.1.1 The semi-group generated by the operator A | 306 15.1.2 Eigenvalues of the operator A | 307 15.1.3 The stability of the semi-group generated by the operator A | 311 15.1.4 Auxiliary operator | 315 15.1.5 Completeness system | 317 15.1.6 Asymptotic behavior of the eigenvalues of the operator A | 319 15.2 The shape memory alloys operator | 321 15.2.1 Elementary results | 321 15.2.2 Denseness of the generalized eigenvectors | 327 15.2.3 15.3 15.3.1 15.3.2 15.3.3 15.3.4 15.4 15.4.1 15.4.2 15.4.3 15.4.4

1 2

Diagonalization of the semi-groups e−tC and e−tC | 327 Expansion of solution for a hyperbolic system | 330 The resolvent expression of 𝒜 | 331 Spectrum of 𝒜 | 334 Spectral radius and essential spectral radius | 338 Riesz basis | 341 Frame of a one-dimensional wave control system | 345 One-dimensional string equation | 345 Adjoint of A | 347 Spectrum of A | 348 Non-orthogonal fusion frame | 352

16 Reggeon field theory | 359 16.1 Gribov operator in Bargmann space | 359 16.1.1 General results | 360 16.1.2 Auxiliary results | 362 16.1.3 Spectral properties of Hλ󸀠 | 365 16.1.4 Cauchy problem | 368 16.1.5 Denseness of the generalized eigenvectors of the operator Hλ󸀠 for λ󸀠 ≠ 0 | 370 16.2 The case of null transverse dimension (n = 1) | 372 16.2.1 Annihilation and creation operators | 373 16.2.2 Subordinate and boundedness | 374 16.2.3 Unconditional basis in Bargmann space | 382 −tH 16.2.4 Generalized diagonalization of the semi-groups e λ󸀠󸀠 ,λ󸀠 ,μ,λ 1

−tH 2󸀠󸀠

and e

λ ,λ󸀠 ,μ,λ

| 384

Contents | XV

16.2.5 16.3

Riesz basis of finite-dimensional Hλ󸀠󸀠 ,λ󸀠 ,μ,λ -invariant subspaces | 386 Schauder basis of Gribov operator in the Bargmann space | 387

Bibliography | 393 Index | 403

1 Introduction In functional analysis, an important place is occupied by the spectral theory of operators, which has applications in several areas of modern mathematical analysis and physics, such as differential and integral equations and quantum theory. In recent years, spectral theory has witnessed an explosive development and has provided a number of approaches to solve many problems that have arisen over the years. In this book, we present a number of achievements in the theoretical field for Riesz bases and frames in Hilbert spaces as well as some of their appearances in applications to some mathematical models. We also investigate the Γ-hypercyclic set and the ν-convergence of linear operators. These theories cover several topics dealing with linear operators. Now, let us describe the main contents of this book.

1.1 Bases and semi-groups In the literature, the theory of one-parameter semi-groups of linear operators on Banach spaces attracted the attention of many mathematicians. Some progress has been made by E. Hille and R. S. Phillips [106]. In the 1970s and 1980s, the theory reached a certain state of perfection, which is well represented in the monographs by E. B. Davies [70], J. A. Goldstein [97], A. Pazy [169], and others. This theory is characterized by a large manifold of applications that have become important tools for integro-differential equations, functional differential equations, and quantum mechanics and in infinite-dimensional control theory. This theory is applied with great success to concrete equations arising, e. g., in population dynamics or transport theory. Among the many properties of these semi-groups that are well studied in the literature, we recall in this book some concepts, such as dissipativity and diagonalization of semi-groups, and we show that these are the key to a deep and beautiful theory. In fact, around these concepts, we develop some techniques that allow us to obtain some important spectral properties for many concrete equations. On the other hand, bases are of great importance to study the geometry of Banach spaces, which represent a rich and beautiful classical subject in analysis. Moreover, they are used in classical and applied harmonic analysis, where they play an important role for decomposing and manipulating functions, operators, signals, images, and other objects. We also present in this book the abstract theory of bases in Banach and Hilbert spaces. We begin with the classical topics of Hamel, Schauder, and orthonormal bases and conclude with more modern topics, such as Riesz bases, bases of subspaces, unconditional bases, and Riesz bases of finite-dimensional invariant subspaces in https://doi.org/10.1515/9783110493863-001

2 | 1 Introduction Hilbert spaces. Such bases represent a natural extension to orthonormal bases and, if required, give rise to biorthonormal systems. This will be treated extensively and for those bases we give some concrete applications in Reggeon field theory and some mathematical models.

1.2 Discrete operator and denseness of the generalized eigenvectors Let A be a closed densely defined linear operator in a Hilbert space X and let sp(A) be the closed subspace of X spanned by the generalized eigenvectors of A. In their celebrated treatise [76–78], N. Dunford and J. T. Schwartz used Carleman’s inequality and the Phragmén–Lindelöf theorem to establish sufficient conditions for sp(A) to coincide with X. An essential ingredient in their theorems is the requirement of a growth or decay rate for the resolvent of the form 󵄩󵄩 −1 󵄩 −N 󵄩󵄩(λ − A) 󵄩󵄩󵄩 = O(|λ| ) along arcs in the complex plane, where N is a positive integer. Their results can be used to develop the spectral properties of some differential operators with regular boundary conditions. However, for irregular boundary conditions, the situation is more complicated and it is impossible to obtain a decay rate associated with the resolvent. For this reason, P. Lang and J. Locker [144–146] gave a generalization to permit weaker decay rates for the resolvent and, in particular, to obtain a theorem which is sufficiently general to handle certain problems. More precisely, the authors considered a closed densely defined linear operator A acting in a Hilbert space X and assumed that there exists ξ0 ∈ ρ(A) such that (ξ0 − A)−1 is a Hilbert–Schmidt operator. The operator A is a special type of discrete operator, a so-called HS discrete operator, which is shown to be a Fredholm operator in X with the Fredholm set equal to the whole complex plane. Let σ(A) = {λi }∞ i=1 , let mi (0 < mi < ∞) denote the ascent of the operator λi −A for i = 1, 2, . . . , let Pi , i = 1, 2, . . . , denote the projection of X onto the generalized eigenspace N([λi − A]mi ) along R([λi − A]mi ), and let S∞ and M∞ be the subspaces of X consisting of all x ∈ X such that ∞

x = ∑ Pi x i=1

and such that Pi x = 0 for i = 1, 2, . . . , respectively. Sufficient conditions are introduced in [146], which guarantee that S∞ = X and M∞ = {0}. These conditions require that ‖(λ − A)−1 ‖ be bounded on certain rays in the complex plane and 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − A) 󵄩󵄩󵄩 → 0

1.3 ν-convergence operators |

3

as λ → ∞ on at least one of the rays, but specific decay rates for ‖(λ − A)−1 ‖ are not necessary.

1.3 ν-convergence operators The concept of ν-convergence emerged as a new mode of convergence introduced by M. Ahues (see [11]). It is a way to approximate the non-compact operators resorting to finite-rank operators. Yet it is a pseudo-convergence given that we can find a sequence of bounded linear operators Un which is ν-convergent to U and ν-convergent to V but U ≠ V. However, the spectra σ(⋅) of U and V are the same (see [11, Exercise 2.12]). In [177], S. Sánchez-Perales and S. V. Djordjevic̀ showed that these findings are also proper of the approximate point spectrum. More precisely, they are interested in studying the ν-convergence properties as well as some spectral properties. Later, a first main finding was developed by A. Ammar (see [16, Theorem 2.2]), in which he proved that the Wolf and Weyl essential spectra of U and V are equal. Then he inspected the relationship between the Wolf and Weyl essential spectra of Un and U for Un being ν-convergent to U. Particularly, A. Ammar proved that, if (Un )n is a sequence of bounded linear operators and U is a bounded linear operator mapping X into X such that Un is ν-convergent to U and if 𝒪 ⊆ ℂ is an open set with 0 ∈ 𝒪, then there exists n0 ∈ ℕ such that, for all n ≥ n0 , we have σi (Un ) ⊆ σi (U) + 𝒪 for i = f , w, where σf (⋅) (respectively σw (⋅)) is the Wolf (respectively Weyl) essential spectrum. Last but not least, inspired by the notion of ν-convergence, A. Ammar introduced the ν-continuity. It is to keep in mind that the function σ(⋅) is upper semicontinuous, i. e., if Un → U, then lim sup σ(Un ) ⊂ σ(U) and that, in non-commutative algebras, it does have points of discontinuity. The work of J. Newburgh [163] includes a very interesting discussion, including illustrative examples, about the fundamental results on spectral continuity in general Banach algebras. J. Conway and B. Morrel in [67] have undertaken a detailed study of spectral continuity in the case where the Banach algebra is the C ∗ -algebra of all operators acting on a complex separable Hilbert space. In [165], K. K. Oberai has postulated that the function σw (⋅) is upper semi-continuous, but not continuous at U. However, A. Ammar showed that, if Un → U and Un U = UUn for all n ∈ ℕ, then σw (⋅) is continuous at U. Furthermore, if U is a bounded Fredholm linear operator, then it is ν-upper semicontinuous at U.

4 | 1 Introduction The relationship between the continuity of 2 × 2 upper block operator matrices and the continuity of its component was investigated in [194] whereas A. Ammar, in his work [16], studied the ν-continuity of some 2 × 2 block operator matrices.

1.4 Γ-hypercyclic set of linear operators First, to settle some terminology, let us recall that, if T is a bounded linear operator acting on some topological vector space, the T-orbit of a vector x ∈ X is the set orbit orb(T, x) = {T n x, n ∈ ℕ}. The operator is said to be hypercyclic if there exists some vector x ∈ X whose T-orbit is dense in X. Such a vector is said to be T-hypercyclic. We note that the notion makes sense only in separable Banach space (see [174]). Clearly, in non-separable Banach space there are no hypercyclic operators. We refer the reader to the book of F. Bayart and É. Matheron [41] for all the notions and notations above. Based on the study of hypercyclic operators, A. Ben Amar, A. Jeribi, B. Krichen, and El. H. Zerouali in [47] discussed some algebraic properties of a new set defined by Γhyp (T) := {λ ∈ ℂ: T − λ is hypercyclic}, for a given bounded linear operator T acting on separable Banach space. More precisely, they proved that, if T is a bounded operator acting in a Banach space X, then Γhyp (T) ⊂ {λ ∈ ℂ such that σ(T − λ) ∩ 𝕊ℂ (0, 1) ≠ 0} and Γhyp (T) ⊂ Δ(0, r(T) + 1), where 𝕊ℂ (0, 1) = {λ ∈ ℂ such that |λ| = 1} and Δ(0, r(T) + 1) is the complex closed disc centered at 0, having radius r(T) + 1. In particular, if λ ∈ Γhyp (T) and |σ(T − λ)| is countable, then σ(T − λ) ⊂ 𝕊ℂ (0, 1). Several fundamental examples are treated in detail: hyponormal operators [50], operators satisfying the Bishop property, and weighted shifts on lp (1 ≤ p < ∞) spaces. The key point here is the criterion of G. Godefroy and J. H. Shapiro [93], who gave in [93] a sufficient condition for hypercyclicity that is a consequence of Kitai’s criterion [141]. The result shows that bounded operators acting on infinite-dimensional separable complex Hilbert space with a sufficiently large supply of eigenvectors are hypercyclic. The authors in [47] conclude by applying the obtained results to Aluthge transforms, to a class of bounded linear operators satisfying ABA = A2 and BAB = B2 (see for instance [180]), and to upper triangular matrix operators.

1.5 Analytic operators in Béla Szökefalvi-Nagy’s sense

| 5

1.5 Analytic operators in Béla Szökefalvi-Nagy’s sense Historically, the perturbation method was developed as an approximation device in classical and quantum mechanics. In the perturbation theory of linear operators, the crucial problem is the study of the behavior of spectral properties of linear operators undergoing a small change. The foundation of the mathematical theory including a complete convergence proof of perturbation series has been developed by several authors. The most complete results obtained by A. Jeribi [124], T. Kato [139], and F. Rellich [172] are mainly concerned with the regular perturbation of self-adjoint operators of a Hilbert space, while some attempts have also been made towards the treatment of non-regular cases which are no less important in applications. Another generalization of the theory was given by B. Sz.-Nagy [161]. By his elegant and powerful method of contour integration, he has been able to transfer most of the theorems for self-adjoint operators to a wider class of closed linear operators in Banach space. More precisely, in [161] B. Sz.-Nagy has considered the perturbed operator T(ε) := T0 + εT1 + ε2 T2 + ⋅ ⋅ ⋅ + εk Tk + ⋅ ⋅ ⋅ ,

(1.5.1)

where ε ∈ ℂ and T0 , T1 , T2 , T3 , . . . are linear operators on X, having the same domain 𝒟 and satisfying the condition ‖Tk φ‖ ≤ qk−1 (a‖φ‖ + b‖T0 φ‖)

(1.5.2)

for all φ ∈ 𝒟 and for all k ≥ 1, where a, b, q > 0. Mainly, B. Sz.-Nagy has proved in [161] that the series T0 φ + εT1 φ + ε2 T2 φ + ⋅ ⋅ ⋅ + εk Tk φ + ⋅ ⋅ ⋅ converges for all φ ∈ 𝒟(T0 ) and for all |ε| < q−1 . Let T(ε)φ be its limit. Then T(ε) is a linear operator with domain 𝒟(T0 ). Moreover, it was shown that, if T0 is closed, then, for |ε| < (q + b)−1 , T(ε) is closed. On the other hand, if (λn )n denote the isolated eigenvalues of the operator T0 with multiplicity one associated to the eigenvectors (φn )n , then, for each λn , there exists a neighborhood of λn in which there is a unique eigenvalue λn (ε) of the operator T(ε) with multiplicity one. Moreover, for |ε| small enough, λn (ε) and the corresponding eigenvector φn (ε) of T(ε) can be developed as an entire series of ε as follows: λn (ε) = λn + ελn1 + ε2 λn2 + ⋅ ⋅ ⋅ ,

φn (ε) = φn + εφ1n + ε2 φ2n + ⋅ ⋅ ⋅ . Motivated by some mathematical models which occur for a problem of radiation of a vibrating structure in a light fluid [90] and in Reggeon field theory due to V. Gribov [98], I. Feki, A. Jeribi, and R. Sfaxi studied the above operator T(ε), where ε ∈ ℂ and T0 is a closed densely defined linear operator on a separable Banach space X with domain

6 | 1 Introduction 𝒟(T0 ), while T1 , T2 , . . . are linear operators on X with the same domain 𝒟 ⊃ 𝒟(T0 ) and

satisfying the following growing inequality:

‖Tk φ‖ ≤ qk−1 (a‖φ‖ + b‖T0 φ‖β ‖φ‖1−β ),

(1.5.3)

for all φ ∈ 𝒟(T0 ) and for all k ≥ 1, with β ∈ ]0, 1] and a, b, q > 0. This new condition leads to a considerable improvement to the estimation of the convergence radius for λn (ε) and φn (ε) and to the estimation of the coefficients (λni )i≥1 and (φin )i≥1 .

1.6 Bases of the perturbed operator T (ε) Recently, the perturbed operator T(ε) drew the attention of A. Jeribi and his collaborators, which led to a new progress in the theory of non-self-adjoint operators. In fact, N. Ben ali and A. Jeribi studied in their paper [42] the completeness and the existence of Riesz bases of generalized eigenvectors in a separable Hilbert space when condition (1.5.2) holds. Later, taking account of the growing inequality (1.5.3), I. Feki, A. Jeribi, and R. Sfaxi were interested in the completeness of generalized eigenvectors of the perturbed operator T(ε) and proved in [87] that the system of eigenvectors of the perturbed operator T(ε) forms a Schauder basis in a separable Banach space. On the other hand, it is known mathematically that the family of exponentials {eint }n∈ℤ forms an orthonormal basis in L2 (0, 2π). The natural question that arises is what happens if we replace it by the classical system of exponentials {eiλn t }n∈ℤ . This question finds its origin in the celebrated work of R. Paley and N. Wiener [166] on L2 (0, T), where T > 0. They proved that, if λn ∈ ℝ, n ∈ ℤ, and sup |n − λn | < π −2 , n∈ℤ

then the system {eiλn t }n∈ℤ forms a Riesz basis in L2 (0, 2π). Some progress has been made by many mathematicians such as A. E. Ingham [108] in 1936 and M. I. Kadeč [137] in 1964. They have established that, if |λn − n| ≤ δ,

n ∈ ℤ,

then the precedent family forms a Riesz basis in L2 (0, 2π) if and only if δ < 41 . However, all these researches are based on the fact that {eiλn t }n∈ℤ is close to the orthonormal basis {eint }n∈ℤ . A different approach to the description of the Riesz bases of exponentials was suggested by B. J. Levin [152]. In this approach, the main role is played by the generating function of {eiλn t }∞ 1 and the notion of sine-type functions. In other words, if {λn }∞ is the set of zeros of a sine-type generating function of the exponential system 1 2 2 {eiλn t }∞ in L (0, T) and is separated, then {eiλn t }∞ 1 1 forms a Riesz basis in L (0, T).

1.7 Frames in Hilbert spaces | 7

Over the last 80 years, the research in this area has been greatly increased. New approaches to old problems have led to significant advances in the theory. In fact, a powerful approach to the Riesz basis concept was developed by B. S. Pavlov [168]. He used a geometrical approach to derive a precise description of Riesz bases of {eiλn t }∞ 1 in terms of the generating function. The famous theorem due to B. S. Pavlov on the Riesz basis property of exponential families opens up many problems, such as the problem of radiation of a vibrating structure in a light fluid initially motivated by P. J. T. Filippi et al. [90]. This problem was considered in [58], where the authors proved that there exists a sequence of complex numbers (εn )n such that the sequence of the exponential family associated to the eigenvalues of the operator (I + εn K)−1

d2 dx2

forms a Riesz basis in L2 (0, T), for some T > 0, where K is the integral operator with kernel the Hankel function of the first kind and order 0. More precisely, the authors studied the existence of a Riesz basis of exponentials of the perturbed operator T(ε) where inequality (1.5.2) is satisfied and they applied their obtained results in the problem of radiation of a vibrating structure in a light fluid, but the weakness of these results relies on the fact that the Riesz basis of exponentials given in [58] depends on the sequence (εn )n and is not related to the exact eigenvalue problem considered in [90]. It is along this line of thoughts that the authors H. Ellouz, I. Feki, and A. Jeribi in [81, 83] tried to give some supplements to the results developed in [58] in order to prove the existence of a Riesz basis of exponentials of the perturbed operator T(ε) where inequality (1.5.3) is fulfilled. Their results lead to a Riesz basis of exponentials of the integro-differential operator (I + εK)−1

d2 , dx2

for a fixed ε. Recently, some progress has been made for the existence of an unconditional basis with parentheses of generalized eigenvectors associated to the analytic operator T(ε). In fact, inspired by some important results established in [15], I. Feki, A. Jeribi, and R. Sfaxi provided in [86] some necessary conditions which ensure the existence of an unconditional basis with parentheses of T(ε) where condition (1.5.3) is fulfilled.

1.7 Frames in Hilbert spaces Frames were first introduced in 1952 by R. J. Duffin and A. C. Schaeffer [74], reintroduced in 1986 by I. Daubechies et al. [69], and developed later by several authors,

8 | 1 Introduction such as O. Christensen [62–65] and R. M. Young [194], to study some deep problems in non-harmonic Fourier series. Frames, which are systems that provide robust, stable, and usually non-unique representations of vectors, have found many practical applications both in mathematics and engineering where redundancy plays a vital and useful role. Moreover, they also have important stability properties, such as unconditional convergence and the existence of an equivalent discrete-type norm for the space via the analysis operator. Actually, a sequence of vectors is a Riesz basis if and only if it is a frame and is ω-linearly independent. Frames are a much more flexible tool than orthonormal bases since the basis condition is very strong. Indeed, it might be difficult to find a basis satisfying extra conditions that a certain application requires, while the frame condition is weaker. Therefore, one can find a frame enjoying special properties which are impossible for a basis. Further, a frame can be viewed as an overcomplete set. More precisely, a frame is a redundant set of vectors in a separable Hilbert space X with the property that provides usually non-unique representations of vectors in terms of the frame elements. This property is desirable for many situations, such as wavelet and frequency analysis theories, filter bank theory, and signal and image processing, but it contradicts the one of the bases. In fact, the lack of uniqueness opens up the possibility of choosing the coefficients that fit a certain application best. One can ask the question when a frame can be considered as a basis. Actually, many authors, such as R. M. Young [194], have proved that frames can be divided into two classes, called exact frames and inexact frames. Recall that a frame is said to be exact if it ceases to be a frame whenever any of its elements is removed. Further, they showed that Riesz bases are equivalent to exact frames. To handle these properties, H. Ellouze, I. Feki, and A. Jeribi had the idea in their paper [82] to extend the results developed in [42, 81, 83, 87, 127] to the concept of frames. More precisely, in [82], the authors prove, under some conditions, the existence of a frame of eigenvectors related to the considered integro-differential operator and they generalized this result to the analytic operator (1.5.1) introduced by B. Sz.-Nagy and acting on a separable Hilbert space X in order to ameliorate some results of [42], where the authors proved the existence of a Riesz basis of eigenvectors related to the perturbed operator T(εn ). Since Riesz bases are frames with the property of ω-linearly independence, the authors in [84] gave some conditions to guarantee the existence of frames of eigenvectors of T(εn ) and hence they extended the results of [42]. However, the frame of eigenvectors {φn (εn )}∞ n=1 depends on a sequence of a complex number, (εn )∞ , and it is related to a sequence of operators (T(εn ))∞ n=1 n=1 and not to the operator T(ε). Hence, the authors in [82] also ensure the existence of a fixed complex number ε for which the family {φn (ε)}∞ n=1 of the perturbed operator T(ε) forms a frame for X. On the other hand, the concept of a non-orthogonal fusion frame was initially motivated by J. Cahill et al. [52] as a slight modification of the fusion frame (or frame of

1.7 Frames in Hilbert spaces | 9

subspaces) established by P. G. Cassaza and G. Kutyniok [54] as a natural generalization of frame theory. As remarked in [54], the relevance of this notion is a consequence of the fact that it gives criteria for constructing a frame for X by joining sequences of frames for subspaces of X. This notion has been intensely studied during the last years and several new applications have been discovered. The difference between the nonorthogonal fusion frames and the fusion frames is the use of non-orthogonal projections instead of orthogonal projections. Recently, H. Ellouze, I. Feki, and A. Jeribi in [84] discussed the perturbed operator (1.5.1) in order to study the existence of related non-orthogonal fusion frames. Indeed, in [161], B. Sz.-Nagy proved that, if we denote by Pn the spectral projection of T0 related to the eigenvalue λn , then, for |ε| small enough, there exists a sequence of spectral projections {Pn (ε)}∞ n=1 of T(ε) that can be developed as an entire series of ε as follows: Pn (ε) = Pn + εPn,1 + ε2 Pn,2 + ⋅ ⋅ ⋅ .

(1.7.1)

Based on the estimates given in [161], the authors establish in [84], under sufficient conditions, the existence of a sequence of complex numbers (εn )∞ n=1 and a sequence of ∞ spectral projections {Pn (εn )}∞ of (T(ε )) having the form (1.7.1) such that the system n n=1 n=1 {Pn (εn ), vn }∞ forms a non-orthogonal fusion frame for X, where (vn )∞ n=1 n=1 is a family of weight. Note here that equation (1.7.1) plays a crucial role in the existence of the nonorthogonal fusion frame related to the perturbed operator (1.5.1). In fact, it allows one to get a considerable improvement to the results developed in [42] and [87] since the eigenvalue λn of T0 does not necessarily have multiplicity one. ∞ However, the non-orthogonal fusion frame {Pn (εn ), vn }∞ n=1 depends on (εn )n=1 . Fur∞ ther, it is related to a sequence of operators (T(εn ))n=1 and not to the operator (1.5.1). In this context and in order to get such improvements, the authors studied in [84] the existence of a fixed ε for which the families N

∞

{Pn (ε), vn }1 ∪ {Pn (εn ), vn }N+1 and N

{Pn (ε), vn }1 ∪ {Pn , vn }∞ N+1

(N ≥ 1)

form non-orthogonal fusion frames for X. More precisely, they show that, for |ε| small enough, the first N projections coincide with a sequence of spectral projections (Pn (ε))1≤n≤N of T(ε) that can be developed as an entire series of ε. Again, these families more or less rely on ε. Indeed, it is clear here that either the first N projections are associated with the perturbed operator T(ε) or all the projections are related to a sequence of operators (T(εn ))∞ n=1 , so the authors ensure in [84] the existence of a fixed complex number ε so that the family {Pn (ε), vn }∞ n=1 is a non-orthogonal fusion frame for X.

10 | 1 Introduction

1.8 Applications 1.8.1 Perturbation method for sound radiation by a vibrating plate in a light fluid We consider an elastic membrane lying in the domain −a < x < a of the plane y = 0. It is embedded along the two straights x = −a and x = a, in the two perfectly rigid half-planes (x < −a, y = 0) and (x > a, y = 0). The two half-spaces y < 0 and y > 0 are filled with gas. Finally, the membrane is excited by a harmonic force exp(−iwt) with an amplitude f (x) which is independent of the third space variable. Thus, the equation of the motion of the membrane is reduced to the equation of the vibrant cord. The fluid motion is described by a Helmholtz equation in ℝ2 . The physical characteristics of the system are: – ρ0 : fluid density; – c0 : fluid sound speed; – ρ1 : surface density of the membrane; – T1 : the membrane tightness; and 1 ρ – c1 = ( T1 )− 2 : the flexion wave speed in the membrane. 1

Let u denote the displacement of the membrane and p the acoustic pressure in the fluid. The motion equations are given by (Δ + (

ω2 )p(M) = 0 c02

in y < 0 and y > 0,

(1.8.1)

ρ d2 1 + ω2 1 )u(x) = − (f (x) − p+ (x) + p− (x)), T1 T1 dx 2

p+ (x) =

lim p[M(x, y)],

y>0 y→0

p− (x) =

(1.8.2)

lim p[M(x, y)],

y a, 𝜕y

(1.8.4)

u(−a) = u(a) = 0.

(1.8.6)

(1.8.5)

Using equations (1.8.1), (1.8.4), (1.8.5), and (1.8.6), the pressure p(M) is given by the following integral: a

p(M) = −iδ p(M) = iδ

ω2 ρ0 2 ∫ H0 (k0 √(x − x 󸀠 ) + y2 )u(x󸀠 )dx󸀠 , 2 2

for y > 0,

(1.8.7)

−a a

ω ρ0 2 ∫ H0 (k0 √(x − x 󸀠 ) + y2 )u(x 󸀠 )dx󸀠 , 2 −a

for y < 0,

(1.8.8)

1.8 Applications | 11

where H0 (z) = J0 (z) + iY0 (z) is the Hankel function of the first kind and order 0 and k0 = cω is the wave number of the fluid. Using (1.8.2), (1.8.3), and (1.8.6)–(1.8.8) leads 0 to the following boundary value problem: a

iρ ρ f (x) d2 u (x) + ω2 { 1 u(x) + 0 ∫ H0 (k0 |x − x󸀠 |)u(x󸀠 )dx󸀠 } = , 2 T1 T1 T1 dx

∀ −a < x < a, (1.8.9)

−a

u(−a) = u(a) = 0.

(1.8.10)

In the sequel we shall need the following operators: T0 : 𝒟(T0 ) ⊂ L2 (]−a, a[) 󳨀→ L2 (]−a, a[), { { { { d2 ψ ψ 󳨀→ T0 ψ(x) = − 2 (x), { { dx { { 1 2 {𝒟(T0 ) = H0 (]−a, a[) ∩ H (] − a, a[)

(1.8.11)

K : L2 (]−a, a[) 󳨀→ L2 (]−a, a[) { { { { a i { { ψ 󳨀→ Kψ(x) = H0 (k|x − x 󸀠 |)ψ(x󸀠 )dx 󸀠 . ∫ { { 2 −a {

(1.8.12)

and

From the problem (1.8.9)–(1.8.10), P. J. T. Filippi, in [90], has considered the following eigenvalue problem. Find the values λ ∈ ℂ for which there is a solution u ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[), u ≠ 0, for the following equation: T0 u = λ(I + εK)u, ω2 ρ

(1.8.13)

2ρ

where λ = T 1 and ε = ρ 0 . 1 1 According to the definition given in [160, Chapter 9, Section 4], λ is the eigenvalue and u is the eigenmode. Note that λ and u both depend on the value of ε, so we denote λ := λ(ε) and u := u(ε). 1 For |ε| < ||K|| , the operator (I + εK)−1 is invertible, so the problem (1.8.13) becomes the following. Find the values λ(ε) ∈ ℂ for which there is a solution φ ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[), φ ≠ 0, for the following equation: (I + εK)−1 T0 φ = λ(ε)φ.

(1.8.14)

The last problem (1.8.14) is equivalent to the following. Find the values λ(ε) ∈ ℂ for which there is a solution φ ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[), φ ≠ 0 for the following equation: (T0 − εKT0 + ε2 K 2 T0 − ⋅ ⋅ ⋅ + (−1)n εn K n T0 + ⋅ ⋅ ⋅)φ = λ(ε)φ.

(1.8.15)

12 | 1 Introduction In recent years, important progress has been made in the study of spectral properties of the above operator. Indeed, in [42], the authors are interested in giving some compactness results about the Hankel operator (1.8.12). They also established some spectral properties about the operator T0 , defined in (1.8.11), and proved the convergence of the series (1.8.15) to the perturbed operator T(ε). This operator was shown in [42] to be closed with compact resolvent and its system of generalized eigenvectors is dense and forms a Riesz basis in the suitable Hilbert space. Later, this operator T(ε) attracted the attention of I. Feki, A. Jeribi, and R. Sfaxi [86], who proved the existence of its unconditional basis with parentheses. On the other hand, S. Charfi, A. Jeribi, and I. Walha are concerned, in [58], with the Riesz basis property of families of non-harmonic exponentials of this perturbed operator. Recently, the authors in [82] established some results about the existence of frames.

1.8.2 The shape memory alloys operator We consider A : 𝒟(A) ⊂ L2 (0, 1) 󳨀→ L2 (0, 1), { { { { d4 ψ ψ 󳨀→ Aψ(x) = γ (x), { { dx4 { { 4 󸀠󸀠 󸀠󸀠 {𝒟(A) = {ψ ∈ H (0, 1) such that ψ(0) = ψ(1) = ψ (0) = ψ (1) = 0}, where γ is a positive constant arising in Landau–Ginzburg potentials. A. Intissar in [112] is interested in this operator and study the denseness of the generalized eigenvectors and the diagonalization of some semi-groups related to this operator.

1.8.3 Heat exchanger equation with boundary feedback In this section, we are concerned with the following type of counter-flow heat exchanger equation with boundary feedback: 𝜕θ1 𝜕θ { (t, x) = −v1 1 (t, x) + h1 (θ2 (t, x) − θ1 (t, x)) { { { 𝜕t 𝜕x { { { { 𝜕θ 𝜕θ { 2 (t, x) = v 2 (t, x) + h (θ (t, x) − θ (t, x)) 2 2 1 2 𝜕t 𝜕x { { { { { θ1 (t, 0) = −k1 θ2 (t, 0), θ2 (t, l) = −k2 θ1 (t, l) { { { { {θ1 (0, x) = θ10 (x), θ2 (0, x) = θ20 (x)

for (t, x) ∈ (0, ∞) × [0, l], for (t, x) ∈ (0, ∞) × [0, l],

(1.8.16)

for t ∈ (0, ∞), for x ∈ [0, l],

where v1 , v2 , h1 , h2 , and l are positive physical parameters and k1 and k2 are feedback gains (in [143] they are assumed to be non-negative). For the sake of generality, we

1.8 Applications | 13

assume that k1 , k2 ∈ ℝ and k1 k2 ≠ 0. Let ℋ := L2 [0, l] × L2 [0, l] with inner product l

⟨f , g⟩ := ∫[f1 (x)g1 (x) + f2 (x)g2 (x)]dx, 0 T

where f := [f1 , f2 ] and g := [g1 , g2 ]T ∈ ℋ. We write system (1.8.16) as follows: 1 (t, x) −v1 𝜕θ −h 𝜕 θ1 (t, x) ( ) = ( 𝜕θ𝜕x )+( 1 2 h2 𝜕t θ2 (t, x) v2 (t, x)

𝜕x

θ (0, x) θ (x) ( 1 ) = ( 10 ) , θ2 (0, x) θ20 (x)

θ1 (t, 0) = −k1 θ2 (t, 0),

h1 θ (t, x) )( 1 ), −h2 θ2 (t, x)

θ2 (t, l) = −k2 θ1 (t, l).

In ℋ, the operator A is defined by −v1 f1󸀠 −h )+( 1 h2 v2 f2󸀠

Af := (

T

f h1 ) ( 1) −h2 f2

for f ∈ 𝒟(A), 󸀠

󸀠

2

𝒟(A) := {f := [f1 , f2 ] ∈ ℋ such that f1 , f2 ∈ L [0, l], f1 (0) = −k1 f2 (0), f2 (l) = −k2 f1 (l)}.

Then (1.8.16) can be written into an evolutionary equation in ℋ as follows: d { Θ(t) = AΘ(t), dt { {Θ(0) = Θ0 ,

t > 0,

with Θ(t) := [θ1 (t, x), θ2 (t, x)]T and Θ0 = (θ10 (x), θ20 (x))T . Set α1 = 1 . v2

h1 , v1

β1 =

h2 , v2

α2 =

1 , v1

and β2 = A complete spectral analysis of the above operator A was carried out in [143, 191]. In fact, it was shown that the operator A has compact resolvent and generates a uniformly bounded C0 -semi-group. The localization of the eigenvalues is well studied, as well as the stability of the semi-group generated by the operator A. The authors in [191] showed that the generalized eigenvectors of A are complete in a Hilbert space setting and studied the asymptotic behavior of the eigenvalues of the operator A. 1.8.4 Expansion of solution for a hyperbolic system We concentrate our attention in this subsection to give a unified treatment for the following general class of linear hyperbolic systems: u(x, t) h(x, t) d u(x, t) { ( )+𝒞( )=( ), { { { w(x, t) w(x, t) k(x, t) dt { { { u(0, t) = α1 w(0, t), { { { {u(L, t) = β1 w(L, t), α1 , β1 ≠ {−1, 0, 1},

14 | 1 Introduction where 0 d −c(x) dx

𝒞 := (

d −c(x) dx ), 0

(1.8.17)

with a strictly positive function c(x), which was first studied by A. Intissar in [110] in the case where the function c(x) is considered a constant equal to one with a unique restrictive termination. A nice summary of the later development can be found in [140] under some transformations on the operator 𝒞 . Among them, a powerful concept of Riesz bases was introduced to prove that the eigenvectors of 𝒞 form a Riesz basis in Hilbert space. However, it appears that a mathematical treatment of this problem has not yet been developed in the case where the operator 𝒞 is given in the above form (1.8.17). The authors A. Intissar, A. Jeribi, and I. Walha showed in [113] that the Riesz basis property, as well as the spectrum-determined growth condition, holds for the studied system. 1.8.5 Frame of a one-dimensional wave control system We consider in this subsection a controlled wave system given by wtt (x, t) = wxx (x, t), 0 < x < 1, t > 0, { { { { { { wx (0, t) = wt (0, t) + γw(0, t), { { wx (1, t) = −k2 wt (1, t) − δw(1, t), { { { { { w(x, 0) = w0 (x), wt (x, 0) = w1 (x),

(1.8.18)

where γ, δ > 0 and k2 ≥ 0. The abstract formulation of equation (1.8.18) is equivalent to considering the Hilbert space H 1 (0, 1) × L2 (0, 1) and the operator A given by A(u, v) := (v, u󸀠󸀠 ),

(u, v) ∈ 𝒟(A),

where 2

1

󸀠

𝒟(A) := {(u, v) ∈ H (0, 1) × H (0, 1) such that u (0) = γu(0) + v(0),

u󸀠 (1) = −δu(1) − k2 v(1)}. Then our initial problem can be written as d { W(t) = AW(t), dt { { W(0) = W0 ,

t > 0,

where W(t) := (w(x, t), wt (x, t)) and W0 := (w0 (x), w1 (x)).

1.9 Outline of contents |

15

For this one-dimensional wave control equation, H. Ellouze, I. Feki, and A. Jeribi established in [84] some spectral properties and confirmed the existence of a nonorthogonal fusion frame.

1.8.6 Reggeon field theory It is well known that quantum Hamiltonians are constructed as self-adjoint operators. For certain situations, however, non-self-adjoint Hamiltonians are also of importance. In particular, the Reggeon field theory invented by V. Gribov [98] is an attempt to predict the high-energy behavior of soft processes and is governed by the following magic non-self-adjoint Gribov operator: n

n

n

n

j=1

j=1

j=1

j=1

2 ∗ ∗ ∗ ∗ ∗ ∗ Hλ󸀠 ,μ,α,λ = λ󸀠 ∑ A∗2 j Aj + μ ∑ Aj Aj + iλ ∑ Aj Aj (Aj + Aj )Aj + α ∑[Aj+1 Aj + Aj Aj+1 ].

The complete spectral properties of the non-symmetrical operator Hλ󸀠 ,μ,α,λ arising in the Gribov theory are studied in [109]. Later, the case of null transverse dimension (n = 1) is given by Hλ󸀠󸀠 ,λ󸀠 ,μ,λ = λ󸀠󸀠 A∗3 A3 + λ󸀠 A∗2 A2 + μA∗ A + iλA∗ (A∗ + A)A. This operator has attracted the attention of many researchers to study the existence of some bases of generalized eigenvectors (see [13–15, 59, 110, 112]).

1.9 Outline of contents Moving on to a brief indication of the contents of the individual chapters, we mention first of all that our book consists of 16 chapters. We begin this book with a chapter giving some fundamentals about linear operators and we present the background material that is drawn from functional analysis and operator theory. We continue in Chapter 3 by giving some definitions, notations, and basic results on a functional analysis that underlies most of the concepts presented in this book. We focus in Chapter 4 on the abstract theory of bases in Banach and Hilbert spaces and we recall different bases as well as their basic properties. In Chapter 5, we introduce the strongly continuous semi-groups. With these semigroups we associate a generator and characterize these generators in the Hille–Yosida generation. We also characterize the eigenvalues related to this generator and we study the diagonalization of semi-groups. Chapter 6 is devoted to the study of the Hilbert–Schmidt discrete operators and the denseness of the generalized eigenvectors. In Chapter 7, we introduce the notion

16 | 1 Introduction of frames in Hilbert spaces and we give some perturbation results related to this notion. Moreover, we discuss the notion of frames of subspace or fusion frame and we discuss some properties of synthesis operators. We end this chapter with the study of oblique projections and biorthogonality and we provide details on some concepts such as duals of fusion frames, relations between fusion frames and operators, and non-orthogonal fusion frames. In Chapter 8, we recall some basic facts about the summability of series in the principal vectors of non-self-adjoint operators. An important consideration is given to the Jordan chain, a summation of series by Abel’s method, and we focus on the coefficients of a series in principal vectors. In Chapter 9, considerable attention is devoted to the ν-convergence property of operators. We begin by studying some spectral properties of this concept and we discuss the Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators ν-convergent in a Banach space. We also study the ν-continuity of the Wolf and Weyl essential spectra and we focus on the relationship between the ν-continuity of 2 × 2 block operator matrices and the ν-continuity of its components. The purpose of Chapter 10 is to give some properties of the Γ-hypercyclic set of linear operators and to introduce some conditions that ensure operators to be essentially quasi-nilpotent. We also study operators with a small and big Γ-hypercyclic set, we introduce the Aluthge transforms, and we study some operator equations and the Γ-hypercyclic set of upper triangular matrices. Chapter 11 is devoted to the generalization of some results due to B. Sz.-Nagy in [161] and we establish the invariance of the closure of the perturbed operator given by Béla Szökefalvi-Nagy’s development. Moreover, we characterize the eigenvalues and the eigenvectors of the associated operator and we continue our study in Chapter 12 by proving the completeness and the Riesz basis of generalized eigenvectors. We establish not only the existence of a Schauder basis in a separable Banach space but also the Riesz basis property of families of non-harmonic exponentials and the unconditional basis with parentheses. Interesting details are given in Chapter 13 on the existence of frames of the perturbed operator. We devote ourselves in Chapter 14 exclusively to the investigation of some spectral properties of the operator governing the perturbation method for sound radiation by a vibrating plate in a light fluid, such as compactness and completeness results, closeness operators, denseness and the Riesz basis of eigenvectors or unconditional bases with parentheses, the Riesz basis property of families of non-harmonic exponentials, and finally the existence of frame. Chapter 15 concentrates on a selection of applications to mathematical models. We begin by the heat exchanger equation with a boundary feedback to investigate the semi-group generated by this operator, we determine its eigenvalues in order to study the stability of the semi-group, and we investigate the Riesz basis and the completeness of the system as well as the asymptotic behavior of the eigenvalues of the operator describing the heat exchanger equation. As a second application in this chapter, we

1.9 Outline of contents |

17

study the shape memory alloys operator and we check the denseness of the generalized eigenvectors and the diagonalization of some semi-groups. After that, we focus exclusively on studying the expansion of a solution for a hyperbolic system and the frame of a one-dimensional wave control system. Finally, we are interested in Chapter 16 in the Reggeon field theory governed by the Gribov operator in Bargmann space in order to illustrate the applicability of some results of this book, such as the denseness of the generalized eigenvectors, subordinate and boundedness properties, the unconditional basis in Bargmann space, generalized diagonalization of some semi-groups, and the Riesz basis of finite-dimensional invariant subspaces.

2 Linear operators Let X and Y be two Banach spaces. A mapping A which assigns to each element x of a set 𝒟(A) ⊂ X a unique element y ∈ Y is called an operator (or transformation). We use the notation A(X 󳨀→ Y). The set 𝒟(A) on which A acts is called the domain of A. The operator A is called linear if 𝒟(A) is a subspace of X and if A(λx + βy) = λAx + βAy for all scalars λ, β and all elements x, y in 𝒟(A). The operator A is called bounded if there is a constant M such that ‖Ax‖ ≤ M‖x‖,

x ∈ X.

The norm of such an operator is defined as follows: ‖A‖ = sup x =0 ̸

‖Ax‖ . ‖x‖

(2.0.1)

The operator A is norm-preserving or isometric if ‖Ax‖ = ‖x‖ for every x ∈ X. If, in particular, R(A) = X, then a (bounded) isometric operator A is called a unitary operator. If A is linear, bijective, and isometric, then we call A an isometric isomorphism.

2.1 Closed and closable operators The graph G(A) of a linear operator A on 𝒟(A) ⊂ X into Y is the set {(x, Ax) such that x ∈ 𝒟(A)} in the product space X × Y. Then A is called a closed linear operator when its graph G(A) constitutes a closed linear subspace of X × Y. Hence, the notion of a closed linear operator is an extension of the notion of a bounded linear operator. A sequence (xn )n ⊂ 𝒟(A) will be called A-convergent to x ∈X if both (xn )n and (Axn )n are Cauchy sequences and xn → x. A linear operator A on 𝒟(A) ⊂ X into Y is said to be closable if A has a closed extension. It is equivalent to the condition that the graph G(A) is a submanifold (or subspace) of a closed linear manifold (or space) which is at the same time a graph. It follows that A is closable if and only if the closure G(A) of G(A) is a graph. We are thus led to the following criterion. A is closable if and only if no element of the form (0, x), x ≠ 0, is the limit of elements of the form (x, Ax). In other words, A is closable https://doi.org/10.1515/9783110493863-002

20 | 2 Linear operators if and only if (xn )n ∈ 𝒟(A), xn → 0, and Axn → x imply x = 0. When A is closable, there is a closed operator A with G(A) = G(A). A is called the closure of A. It follows immediately that A is the smallest closed extension of A, in the sense that any closed extension of A is also an extension of A. Since x ∈ 𝒟(A) is equivalent to (x, Ax) ∈ G(A), x ∈ X belongs to 𝒟(A) if and only if there exists a sequence (xn )n that is A-convergent to x. In this case, we have Ax = lim Axn . n→∞

By 𝒞 (X, Y) we denote the set of all closed, densely defined linear operators from X into Y and by ℒ(X, Y) we denote the Banach space of all bounded linear operators from X into Y. If X = Y, the sets ℒ(X, Y) and 𝒞 (X, Y) are replaced, respectively, by ℒ(X) and 𝒞 (X). For an introduction to the theory of bounded or unbounded linear operators, we refer to the books of E. B. Davies [71], N. Dunford and J. T. Schwartz [78], I. C. Gohberg, S. Goldberg, and M. A. Kaashoek [94], A. Jeribi [123–125, 135], and T. Kato [139]. By an operator A from X into Y, we mean a linear operator with a domain 𝒟(A) ⊂ X. We denote by N(A) its null space and by R(A) its range. For injective A, the inverse A−1 (Y 󳨀→ X) is an operator with 𝒟(A−1 ) = R(A) and R(A−1 ) = 𝒟(A). We say that a subspace D ⊂ 𝒟(A) is a core for A if, for every x ∈ 𝒟(A), there exists a sequence (xn )n in D such that xn → x and Axn → Ax. If A is closed, a subset D ⊂ 𝒟(A) is called a core for A if A|D = A. It is easy to see that a bounded operator defined on the whole Banach space X is closed. The inverse is also true and this follows from the closed graph theorem, which is the following theorem. Theorem 2.1.1. If X, Y are Banach spaces and A is a closed linear operator from X into Y with 𝒟(A) = X, then: (i) there are positive constants M, r such that ‖Ax‖ ≤ M whenever ‖x‖ < r; (ii) A ∈ ℒ(X, Y).

2.2 Basis Definition 2.2.1. Two inner products are said to be equivalent if they generate equivalent norms. Definition 2.2.2. Two vectors x, y in a Hilbert space are orthogonal if ⟨x, y⟩ = 0. Definition 2.2.3. Let (xn )n be a sequence in a Hilbert space X. If ⟨xm , xn ⟩ = 0, (xn )n is an orthogonal sequence whenever m ≠ n.

2.2 Basis | 21

Definition 2.2.4. Let (xn )n be a sequence in a Hilbert space X. It is called an orthonormal sequence if ⟨xm , xn ⟩ = δmn , where δmn is the symbol of Kronecker or the Kronecker delta, i. e., δmn = {

0 1

if m ≠ n, if m = n,

i. e., (xn )n is orthogonal and ‖xn ‖ = 1 for every n. Definition 2.2.5. Let X be a Hilbert space and {en }∞ n=1 be a sequence of mutually orthogonal vectors of X. The linear span of {en }∞ , denoted by span{en }∞ n=1 n=1 , is the smallest linear subspace of X (the set of finite linear combinations) containing {en }∞ n=1 . Definition 2.2.6. A set of vectors {en }∞ n=1 on a Banach space X is said to be ω-linearly independent if the equality ∞

∑ cn en = 0

n=1

is possible only for cn = 0 (n = 1, 2, 3, . . .). Definition 2.2.7 (Biorthogonal systems). Given a Banach space X and given sequences (xn )n ⊂ X and (an )n ⊂ X ∗ (the adjoint space of X), we say that (an )n is biorthogonal to (xn )n if ⟨xm , an ⟩ = δmn for every m, n ∈ ℕ. We call (an )n a biorthogonal system or a dual system to (xn )n . Let l2 (ℕ) be the space defined by +∞

l2 (ℕ) = {(xn )n≥1 such that xn ∈ ℂ and ∑ |xn |2 < ∞}. n=1

Definition 2.2.8. A sequence (en )n is a basis for a separable Hilbert space X if: (i) it is complete, i. e., ⟨u, en ⟩ = 0 ∀n ∈ ℕ 󳨐⇒ u = 0; and (ii) the en are independent, i. e., if ∞

∑ cn en = 0

n=0

for some sequence (cn )n ∈ l2 (ℕ), then cn = 0 for all n ∈ ℕ.

22 | 2 Linear operators Definition 2.2.9. Let X be a pre-Hilbert space and {en }∞ n=1 be a sequence of mutually orthogonal vectors of X. Then: ∞ (i) {en }∞ n=1 is called a Hilbert basis of X if {en }n=1 is orthonormal (i. e., elements in ∞ {en }∞ n=1 have norm 1 and are pairwise orthogonal) and {en }n=1 is total, i. e., the lin∞ ear span of {en }n=1 is dense in X; and (ii) every vector in X can be written uniquely as an infinite linear combination of the vectors in the basis.

2.3 Fredholm operators Let X and Y be two Banach spaces. By an operator A from X into Y we mean a linear operator with domain 𝒟(A) ⊂ X and range R(A) ⊂ Y. Let A ∈ 𝒞 (X, Y). The nullity, α(A), of A is defined as the dimension of N(A) and the deficiency, β(A), of A is defined as the codimension of R(A) in Y. α(A) = ∞ and β(A) = ∞ when N(A) and X/R(A) are infinitedimensional, respectively. If the range R(A) of A is closed and α(A) < ∞ (respectively β(A) < ∞), then A is called an upper (respectively lower) semi-Fredholm operator denoted by Φ+ (X, Y) (respectively Φ− (X, Y)). A semi-Fredholm operator is an upper or a lower semi-Fredholm operator denoted by Φ± (X, Y). If both α(A) and β(A) are finite, then A is called a Fredholm operator, denoted by Φ(X, Y). For A∈Φ± (X, Y), the number i(A) = α(A)−β(A) is called the index of A. It is clear that, if A∈Φ(X, Y), then i(A) < ∞. If A ∈ Φ+ (X, Y)\Φ(X, Y), then i(A) = −∞ and if A ∈ Φ− (X, Y)\Φ(X, Y), then i(A) = +∞. Let Φb (X, Y), Φb+ (X, Y), and Φb− (X, Y) denote the sets Φ(X, Y) ∩ ℒ(X, Y), Φ+ (X, Y) ∩ ℒ(X, Y), and Φ− (X, Y) ∩ ℒ(X, Y), respectively. If X = Y, the sets Φ(X, Y), Φ+ (X, Y), Φ− (X, Y), Φ± (X, Y), Φb (X, Y), Φb+ (X, Y), and Φb− (X, Y) are replaced, respectively, by Φ(X), Φ+ (X), Φ− (X), Φ± (X), Φb (X), Φb+ (X), and Φb− (X). The set of all points λ for which λ − A is a Fredholm operator from X into X is called the Fredholm set of A and will be denoted by ΦA . As is well known, the Fredholm set of an operator A is open. A complex number λ is in ΦbA if λ − A is a bounded Fredholm linear operator from X into X. Let ε > 0. We denote by 𝔹(λ, ε) the open ball in ℂ, centered at λ and with a radius ε. It is obvious that the sum M + N of two linear subspaces M and N of a vector space X is again a linear subspace. If M ∩ N = {0}, then this sum is called the direct sum of M and N and will be denoted by M ⊕ N. In this case, for every z = x + y in M + N, the components x, y are uniquely determined. If X = M ⊕ N, then N is called an algebraic complement of M. 2.3.1 Bounded operators The following theorem is developed by A. Lebow and M. Schechter in [150, Lemma 4.5]. Theorem 2.3.1. Let U ∈ Φb (X) and V ∈ ℒ(X). If there exists η > 0 such that ‖U − V‖ < η, then V ∈ Φb (X) and i(U) = i(V).

2.3 Fredholm operators | 23

Theorem 2.3.2 (M. Schechter [178]). Let A ∈ ℒ(X, Y) and B ∈ ℒ(Y, Z), where X, Y, and Z are Banach spaces. If A and B are Fredholm operators, upper semi-Fredholm operators, lower semi-Fredholm operators, then BA is a Fredholm operator, an upper semiFredholm operator, a lower semi-Fredholm operator, respectively, and i(BA) = i(B) + i(A). Theorem 2.3.3 (S. Sánchez-Perales and S. V. Djordjevic̀ [177, Theorem 1.1]). Let S, T ∈ ℒ(X). Then: (i) if β(T) = ∞, then β(TS) = ∞; and (ii) if β(S) < ∞ and α(T) = ∞, then α(TS) = ∞. Proof. (i) Since R(TS) ⊂ R(T), we have ∞ = dim X/R(T) = dim X/R(TS). (ii) Let X0 = N(T) ∩ R(S) and define U : N(TS) 󳨀→ X0 by Ux = Sx,

x ∈ N(TS).

Clearly, U is a surjective (or onto) linear operator and N(U) = N(S). Then dim N(TS) = dim N(U) + dim R(U) = dim N(S) + dim X0 . Now, since β(S) < ∞, there exists a finite-dimensional subspace E of X such that X = R(S) ⊕ E. Suppose that dim X0 < ∞. Then there exists X1 such that N(T) = X0 ⊕ X1 . Let {xk }k∈J be a basis of X1 , where J is a countable index set. For each k ∈ J, there exist unique rk ∈ R(S) and vk ∈ E such that xk = rk + vk . Note that {vk }k∈J is a linearly independent subset of E. Indeed, if ∑ ak vk = 0,

k∈J

24 | 2 Linear operators then N(T) ⊃ X1 ∋ ∑ ak xk = ∑ ak rk ∈ R(S), k∈J

k∈J

which implies ∑ ak xk ∈ X0 ∩ X1 = {0},

k∈J

so ak = 0 for all k ∈ J. Consequently, dim X1 ≤ dim E. Thus, dim N(T) = dim X0 + dim X1 < ∞, contradicting the hypothesis α(T) = ∞. Therefore, dim X0 = ∞ and, hence, dim N(TS) = dim N(S) + dim X0 = ∞. 2.3.2 Unbounded operators Theorem 2.3.4 (S. R. Caradus, W. E. Pfaffenberger, and B. Yood [53, Theorem 4.4.1]). If T ∈ Φ(X), then there exists ε > 0 such that, for each U ∈ ℒ(X) with ‖U‖ < ε, we have T + U ∈ Φ(X) and i(T + U) = i(T). Therefore, the index is continuous on the open semi-group Φ(X). Theorem 2.3.5 (S. R. Caradus, W. E. Pfaffenberger, and B. Yood [53, Theorem 4.4.2]). If T ∈ Φ− (X), then there exists ε > 0 such that, for each U ∈ ℒ(X) with ‖U‖ < ε, we have T + U ∈ Φ− (X). Moreover, ε can be chosen so that β(T + U) ≤ β(T) and, if α(T) = ∞, then α(T + U) = ∞.

2.4 Fredholm perturbations | 25

2.4 Fredholm perturbations Definition 2.4.1. Let X and Y be two Banach spaces and let F ∈ ℒ(X, Y). Then F is called a Fredholm perturbation, if U + F ∈ Φ(X, Y) whenever U ∈ Φ(X, Y). The set of Fredholm perturbations is denoted by ℱ (X, Y). This class of operators is introduced and investigated in [95]. In particular, it is shown in [46] that ℱ (X, Y) is a closed subset of ℒ(X, Y) and, if X = Y, then ℱ (X) := ℱ (X, X) is a closed two-sided ideal of ℒ(X).

2.5 Compact, A-bounded, and A-compact operators Definition 2.5.1. Let X and Y be two Banach spaces. A bounded linear operator K from X into Y is said to be compact (or completely continuous) if it transforms every bounded set of X in a relatively compact set of Y. In a similar way, K is said to be compact if, for any bounded sequence (xn )n of X, the sequence (Kxn )n has a convergent sequence in Y. We denote by 𝒦(X, Y) the set of all compact linear operators from X into Y. If X = Y, then the set 𝒦(X, X) is replaced by 𝒦(X). Lemma 2.5.1 (B. Russo [175]). Let a, b, c, and d ∈ ℝ such that a < b and c < d. We suppose that the following conditions are satisfied: 1 b (i) (∫a |N(x, y)|r dy) r ≤ C1 a. e. x ∈ [c, d], where r > 0; d

1

(ii) (∫c |N(x, y)|σ dx) σ ≤ C2 a. e. y ∈ [a, b], where σ > 0; and (iii) q ≥ p, q > σ, and (1 − σq )p󸀠 < r, where p󸀠 is the conjugate of p (p󸀠 =

p ). p−1

Then the operator b

Kψ(x) = ∫ N(x, y)ψ(y)dy,

x ∈ [c, d],

a

is compact from Lp ([a, b]) into Lq ([c, d]). Definition 2.5.2. Let A and B be linear operators on a Banach space X. We say that B is A-bounded with order p ∈ [0, 1], if 𝒟(A) ⊂ 𝒟(B) and there exist strictly positive constants a, b such that ‖Bφ‖ ≤ b‖Aφ‖p ‖φ‖1−p + a‖φ‖,

for every φ ∈ 𝒟(A).

For p = 1, we say that B is A-bounded. Definition 2.5.3. Let A and B be two linear operators on a Banach space X. Assume that A is closed. The operator B is said to be A-compact of order p∈[0, 1], if 𝒟(A) ⊂ 𝒟(B)

26 | 2 Linear operators and if, for all ε > 0, there is a constant c(ε) > 0 such that ‖Bφ‖ ≤ ε‖Aφ‖p ‖φ‖1−p + c(ε)‖φ‖ for all φ ∈ 𝒟(A). If p = 1, we say that B is A-compact.

2.6 Weakly compact operators Definition 2.6.1. An operator A ∈ ℒ(X, Y) is said to be weakly compact, if A(B) is relatively weakly compact in Y, for every bounded subset B ⊂ X. The family of weakly compact operators from X into Y is denoted by 𝒲 (X, Y). If X = Y, the family of weakly compact operators on X, 𝒲 (X) := 𝒲 (X, X), is a closed two-sided ideal of ℒ(X) containing 𝒦(X) (cf. [78, 96]).

2.7 Strictly singular operators Definition 2.7.1. Let X and Y be two Banach spaces. An operator A ∈ ℒ(X, Y) is called strictly singular if, for every infinite-dimensional subspace M, the restriction of A to M is not a homeomorphism. Let 𝒮 (X, Y) denote the set of strictly singular operators from X into Y. The concept of strictly singular operators was introduced by T. Kato in his pioneering paper [139] as a generalization of the notion of compact operators. For a detailed study of the properties of strictly singular operators, we refer to [96, 139]. For our own use, let us recall the following four facts. The set 𝒮 (X, Y) is a closed subspace of ℒ(X, Y). If X = Y, 𝒮 (X) := 𝒮 (X, X) is a closed two-sided ideal of ℒ(X) containing 𝒦(X). If X is a Hilbert space, then 𝒦(X) = 𝒮 (X) and the class of weakly compact operators on L1 -spaces (respectively C(K)-spaces with K being a compact Hausdorff space) is nothing else but the family of strictly singular operators on L1 -spaces (respectively C(K)-spaces) (see [170, 171]).

2.8 Closable operator perturbation Theorem 2.8.1 (T. Kato [139, Theorem 1.11, p. 194]). Let T, A be operators from X into Y and let A be T-compact. If T is closable, then S = T + A is also closable, the closures of T and S have the same domain, and A is S-compact. In particular, S is closed if T is closed. Lemma 2.8.1. Let A and B be two linear operators from a Banach space X into a Banach space Y, having the same domain 𝒟 ⊂ X. Let A be a closed operator. If there exist positive

2.9 Adjoint operator |

constants θ1 , θ2 , and β satisfying β ∈ ]0, 1] and θ2
. 2

(2.16.1)

Then, on any ray l which starts at the origin and does not belong to the sector (2.16.1), the norm of the resolvent (I − λK)−1 is uniformly bounded. Proof. Suppose that λ ∈ l and let (I − λK)−1 f = φ. Then f = φ − λKφ and, consequently, ⟨f , φ⟩ = ⟨φ, φ⟩ − λ⟨Kφ, φ⟩.

(2.16.2)

Let ℛ denote the set of points ‖φ‖2 and let r be the distance between λ1 and ℛ. If ϕ is the smallest of the angles formed by the ray l and the rays which bound the sector (2.16.1), then, obviously, ⟨Kφ,φ⟩

r≥

sin ϕ . |λ|

Thus, we obtain, from equation (2.16.2), 󵄨󵄨 ⟨f , φ⟩ 󵄨󵄨 󵄨󵄨 1 ⟨Kφ, φ⟩ 󵄨󵄨 sin ϕ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 , 󵄨󵄨 󵄨=󵄨 − 󵄨≥ 󵄨󵄨 λ‖φ‖2 󵄨󵄨󵄨 󵄨󵄨󵄨 λ |λ| ‖φ‖2 󵄨󵄨󵄨 so that 󵄨 󵄨 ‖φ‖2 sin ϕ ≤ 󵄨󵄨󵄨⟨f , φ⟩󵄨󵄨󵄨 ≤ ‖f ‖‖φ‖ and, consequently, ‖φ‖ ≤

‖f ‖ . sin ϕ

Thus, ‖f ‖ 󵄩󵄩 −1 󵄩 󵄩󵄩(I − λK) f 󵄩󵄩󵄩 ≤ sin ϕ for all λ ∈ l and all f , so 1 󵄩󵄩 −1 󵄩 , 󵄩󵄩(I − λK) 󵄩󵄩󵄩 ≤ sin ϕ which was to be proved.

(2.16.3)

2.17 Convergence operators | 37

Proposition 2.16.1 (H. Heuser [105]). The spectrum of the symmetric operator A lies in the closed interval [ inf ⟨Ax, x⟩, sup ⟨Ax, x⟩] ‖x‖=1

‖x‖=1

of the real axis; the bounds inf‖x‖=1 ⟨Ax, x⟩, sup‖x‖=1 ⟨Ax, x⟩ belong to σ(A). If λ is not real, then, for the resolvent operator (λ − A)−1 , the estimate 1 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − A) 󵄩󵄩󵄩 ≤ |Im λ| is valid.

2.17 Convergence operators 2.17.1 Convergence sequence Consider a sequence of operators Un : X 󳨀→ Y, n ∈ ℕ, which converges to a mapping U : X 󳨀→ Y pointwise, i. e., Un x → Ux,

as n → ∞, for all x ∈ X.

We say that Un converges to U in the strong operator topology. The Banach–Steinhaus theorem, also known as the uniform boundedness principle, states the following. Theorem 2.17.1. Let Un : X 󳨀→ Y, n ∈ ℕ, be a sequence of bounded operators, which converges pointwise to a mapping U : X 󳨀→ Y. Then U is linear and bounded. Furthermore, the sequence of norms ‖Un ‖ is bounded and ‖U‖ ≤ lim inf ‖Un ‖. Definition 2.17.1. We say that an infinite series ∑∞ n=1 αn en is convergent with sum x ∈ X if 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩 lim 󵄩x − ∑ αk ek 󵄩󵄩󵄩 = 0. n→∞󵄩 󵄩󵄩 󵄩󵄩 k=1 󵄩 󵄩 Let {fk }∞ k=1 be a sequence in X and suppose that ∞

∑ ck fk

k=1

2 is convergent for all {ck }∞ k=1 ∈ l (ℕ). Consider the sequence of linear operators

Tn : l2 (ℕ) 󳨀→ X,

n

Tn {ck }∞ k=1 = ∑ ck fk . k=1

38 | 2 Linear operators Clearly, (Tn )n is bounded and Tn → T pointwise as n → ∞, where T is given by ∞

T : l2 (ℕ) 󳨀→ X,

T{ck }∞ k=1 = ∑ ck fk . k=1

(2.17.1)

The results obtained by O. Christensen, appearing in the remaining part of this section, can be found in [66]. ∞ Lemma 2.17.1. Let {fk }∞ k=1 be a sequence in X and suppose that ∑k=1 ck fk is convergent ∞ 2 for all {ck }k=1 ∈ l (ℕ). Let T be the operator given in (2.17.1). Then T is bounded.

Proof. By using Theorem 2.17.1, we infer that T is bounded. Let {fk }∞ k=1 be a sequence in X and suppose that ∞

∑ ck fk

k=1

2 is convergent for all {ck }∞ k=1 ∈ l (ℕ). Let T be the operator given in (2.17.1). Using Lemma 2.17.1, we deduce that T is bounded. In order to find the expression for T ∗ , 2 let f ∈ X, {ck }∞ k=1 ∈ l (ℕ). Then ∞

∞

k=1

k=1

⟨f , T{ck }∞ k=1 ⟩ = ⟨f , ∑ ck fk ⟩ = ∑ ⟨f , fk ⟩ck .

(2.17.2)

We mention two ways to find T ∗ f from this. (i) The convergence of the series ∞

∑ ⟨f , fk ⟩ck

k=1

2 ∞ 2 for all {ck }∞ k=1 ∈ l (ℕ) implies that {⟨f , fk ⟩}k=1 ∈ l (ℕ). Thus, we write

⟨f , T{ck }∞ k=1 ⟩ = ⟨{⟨f , fk ⟩}, {ck }⟩l2 (ℕ) and conclude that ∞

T ∗ f = {⟨f , fk ⟩}k=1 . (ii) Alternatively, when T : l2 (ℕ) 󳨀→ X is bounded, we already know that T ∗ is a bounded operator from X into l2 (ℕ). Therefore, the kth coordinate function is bounded from X into ℂ; by Riesz’s representation theorem, T ∗ has the form ∞

T ∗ f = {⟨f , gk ⟩}k=1 for some {gk }∞ k=1 in X. Thus, we have the following lemma.

2.17 Convergence operators | 39

∞ Lemma 2.17.2. Let {fk }∞ k=1 be a sequence in X and suppose that ∑k=1 ck fk is convergent ∞ 2 for all {ck }k=1 ∈ l (ℕ). Let T be the operator given in (2.17.1). Then the adjoint operator of T is given by

T ∗ : X 󳨀→ l2 (ℕ),

∞

T ∗ f = {⟨f , fk ⟩}k=1 .

∞ Lemma 2.17.3. Let {fk }∞ k=1 be a sequence in X and suppose that ∑k=1 ck fk is convergent 2 for all {ck }∞ k=1 ∈ l (ℕ). Let T be the operator given in (2.17.1). Then ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 ≤ ‖T‖2 ‖f ‖2 ,

for all f ∈ X.

k=1

(2.17.3)

Proof. By the definition of T ∗ , given in Lemma 2.17.2, equation (2.17.2) now shows that ∞

∞

k=1

k=1

∑ ⟨f , gk ⟩ck = ∑ ⟨f , fk ⟩ck ,

for all

{ck }∞ k=1

2

∈ l (ℕ) and for all f ∈ X. Hence, gk = fk .

The adjoint of a bounded operator T is itself bounded and 󵄩 󵄩 ‖T‖ = 󵄩󵄩󵄩T ∗ 󵄩󵄩󵄩. Under the assumption in Lemma 2.17.2, we therefore have 󵄩󵄩 ∗ 󵄩󵄩2 2 2 󵄩󵄩T f 󵄩󵄩 ≤ ‖T‖ ‖f ‖ ,

for all f ∈ X,

which leads to (2.17.3). 2.17.2 Limit inferior and superior Let 𝒮 be the collection of all non-empty compact subsets of ℂ. It is well known that the convergence of a sequence in 𝒮 with respect to the Hausdorff metric can be characterized through the concepts of limit inferior and superior. Let {En }n be a sequence of arbitrary subsets of ℂ and define the limits inferior and superior of {En }n , denoted, respectively, by lim inf En and lim sup En , as follows: lim inf En = {λ ∈ ℂ, for every ε > 0, there exists N ∈ ℕ such that 𝔹(λ, ε) ∩ En ≠ 0 for all n ≥ N}

and lim sup En = {λ ∈ ℂ, for every ε > 0, there exists J ⊂ ℕ infinite such that 𝔹(λ, ε) ∩ En ≠ 0 for all n ∈ J}.

We recall the following properties of limit inferior and superior.

40 | 2 Linear operators Theorem 2.17.2. Let {En }n be a sequence of non-empty subsets of ℂ. The following properties of limit inferior and superior are known: (i) lim inf En and lim sup En are closed subsets of ℂ; (ii) λ ∈ lim sup En if and only if there exists an increasing sequence of natural numbers n1 < n2 < n3 < ⋅ ⋅ ⋅ and points λnk ∈ Enk , for all k ∈ ℕ∗ , such that lim λnk = λ; (iii) λ ∈ lim inf En if and only if there exists a sequence {λn }n such that λn ∈ En for all n ∈ ℕ and lim λn = λ; (iv) suppose E, En ∈ 𝒮 for all n ∈ ℕ and there exists K ∈ 𝒮 such that En ⊂ K, for all n ∈ ℕ. Then En → E in the Hausdorff metric if and only if lim sup En ⊂ E and E ⊂ lim inf En . 2.17.3 ν-convergence Definition 2.17.2. A sequence (Un )n of bounded linear operators mapping from X into X is said to be strongly convergent to U, denoted by Un →U, if ‖Un − U‖ → 0. Definition 2.17.3. Let U ∈ ℒ(X). A function τ, defined on ℒ(X), whose values are nonempty compact subsets of ℂ is said to be upper (respectively lower) semi-continuous at U, if Un →U implies lim sup τ(Un ) ⊂ τ(U)

(respectively τ(U) ⊂ lim inf τ(Un )). It is known that, if τ(⋅) is bounded on convergent sequences, then τ(⋅) is continuous in the Hausdorff metric if and only if τ(⋅) is both upper and lower semi-continuous at U.

2.17 Convergence operators | 41

Definition 2.17.4. A sequence (Un )n of bounded linear operators mapping from X ν

into X is said to be ν-convergent to U, denoted by Un → U, if (‖Un ‖)n is bounded, 󵄩 󵄩󵄩 󵄩󵄩(Un − U)U 󵄩󵄩󵄩 → 0,

and 󵄩 󵄩󵄩 󵄩󵄩(Un − U)Un 󵄩󵄩󵄩 → 0. Simple examples show that none of these conditions on the norms of the operators Un , (Un − U)U, and (Un − U)Un is implied by the remaining two. For example, let X := ℂ2×1 with any norm and 0 0

An := (

n ) 0

for n = 1, 2, . . . . Then 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩(An − A1 )A1 󵄩󵄩󵄩 = 0 = 󵄩󵄩󵄩(An − A1 )An 󵄩󵄩󵄩, but (‖An ‖)n is unbounded.

ν

The following examples, taken from [11], show that, if Un → U, this does not generally imply that Un → U. In fact, take X = l2 (ℕ) with canonical Hilbert basis e1 , e2 , . . . . Let Un be defined by Un x := x(n + 1)en

for n = 1, 2, . . .

and ∞

x = ∑ x(k)ek ∈ X. k=1

ν

Then Un → 0 but Un 󴀀󴀂󴀠 0. 2.17.4 ν-continuity Inspired by the notion of ν-convergence, we examine the following definition. Definition 2.17.5. A mapping τ on ℒ(X) whose values are compact subsets of ℂ is said to be ν-upper semi-continuous at U when ν

Un → U

󳨐⇒

lim sup τ(Un ) ⊂ τ(U)

42 | 2 Linear operators and to be ν-lower semi-continuous at U when ν

Un → U

󳨐⇒

τ(U) ⊂ lim inf τ(Un ).

If τ is both ν-upper and ν-lower semi-continuous, then it is said to be ν-continuous. Remark 2.17.1. (i) If τ is ν-lower semi-continuous at U, then τ is lower semi-continuous at U. (ii) If τ is ν-upper semi-continuous at U, then τ is upper semi-continuous at U. (iii) If τ is bounded on ν-convergent sequences and τ is ν-continuous at U, then τ is continuous in the Hausdorff metric at U. (iv) Generally, the converse of (i), (ii), and (iii) is not true, but if 0 ∈ ρ(U), we have: (iv1 ) τ is ν-lower semi-continuous at U if and only if τ is lower semi-continuous at U; (iv2 ) τ is ν-upper semi-continuous at U if and only if τ is upper semi-continuous at U; and (iv3 ) if τ is bounded on ν-convergent sequences, then τ is ν-continuous at U if and only if τ is continuous in the Hausdorff metric at U. (v) It is worth noticing that σ(⋅) is not ν-continuous in general. For example, we take 𝒰n and 𝒰0 as operators defined on l2 (ℕ) ⊕ l2 (ℕ) by I−UU ∗ n ) ∗

U 0

𝒰n = (

U

and U 0

𝒰0 = (

0 ), U∗

where U ∈ ℒ(l2 (ℕ)) is the forward unilateral shift defined by U : l2 (ℕ) 󳨀→ l2 (ℕ), { (x0 , x1 , x2 , . . .) 󳨀→ (0, x0 , x1 , x2 , . . .), in terms of the standard basis in l2 (ℕ), that is, Uej = ej+1 and ‖U‖ = 1. Since 𝒰n → 𝒰 , we have ν

𝒰n → 𝒰 .

However, σ(𝒰n ) = {λ ∈ ℂ such that |λ| = 1} and σ(𝒰 ) = {λ ∈ ℂ such that |λ| ≤ 1}.

2.18 Normal, hyponormal, and pseudo-inverse operator

| 43

2.18 Normal, hyponormal, and pseudo-inverse operator 2.18.1 Normal and hyponormal operator Definition 2.18.1. A closed densely defined operator A on a Banach space X is called normal if it satisfies the equivalent conditions AA∗ = A∗ A, i. e., 𝒟(AA∗ ) = 𝒟(A∗ A) and AA∗ u = A∗ Au for all u ∈ 𝒟(A∗ A). Let A be a normal operator. Then 1 󵄩󵄩 −1 󵄩 , 󵄩󵄩(λ − A) 󵄩󵄩󵄩 = dist(λ, σ(A))

(2.18.1)

where dist(λ, σ(A)) = minz∈σ(A) |λ − z|. Corollary 2.18.1 (A. S. Markus [155, Corollary 3.7]). Let L be a normal operator with compact resolvent whose spectrum lies on a finite number of rays arg λ = αk (k = 1, . . . , n). If the operator T − L is compact relative to L, then the operator T also has a compact resolvent and, for any δ > 0, only finitely many eigenvalues of T lie outside the angle {λ ∈ ℂ such that | arg λ − αk | < δ} (k = 1, . . . , n). Definition 2.18.2. Let T be a bounded linear operator acting in a Hilbert space X. The operator T is said to be hyponormal if T ∗ T − TT ∗ ≥ 0 or, equivalently, 󵄩󵄩 ∗ 󵄩󵄩 󵄩󵄩T x󵄩󵄩 ≤ ‖Tx‖ for any x ∈ X. As usual, we have used T ∗ to denote the Hilbert space adjoint of T. In particular, normal (T ∗ T − TT ∗ = 0) and subnormal (T has a normal extension) operators are hyponormal.

2.18.2 Pseudo-inverse operator Let X and Y be two Hilbert spaces. Suppose that the operator A ∈ ℒ(X, Y) has a closed range. Then there exists a unique bounded operator A† : Y 󳨀→ X satisfying AA† A = A,

A† AA† = A† ,

∗

(A† A) = A† A,

∗

and (AA† ) = AA† .

(2.18.2)

The operator A† is called the pseudo-inverse operator of A. If A is a bounded invertible operator, then A† = A−1 . Lemma 2.18.1. Let X, Y be two Hilbert spaces and let A ∈ ℒ(X, Y). Then ∗

†

(A† ) = (A∗ ) .

44 | 2 Linear operators Definition 2.18.3. The reduced minimum modulus γ(A) of an operator A ∈ ℒ(X, Y) is defined by γ(A) = inf{‖Af ‖ such that ‖f ‖ = 1 and f ∈ N(A)⊥ }. It is well known that 1

γ(A) = γ(A∗ ) = γ(A∗ A) 2 . It was proved in [73] that an operator A has a closed range if and only if γ(A) > 0. In Hilbert spaces, 󵄩 󵄩−1 γ(A) = 󵄩󵄩󵄩A† 󵄩󵄩󵄩 .

(2.18.3)

2.19 Dunford–Pettis property Definition 2.19.1. Let X be a Banach space. The space X is said to have the Dunford– Pettis property if, for each Banach space Y, every weakly compact operator T : X 󳨀→ Y takes weakly compact sets in X into norm compact sets of Y. The Dunford–Pettis property, as defined above, was explicitly defined by A. Grothendieck [101], who performed an extensive study about it and, also, about some related properties. It is well known that any L1 -space has the Dunford–Pettis property [75]. Moreover, if Ω is a compact Hausdorff space, then C(Ω) has the Dunford– Pettis property [101]. For further examples, we refer the reader to [72] and [78, pp. 494, 479, 508, and 511]. Let us notice that the Dunford–Pettis property is not conserved under conjugation. However, if X is a Banach space whose dual has the Dunford–Pettis property, then X has also the Dunford–Pettis property (see, e. g., [101]). For more information, we refer to Diestel’s paper [72], which contains a real survey of the Dunford– Pettis property, as well as some related topics.

2.20 Jeribi, Wolf, and Weyl essential spectra When dealing with the essential spectra of closed, densely defined linear operators on Banach spaces, one of the main problems consists of studying the invariance of the essential spectra of these operators which are subject to various kinds of perturbation. There are many ways to define the essential spectrum of a closed densely defined linear operator in a Banach space. Indeed, several definitions can be found in the literature. Particularly, various notions of essential spectra appear in the applications of spectral theory (see, for example, [20, 115–123, 126, 128–134, 139, 179]). In a Banach space X, the Jeribi essential spectrum of the operator A ∈ 𝒞 (X) is defined by σj (A) :=

⋂

K∈𝒲∗ (X)

σ(A + K),

2.20 Jeribi, Wolf, and Weyl essential spectra | 45

where 𝒲∗ (X) stands for each one of the sets 𝒲 (X) and 𝒮 (X). More important contributions concerning the Jeribi essential spectrum and its applications to transport operators were due to A. Jeribi and his collaborators (see [1–9, 17–19, 21–32, 36, 37, 43–45, 48, 55–57, 60, 61, 68, 79, 80, 118–122, 129, 131, 136, 147–149, 159]). The Wolf essential spectrum of the operator A ∈ 𝒞 (X) is defined by σf (A) := ℂ\{λ ∈ ℂ such that λ − A ∈ Φ(X)}. The Weyl essential spectrum of the operator A ∈ 𝒞 (X) is defined by σw (A) :=

⋂ σ(A + K).

K∈𝒦(X)

We have the following inclusions: σr (A) ∪ σc (A) ⊂ σw (A), σj (A) ⊂ σw (A), and σf (A) ⊂ σw (A). For A ∈ ℒ(X), let 󵄩 󵄩 σap (A) = {λ ∈ ℂ such that there is {xn }n ⊂ X, ‖xn ‖ = 1 for all n and 󵄩󵄩󵄩(λ − A)xn 󵄩󵄩󵄩 → 0}. It is well known that σap (A) = {λ ∈ ℂ such that λ − A is not bounded below}. The following proposition gives a characterization of the Weyl essential spectrum by means of Fredholm operators. Proposition 2.20.1 (M. Schechter [179, Theorem 7.27, p. 172]). Let A ∈ 𝒞 (X). Then λ ∉ σw (A) if and only if λ − A ∈ Φ(X) and i(λ − A) = 0. Remark 2.20.1. (i) Proposition 2.20.1 gives insight into the relationship between the Weyl essential spectrum and the Wolf essential spectrum of A by σw (A) = σf (A) ∪ {λ ∈ ℂ such that i(λ − A) ≠ 0}. (ii) We recall that, for Θ ∈ {σj (A), σf (A), σw (A)}, A ∈ ℛ(X) if and only if Θ = {0} (see [12, Theorem 3.111]). Theorem 2.20.1. If X satisfies the Dunford–Pettis property, 𝒲∗ (X) = 𝒲 (X) and if A is a closed, densely defined, and linear operator on X, then σj (A) = σw (A).

46 | 2 Linear operators Theorem 2.20.2. Let A be a closed, densely defined, and linear operator on Lp (Ω, dμ) and let p ∈ [1, ∞). In the case where 𝒲∗ (Lp (Ω, dμ)) = 𝒮 (Lp (Ω, dμ)), we have σw (A) = σj (A). Proposition 2.20.2. Let A ∈ 𝒞 (X). Then we have the following results: (i) ΦA is open; (ii) i(λ − A) is constant on any component of ΦA ; and (iii) α(λ − A) and β(λ − A) are constant on any component of ΦA , except on a discrete set of points on which they have larger values. Theorem 2.20.3. Let A ∈ 𝒞 (X) such that ρ(A) is not empty. If ℂ\σf (A) is connected, then σf (A) = σw (A). Proof. Since the inclusion σf (A) ⊂ σw (A) is known, it is sufficient to show that σw (A) ⊂ σf (A), which is equivalent to [ℂ\σf (A)] ∩ σw (A) = 0. Suppose that [ℂ\σf (A)] ∩ σw (A) ≠ 0 and let λ0 ∈ [ℂ\σf (A)] ∩ σw (A). Since ρ(A) ≠ 0, there exists λ1 ∈ ℂ such that λ1 ∈ ρ(A) and, consequently, λ1 − A ∈ Φ(X) and i(λ1 − A) = 0. Moreover, since ℂ\σf (A) is connected, from Proposition 2.20.2 (ii) it follows that i(λ − A) is constant on any component of ΦA . Therefore, i(λ1 − A) = i(λ0 − A) = 0. In this way, we see that λ0 ∈ ̸ σw (A), which is a contradiction. This proves that [ℂ\σf (A)] ∩ σw (A) = 0 and this completes the proof of the theorem. Let A ∈ 𝒞 (X) and let J be an A-bounded operator on X. The following result gives some information about the eigenvalue of the perturbed operator A + J knowing some information about the spectrum of A.

2.21 Hypercyclic operators | 47

Theorem 2.20.4. Let X be a Banach space and ℐ (X) be an arbitrary non-zero two-sided ideal of ℒ(X) satisfying the condition 𝒦(X) ⊆ ℐ (X) ⊆ 𝒥 (X), where b

𝒥 (X) := {F ∈ ℒ(X) such that I − F ∈ Φ (X) and i(I − F) = 0}.

Let A ∈ 𝒞 (X) such that σw (A) = 0, i. e., σ(A) = σp (A). Let J ∈ 𝒞 (X) such that J is A-bounded and J(λ − A)−1 ∈ ℐ (X) for some λ ∈ ρ(A). Then σ(A + J) = σp (A + J). Let A ∈ 𝒞 (X) and K ∈ ℒ(X). The following theorem gives some information about the eigenvalues of the perturbed operator A + K knowing some information about the spectrum of A. Theorem 2.20.5. Let X be a Banach space and let A ∈ 𝒞 (X) such that σw (A) = 0 (i. e., σ(A) = σp (A)). If K(λ − A)−1 or (λ − A)−1 K is in ℐ (X), where ℐ (X) is any subset of ℒ(X) (not necessarily an ideal) satisfying the condition 𝒦(X) ⊂ ℐ (X) ⊂ ℱ (X), then σ(A + K) = σp (A + K).

2.21 Hypercyclic operators Definition 2.21.1. A vector x ∈ X is called hypercyclic for A ∈ ℒ(X) if the set Orb(A, x) := {An x, n ∈ ℕ} is dense in X. Definition 2.21.2. An operator A is called hypercyclic if there is a vector hypercyclic for A. We denote by HC(X) the set of all hypercyclic operators in ℒ(X) (see [100] for an exhaustive survey on hypercyclicity). The notion has sense only in separable Banach space (see [174]). Clearly, in non-separable Banach space, there are no hypercyclic operators. It is easy to find an operator that has no hypercyclic vectors. For example, if ‖A‖ ≤ 1, then all orbits are bounded and, therefore, not dense. On the other hand, if A is hypercyclic, then almost all vectors are hypercyclic for A. Proposition 2.21.1 ([141]). Let A ∈ ℒ(X) be a hypercyclic operator. Then the set HC(A) of all vectors x ∈ X that are hypercyclic for A is a Gδ -set, i. e., HC(A) is a countable intersection of open sets of X and, hence, dense in X. Proposition 2.21.2. Let X be a separable Banach space and let A∈ ℒ(X) be a hypercyclic operator. Then σp (A∗ ) = 0.

48 | 2 Linear operators Corollary 2.21.1. If X is a ℂ-vectorial space and dim X < ∞, then there are no hypercyclic operators acting in X. For X = lp , (p ∈ [1, ∞)) or X = c0 , the Banach spaces of all sequences converge to 0. Consider the backward shift operator Sl , defined by Sl (x0 , x1 , x2 , . . .) = (x1 , x2 , . . .). It was shown in [174] that λSl ∈ HC(X) for any λ ∈ ℂ such that |λ| > 1. We end this section by introducing the following definitions. Definition 2.21.3. Let X be a separable Banach space and let A ∈ ℒ(X). The Γ-hypercyclic set of the operator A is defined by Γhyp (A) := {λ ∈ ℂ such that A − λ is hypercyclic}.

(2.21.1)

Definition 2.21.4. Let X be a Banach space. An operator A ∈ ℒ(X) is said to be essentially quasi-nilpotent if σf (A) = {0}.

2.22 Closed subspace Throughout, X is a Banach space with norm ‖ ⋅ ‖X , W is a subspace of X such that W is a Banach space with norm ‖ ⋅ ‖W , and W is continuously embedded in X (i. e., there is M > 0 such that ‖w‖X ≤ M‖w‖W for all w ∈ W). When W is invariant for T ∈ ℒ(X), i. e., T(W) ⊂ W, we denote the restriction of T to W by T|W . In this case, it is straightforward to check, using the closed graph theorem, that T|W ∈ ℒ(W). Proposition 2.22.1 (B. Barnes [40, Proposition 2]). Assume that S∈ ℒ(X) and W is S-invariant. The following statements are equivalent: (i) S−1 [W] := {x ∈ X such that S(x) ∈ W} = W + N(S); and (ii) R(S|W ) = R(S) ∩ W. When (ii) holds, R(S) is closed 󳨐⇒ R(S|W ) is closed. Proof. Suppose that (i) holds. Then R(S|W ) ⊂ R(S) ∩ W.

2.22 Closed subspace

| 49

We prove the reverse inclusion. Suppose y ∈ X and S(y) = w ∈ W. Then, by (i), y = v + z, where v ∈ W and z ∈ N(S). Then w = S(v), so w ∈ R(S|W ). Now suppose that (ii) is true. If y ∈ X and S(y) = w ∈ W, then there exists v ∈ W such that S(v) = S(y). Then y = v + (y − v) and y − v ∈ N(S). Thus, (i) holds. Assume that (ii) holds and R(S) is closed. Suppose that {wn }n ⊂ W, w0 ∈ W, and 󵄩󵄩 󵄩 󵄩󵄩S(wn ) − w0 󵄩󵄩󵄩W → 0. By our standing hypothesis, 󵄩󵄩 󵄩 󵄩󵄩S(wn ) − w0 󵄩󵄩󵄩X → 0. Then w0 ∈ R(S) ∩ W. By (ii), w0 = S(v) for some v ∈ W. This proves that R(S|W ) is closed. Let n

𝒫 := {S ∈ ℒ(X) such that S(W) ⊂ W and for some n ≥ 1, S (X) ⊂ W}.

Proposition 2.22.2 (B. Barnes [40, Proposition 3]). Assume that T ∈ 𝒫 and fix n, n ≥ 1, such that T n (X) ⊂ W. Then, for all λ ≠ 0, (λ − T)−1 [W] = W. Consequently, for all λ ≠ 0,

R(λ − T) is closed

󳨐⇒

R(λ − T|W ) is closed.

(2.22.1)

50 | 2 Linear operators Proof. Fix λ ≠ 0. Set S = λ−1 T, so Sn (X) ⊂ W. Suppose x ∈ X and (I − S)x = w ∈ W. Now, I − Sn = (I − S)p(S), where p(S) = I + S + ⋅ ⋅ ⋅ + Sn−1 . Therefore, (I − Sn )x = p(S)(I − S)x = p(S)w ∈ W. Since Sn x ∈ W, we have x ∈ W. This proves (λ − T)−1 [W] = (I − S)−1 [W] = W. Then the assertion (2.22.1) follows from Proposition 2.22.1. Proposition 2.22.3 (B. Barnes [40, Proposition 4]). Assume that S ∈ ℒ(X), W is S-invariant, and S(X) ⊂ W. If λ ≠ 0 and R(λ − S|W ) is closed in W, then R(λ − S) is closed. Proof. First note that S ∈ ℒ(X, W) (this follows from the closed graph theorem and the easily established fact that S : X 󳨀→ W is closed). We assume that λ = 1, so our hypothesis is that R(I − S|W ) is closed in W. For the sake of convenience, set N := N(I − S|W ) and also note that N = N(I − S) (this follows from S(X) ⊂ W). Define Î − S|W : W/N 󳨀→ R(I − S|W ) by (Î − S|W )(w + N) = (I − S|W )w. Let L : R(I − S|W ) 󳨀→ W/N be the bounded inverse of Î − S|W (the open mapping theorem), so L(Î − S|W )(w + N) = w + N for all w ∈ W. Now, we complete the proof that R(I − S) is closed by showing that the linear map (I − S)x 󳨀→ x + N from R(I − S) onto X/N is continuous. Assume that {xn } ⊂ X and 󵄩󵄩 󵄩 󵄩󵄩xn − S(xn )󵄩󵄩󵄩X → 0.

2.22 Closed subspace |

51

Then 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩(I − S|W )S(xn )󵄩󵄩󵄩W = 󵄩󵄩󵄩S(xn − S(xn ))󵄩󵄩󵄩W → 0. Thus, S(xn ) + N = L(Î − S|W )(S(xn ) + N) → 0 in W/N and, therefore, also in X/N. It follows that xn + N = [xn − S(xn ) + N] + [S(xn ) + N] → 0 in X/N. Lemma 2.22.1 (B. Barnes [40, Lemma 5]). Assume that T ∈ ℒ(X) and T(W) ⊂ W. Set V = T −1 [W], so W ⊂ V ⊂ X. Define a norm on V by ‖v‖V = ‖v‖X + ‖Tv‖W . Then we have the following: (i) ‖v‖V is a complete norm on V; (ii) W is continuously embedded in V and V is continuously embedded in X; and (iii) V is T-invariant and T(V) ⊂ W. Proof. The proofs of the assertions (i), (ii), and (iii) are all straightforward. We provide the proof of (i). Assume that {vn }n ⊂ V is a Cauchy sequence in the norm ‖ ⋅ ‖V . Then {vn }n is Cauchy in ‖ ⋅ ‖X and {T(vn )}n is Cauchy in ‖ ⋅ ‖W . There exist v0 ∈ X and w0 ∈ W such that ‖vn − v0 ‖X → 0 and 󵄩󵄩 󵄩 󵄩󵄩T(vn ) − w0 󵄩󵄩󵄩W → 0. Therefore, 󵄩󵄩 󵄩 󵄩󵄩T(vn ) − T(v0 )󵄩󵄩󵄩X → 0 and 󵄩󵄩 󵄩 󵄩󵄩T(vn ) − w0 󵄩󵄩󵄩X → 0.

52 | 2 Linear operators Thus, T(v0 ) = w0 . This proves that 󵄩 󵄩 ‖vn − v0 ‖V = ‖vn − v0 ‖X + 󵄩󵄩󵄩T(vn − v0 )󵄩󵄩󵄩W → 0. Theorem 2.22.1 (B. Barnes [40, Theorem 6]). Assume that T ∈ 𝒫 and fix n, n ≥ 1, such that T n (X) ⊂ W. Then, for all λ ≠ 0, (i) R(λ − T) is closed if and only if R(λ − T|W ) is closed; (ii) N(λ − T) = N(λ − T|W ); and (iii) λ − T is bounded below if and only if λ − T|W is bounded below. Proof. For the sake of convenience, set V0 = W, Vn = X, and ‖w‖0 = ‖w‖W for w ∈ W. Let V1 = T −1 [V0 ] be equipped with the complete norm ‖v‖1 = ‖v‖X + ‖Tv‖0 . By Lemma 2.22.1, V0 ⊂ V1 ⊂ X, the embeddings are continuous, and T(V1 ) ⊂ V0 . Continuing inductively in this fashion, using Lemma 2.22.1, we construct Vk , 1 ≤ k ≤ n − 1, with the norm ‖v‖k = ‖v‖X + ‖Tv‖k−1 , having the following properties: (a) (Vk , ‖v‖k ) is a Banach space, 1 ≤ k ≤ n − 1; (b) W = V0 ⊂ V1 ⊂ V2 ⊂ ⋅ ⋅ ⋅ ⊂ Vn−1 ⊂ Vn = X and each of the embeddings are continuous; (c) T(Vk ) ⊂ Vk−1 for 1 ≤ k ≤ n. Let Tk be the restriction of T to Vk , 0 ≤ k ≤ n. It follows from Propositions 2.22.2 and 2.22.3 that, for all λ ≠ 0 and 1 ≤ k ≤ n, R(λ − Tk−1 ) is closed if and only if R(λ − Tk ) is closed. Therefore, R(λ − T) is closed if and only if R(λ − T|W ) is closed. This proves (i). To verify (ii), first note that N(λ − T|W ) ⊂ N(λ − T).

2.22 Closed subspace |

Now suppose that λ ≠ 0 and x ∈ N(λ − T). Then Tx = λx,

T n x = λn x, and, since T n (X) ⊂ W, x ∈ W. Therefore, N(λ − T) = N(λ − T|W ). Statement (iii) follows from (i) and (ii). For S ∈ ℒ(X), we set 󸀠 σap (S) = σap (S)\{0} (set difference). 󸀠 󸀠 Corollary 2.22.1. Assume that T ∈ 𝒫 . Then σap (T) = σap (T|W ).

Corollary 2.22.2. For all λ ≠ 0, λ − T ∈ Φb (X) if and only if λ − T|W ∈ Φb (W).

53

3 Basic notations and results 3.1 Bessel sequences in Hilbert spaces This section concerns sequences in Hilbert spaces. For the sake of convenience we index all sequences by the natural numbers in this section. We shall soon see that all results actually hold with arbitrary countable index sets. Bessel sequences are defined as follows. Definition 3.1.1. A sequence {fk }∞ k=1 in X is called a Bessel sequence if there exists a constant B > 0 such that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 ≤ B‖f ‖2 ,

for all f ∈ X.

k=1

(3.1.1)

Every number B satisfying (3.1.1) is called a Bessel bound for {fk }∞ k=1 . Let {fk }∞ be a Bessel sequence in X with Bessel bound B. Then the operator k=1 ∞

T : {ck }∞ k=1 󳨀→ ∑ ck fk k=1

is a well-defined one from l2 (ℕ) into X, i. e., ∑∞ k=1 ck fk is convergent. Indeed, consider n, m ∈ ℕ, n > m. Then 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 m 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ ck fk − ∑ ck fk 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ ck fk 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩k=1 󵄩󵄩 󵄩󵄩k=m+1 󵄩󵄩 k=1 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨 = sup 󵄨󵄨⟨ ∑ ck fk , g⟩󵄨󵄨󵄨 󵄨 󵄨󵄨 ‖g‖=1󵄨󵄨 k=m+1 󵄨 n

󵄨 󵄨 ≤ sup ∑ 󵄨󵄨󵄨ck ⟨fk , g⟩󵄨󵄨󵄨 ‖g‖=1 k=m+1

1 2

n

n

󵄨2

󵄨 ≤ ( ∑ |ck |2 ) sup ( ∑ 󵄨󵄨󵄨⟨fk , g⟩󵄨󵄨󵄨 ) ‖g‖=1 k=m+1

k=m+1

n

2

1 2

≤ √B( ∑ |ck | ) . k=m+1

2 Since {ck }∞ k=1 ∈ l (ℕ), we know that n

∞

k=1

n=1

{ ∑ |ck |2 } https://doi.org/10.1515/9783110493863-003

1 2

(3.1.2)

56 | 3 Basic notations and results is a Cauchy sequence in ℂ. Equation (3.1.2) now shows that n

∞

k=1

n=1

{ ∑ ck fk }

is a Cauchy sequence in X and therefore convergent. Thus, T{ck }∞ k=1 is well defined. Above we insisted on a fixed ordering of the sequence {fn }∞ n=1 . It is very important to notice that the convergence properties of ∞

∑ cn fn

n=1

∞ not only depend on the sequence {fn }∞ n=1 and the coefficients {cn }n=1 , but also on the ∞ ordering. Even if {fn }n=1 is a sequence in the simplest possible Banach space, namely, ℝ, it can happen that ∞

∑ fn

n=1

is convergent, but that ∞

∑ fσ(n)

n=1

is divergent for a certain permutation σ of the natural numbers. This observation leads to a second definition of convergence. If ∞

∑ fσ(n)

n=1

is convergent for all permutations σ, we say that ∞

∑ fn

n=1

is unconditionally convergent. In that case, the limit is the same regardless of the order of summation. Theorem 3.1.1 (O. Christensen [66, Theorem 3.2.3]). Let {fk }∞ k=1 be a sequence in X. Then {fk }∞ is a Bessel sequence with Bessel bound B if and only if k=1 ∞

T : {ck }∞ k=1 󳨀→ ∑ ck fk k=1

is a well-defined bounded operator from l2 (ℕ) into X and ‖T‖ ≤ √B.

3.1 Bessel sequences in Hilbert spaces | 57

∞ 2 Proof. Let {fk }∞ k=1 be a Bessel sequence with Bessel bound B and let {ck }k=1 ∈ l (ℕ). ∞ Clearly, T{ck }k=1 is well defined and T is linear; since

󵄨 󵄨 󵄩󵄩 ∞ ∞ 󵄩 󵄩󵄩T{ck }k=1 󵄩󵄩󵄩 = sup 󵄨󵄨󵄨⟨T{ck }k=1 , g⟩󵄨󵄨󵄨, ‖g‖=1

a calculation as above shows that T is bounded and that ‖T‖ ≤ √B. For the opposite implication, suppose that T is well defined and that ‖T‖ ≤ √B. Then (2.17.3) shows that {fk }∞ k=1 is a Bessel sequence with Bessel bound B. Lemma 2.17.3 shows that, if we only need to know that {fk }∞ k=1 is a Bessel sequence and the value for the Bessel bound is irrelevant, we can check that the operator T is well defined. Corollary 3.1.1 (O. Christensen [66, Corollary 3.2.4]). If {fk }∞ k=1 is a sequence in X and ∞

∑ ck fk

k=1

is convergent for all 2 {ck }∞ k=1 ∈ l (ℕ),

then {fk }∞ k=1 is a Bessel sequence. The Bessel condition (3.1.1) remains the same, regardless of how the elements {fk }∞ k=1 are numbered. This leads to a very important consequence of Theorem 3.1.1. Corollary 3.1.2 (O. Christensen [66, Corollary 3.2.5]). If {fk }∞ k=1 is a Bessel sequence in X, then ∞

∑ ck fk

k=1

2 converges unconditionally for all {ck }∞ k=1 ∈ l (ℕ).

Lemma 3.1.1 (O. Christensen [66, Lemma 3.2.6]). Suppose that {fk }∞ k=1 is a sequence of elements in X and that there exists a constant B > 0 such that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 ≤ B‖f ‖2

k=1

for all f in a dense subset V of X. Then {fk }∞ k=1 is a Bessel sequence with bound B. Proof. We have to prove that the Bessel condition is satisfied for all elements in X. Let g ∈ X and suppose by contradiction that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨g, fk ⟩󵄨󵄨󵄨 > B‖g‖2 .

k=1

58 | 3 Basic notations and results Then there exists a finite set F ⊂ ℕ such that 󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨g, fk ⟩󵄨󵄨󵄨 > B‖g‖2 .

k∈F

Since V is dense in X, this implies that there exists h ∈ V such that 󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨h, fk ⟩󵄨󵄨󵄨 > B‖h‖2 ,

k∈F

but this is a contradiction. We conclude that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨g, fk ⟩󵄨󵄨󵄨 ≤ B‖g‖2

k=1

for all g ∈ X.

3.2 Orthogonal sequences in Hilbert spaces We first recall the Pythagorean theorem. Lemma 3.2.1. If {x1 , . . . , xN } are orthogonal vectors in an inner product space X, then 󵄩󵄩 N 󵄩󵄩2 N 󵄩󵄩 󵄩 󵄩󵄩 ∑ xn 󵄩󵄩󵄩 = ∑ ‖xn ‖2 . 󵄩󵄩 󵄩 󵄩󵄩n=1 󵄩󵄩󵄩 n=1 Let X be a Hilbert space and (xn )n be an orthonormal sequence in X. If x ∈ X, then, for each N ∈ ℕ, define N

yN = x − ∑ ⟨x, xn ⟩xn . n=1

(3.2.1)

If 1 ≤ m ≤ N, then we have N

⟨yN , xm ⟩ = ⟨x, xm ⟩ − ∑ ⟨x, xn ⟩⟨xn , xm ⟩ = ⟨x, xm ⟩ − ⟨x, xm ⟩ = 0. n=1

(3.2.2)

Therefore, in view of the Pythagorean theorem, we have 󵄩󵄩 󵄩󵄩2 N 󵄩󵄩 󵄩󵄩 󵄩 ‖x‖ = 󵄩󵄩yN + ∑ ⟨x, xn ⟩xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 n=1 󵄩 󵄩 2

N

󵄩 󵄩2 = ‖yN ‖2 + ∑ 󵄩󵄩󵄩⟨x, xn ⟩xn 󵄩󵄩󵄩 n=1 N

󵄨 󵄨2 = ‖yN ‖2 + ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 . n=1

Let us now recall the Bessel inequality.

(3.2.3)

3.2 Orthogonal sequences in Hilbert spaces | 59

Theorem 3.2.1. If (xn )n is an orthonormal sequence in a Hilbert space X, then ∞

󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 ≤ ‖x‖2

n=1

for every x ∈ X. Proof. Let x ∈ X and N ∈ ℕ. Consider yN given in (3.2.1). If 1 ≤ m ≤ N, then, by using equation (3.2.2), we infer that yN ⊥ x1 , . . . , xN . Therefore, by equation (3.2.3), N

󵄨 󵄨2 ‖x‖2 ≥ ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 . n=1

Letting N → ∞, we obtain Bessel’s inequality. Theorem 3.2.2. If (xn )n is an orthonormal sequence in a Hilbert space X, then ∞

∑ cn xn

n=1

converges if and only if ∞

∑ |cn |2 < ∞.

n=1

2 Proof. Suppose that ∑∞ n=1 |cn | < ∞. Set N

sN = ∑ cn xn n=1

and N

tN = ∑ |cn |2 . n=1

We know that (tN )N is a convergent (hence Cauchy) sequence of scalars and we must show that (sN )N is a convergent sequence of vectors. We have, for N > M, 󵄩󵄩 N 󵄩󵄩2 󵄩󵄩 󵄩󵄩 ‖sN − sM ‖2 = 󵄩󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩n=M+1 󵄩 N

= ∑ ‖cn xn ‖2 n=M+1

60 | 3 Basic notations and results N

= ∑ |cn |2 n=M+1

= |tN − tM |. Since (tN )N is Cauchy, we conclude that (sN )N is a Cauchy sequence in X and hence converges. If x ∈ span{xn }, then, by Bessel’s inequality, ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨x, xn ⟩󵄨󵄨󵄨 < ∞

n=1

and, therefore, by Theorem 3.2.2, we know that the series ∞

∑ ⟨x, xn ⟩xn

(3.2.4)

n=1

converges. Now, we give the continuity of the inner product. Theorem 3.2.3. Let X be a Hilbert space. If (xn )n → x and (yn )n → y in X, then ⟨xn , yn ⟩ → ⟨x, y⟩. Proof. Suppose (xn )n → x and (yn )n → y. Since all convergent sequences are bounded, we have C = sup ‖xn ‖ < ∞. Therefore, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨⟨x, y⟩ − ⟨xn , yn ⟩󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨⟨x − xn , y⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨xn , y − yn ⟩󵄨󵄨󵄨 ≤ ‖x − xn ‖‖y‖ + ‖xn ‖‖y − yn ‖

≤ ‖x − xn ‖‖y‖ + C‖y − yn ‖ → 0

as n → ∞,

which completes the proof. Theorem 3.2.4. Let X be a Hilbert space. If the series x = ∑∞ n=1 xn converges in X, then, for any y ∈ X, we have ∞

∞

n=1

n=1

⟨x, y⟩ = ⟨ ∑ xn , y⟩ = ∑ ⟨xn , y⟩. Proof. Suppose that the series x = ∑∞ n=1 xn converges in X and let N

sN = ∑ xn n=1

3.2 Orthogonal sequences in Hilbert spaces | 61

denote the partial sums of this series. Then, by definition, sN → x in X. Hence, given y ∈ X and using the continuity of the inner product (see Theorem 3.2.3), we have N

∞

∑ ⟨xn , y⟩ = lim ( ∑ ⟨xn , y⟩)

n=1

N→∞

n=1 N

= lim ⟨ ∑ xn , y⟩ N→∞

n=1

= lim ⟨sN , y⟩ N→∞

= ⟨x, y⟩, which completes the proof. Theorem 3.2.5. If (xn )n is an orthonormal sequence in a Hilbert space X, then x ∈ span{xn } if and only if ∞

x = ∑ ⟨x, xn ⟩xn . n=1

Proof. Choose x ∈ span{xn }. From equation (3.2.4), we know the series ∞

y = ∑ ⟨x, xn ⟩xn n=1

converges. Given any particular m ∈ ℕ, by applying Theorem 3.2.4, we have ∞

⟨x − y, xm ⟩ = ⟨x, xm ⟩ − ⟨ ∑ ⟨x, xn ⟩xn , xm ⟩ ∞

n=1

= ⟨x, xm ⟩ − ∑ ⟨x, xn ⟩⟨xn , xm ⟩ n=1

= ⟨x, xm ⟩ − ⟨x, xm ⟩ = 0. Thus, x − y ∈ {xn }⊥ = span{xn }⊥ . However, we also have x − y ∈ span{xn }, so x − y = 0. We close this section by the Riesz representation theorem. Theorem 3.2.6. Let X be a Hilbert space and f : X 󳨀→ ℂ be a continuous linear mapping. Then there exists a unique y ∈ X such that f (x) = ⟨x, y⟩.

62 | 3 Basic notations and results

3.3 Projection 3.3.1 Generalities Definition 3.3.1. An operator P ∈ ℒ(X) is called a projection if P 2 = P. If, in addition, we have P ∗ = P, then P is called an orthogonal projection. Remark 3.3.1. If P is an orthogonal projection, we have 0 ≤ P ≤ I and ‖P‖ = 1 if P ≠ 0. Definition 3.3.2. Let X, Y be normed linear spaces. (i) A linear operator T : X 󳨀→ Y is a topological isomorphism if T is a bijection and both T and T −1 are continuous. (ii) We say that X and Y are topologically isomorphic if there exists a topological isomorphism T : X 󳨀→ Y. Every isometric isomorphism is a topological isomorphism, but the converse does not necessarily hold. Proposition 3.3.1. Let P and Q be two projections on X of the subspaces F and D, respectively. If ‖P − Q‖ < 1, then dim F = dim D. Proof. Since ‖P −Q‖ < 1, the bounded operator I −(P −Q) is invertible. Hence, applying Theorem 2.10.1, we conclude that H := (I − (P − Q))−1 is bounded. Hence, 𝒟(H) = X. Since 𝒟(H) = (I − P + Q)(X), we have (I − P + Q)(X) = X, so P(I − P + Q)(X) = P(X), that is, (P − P 2 + PQ)(X) = P(X). Since P 2 = P, we have PQ(X) = P(X), or P(D) = F. ̃ the restriction of P on D, then we have, for all x ∈ D, Now, if we denote by P ̃ = 󵄩󵄩󵄩Qx + (P ̃ − Q)x󵄩󵄩󵄩 ‖Px‖ 󵄩 󵄩

3.3 Projection

| 63

󵄩 ̃ − Q)x󵄩󵄩󵄩 = 󵄩󵄩󵄩x + (P 󵄩 󵄩 󵄩 ̃ − Q)x󵄩󵄩󵄩 ≥ ‖x‖ − 󵄩󵄩󵄩(P ̃ − Q‖)‖x‖. ≥ (1 − ‖P From the above inequality and in virtue of the assumption ̃ − Q‖ ≤ ‖P − Q‖ < 1, ‖P ̃ is one-to-one. It therefore makes a bijection from D into F. Thus, P ̃ is a we know P bicontinuous isomorphism. Therefore, dim F = dim D. 3.3.2 Spectral projection Let A be a closed linear operator on X. Suppose that the spectrum σ(A) can be decomposed into two isolated parts σ and σ 󸀠 so that we can draw a closed curve Γ passing on the resolvent set ρ(A) and having σ in its interior and σ 󸀠 in its exterior. Define P := E(λ, A) =

1 ∮ R(λ, A)dλ 2πi

(3.3.1)

Γ

to be the spectral (or Riesz) projection of X on F in the direction F 󸀠 , where one traverses Γ in the positive direction. We have F = PX and F 󸀠 = (I − P)X. The subsets F and F 󸀠 are called the spectral spaces corresponding, respectively, to the isolated parts σ and σ 󸀠 of the spectrum. The subspace F includes, in particular, all own elements corresponding to eigenvalues λ interior to σ, i. e., the solutions f of the equation (A − λ)f = 0. Proposition 3.3.2. P is a projection. Proof. Let Γ̃ be another closed curve passing through ρ(A) and having σ in its interior ̃ Let x ∈ X. Then we have and σ 󸀠 in its exterior such that Γ is included in Γ. P 2 x = P(Px) 1 = ∮ R(λ, A)Pxdλ 2πi Γ

64 | 3 Basic notations and results

=

1 1 ∮ R(λ, A)( ∮ R(μ, A)xdμ)dλ 2πi 2πi Γ

Γ̃

1 = ∮ ∮ R(λ, A)R(μ, A)xdμdλ. (2πi)2 Γ Γ̃

Hence, P2 =

1 ∮ ∮ R(λ, A)R(μ, A)dμdλ. (2πi)2 Γ Γ̃

Applying the identity of the resolvent (Proposition 2.10.1), we get P2 =

1 1 (R(μ, A) − R(λ, A))dμdλ ∮∮ 2 λ−μ (2πi) Γ Γ̃

=

1 1 1 (∮ ∮ R(μ, A)dμdλ − ∮ ∮ R(λ, A)dμdλ). λ−μ λ−μ (2πi)2 Γ Γ̃

Γ Γ̃

On the other hand, we have ∮∮

1 1 R(λ, A)dμdλ = ∮(∮ dμ)R(λ, A)dλ. λ−μ λ−μ Γ

Γ Γ̃

Γ̃

Applying the residue theorem, we have ∮ Γ̃

1 dμ = −2πi. λ−μ

Hence, ∮∮ Γ Γ̃

1 R(λ, A)dμdλ = −2πi ∮ R(λ, A)dλ. λ−μ Γ

Moreover, ∮∮ Γ Γ̃

1 1 R(μ, A)dμdλ = ∮ ∮ R(μ, A)dλdμ λ−μ λ−μ Γ̃ Γ

= ∮(∮ Γ̃

Since the function λ 󳨀→

1 λ−μ

Γ

1 dλ)R(μ, A)dμ. λ−μ

is analytic, by the Cauchy theorem, ∮ Γ

1 dλ = 0. λ−μ

(3.3.2)

3.3 Projection

| 65

Hence, 1 R(μ, A)dμdλ = 0. λ−μ

∮∮ Γ Γ̃

(3.3.3)

Moreover, the equations (3.3.2) and (3.3.3) imply P2 = −

1 (−2πi) ∮ R(λ, A)dλ (2πi)2 Γ

1 = ∮ R(λ, A)dλ 2πi = P.

Γ

Thus, P is a projection. Proposition 3.3.3. Let A be a closed linear operator on X and P, defined in (3.3.1), the corresponding spectral projection. Then AP =

1 ∮ λR(λ, A)dλ. 2πi Γ

Proof. Using AR(λ, A) = [(A − λ) + λ]R(λ, A) = −I + λR(λ, A), we have AP = A(

1 ∮ R(λ, A)dλ). 2πi Γ

Since A is closed, by [161, p. 130], we can bring A under the integral sign. Therefore, AP =

1 ∮ AR(λ, A)dλ 2πi Γ

1 = ∮(−I + λR(λ, A))dλ 2πi Γ

1 = ∮ λR(λ, A)dλ. 2πi Γ

This completes the proof of the proposition. 3.3.3 Infinite sum of projections Let us now recall the following result due to S. Ja. Yakubov and K. S. Mamedov [114].

66 | 3 Basic notations and results Theorem 3.3.1 ([193]). Let A be a closed densely defined linear operator in a Hilbert space X having compact resolvent and assume that there exist a sequence of circles 𝒞 (O, rk ), k = 1, 2, . . ., with radius rk going to infinity, a constant c > 0, and an integer m ≥ −1 such that 󵄩󵄩 −1 󵄩 m 󵄩󵄩(A − λ) 󵄩󵄩󵄩 ≤ c|λ| ,

for |λ| = rk .

Then the spectrum of the operator A is discrete and, for any φ ∈ 𝒟(Am+2 ), there exists a subsequence of partial sums of the series ∑ Pn φ n

converging to φ in the sense of X, where Pn =

1 ∫ (σ − A)−1 dσ 2iπ 𝒞n

and 𝒞n is a positively oriented circle with center λn (an eigenvalue of A) and such that no other eigenvalue lies inside or on the circle.

3.3.4 Commutation of projections Let V ⊂ X be a closed subspace and let πV be the orthogonal projection of X on V. First, we prove the following result on operators. Proposition 3.3.4 (P. Gavruta [91, Proposition 2.1]). Let T ∈ ℒ(X) and V ⊂ X be a closed subspace. Then the following statements are equivalent: (i) πTV T = TπV ; and (ii) T ∗ TV ⊂ V. Proof. (i) 󳨐⇒ (ii) Let h ∈ V ⊥ . Then we have πTV Th = TπV h = 0. Hence, Th ∈ (TV)⊥ = (TV)⊥ . However, ⟨Th, Tv⟩ = 0 for all v ∈ V if and only if ⟨h, T ∗ Tv⟩ = 0 for all v ∈ V if and only if h ∈ (T ∗ TV)⊥ . (ii) 󳨐⇒ (i) If v ∈ V, then πTV Tv = Tv

3.3 Projection

| 67

and TπV v = Tv. If h ∈ V ⊥ , then TπV h = T0 = 0 and, from the hypothesis, we have ⊥

h ∈ (T ∗ TV) . As before, we now have Th ∈ (TV)⊥ . Hence, πTV Th = 0. Let X = ℝ2 , V = {(x, 0) : x ∈ ℝ}, and T : ℝ2 󳨀→ ℝ2 ,

T(x, y) = (x + y, y)

for all (x, y) ∈ ℝ2 . Then the adjoint of T is T ∗ (x, y) = (x, x + y)

for all (x, y) ∈ ℝ2 .

We have T ∗ T(x, 0) = T ∗ (x, 0) = (x, x) ∈ ̸ V

for all x ≠ 0.

From Proposition 3.3.4, we deduce πTV T ≠ TπV . Hence, we have the following. Corollary 3.3.1 (P. Gavruta [91, Corollary 2.2]). There exist Hilbert space X, an invertible operator T ∈ ℒ(X), and V, a closed subspace of X, such that πTV T ≠ TπV . Lemma 3.3.1 (P. Gavruta [91, Lemma 2.3]). Let T ∈ ℒ(X) and V ⊂ X be a closed subspace. Then we have πV T ∗ = πV T ∗ πTV . Proof. If f ∈ X, then f = πTV f + g,

g ∈ (TV)⊥ = (TV)⊥ .

Thus, T ∗ f = T ∗ πTV f + T ∗ g. However, for v ∈ V, we have ⟨T ∗ g, v⟩ = ⟨g, Tv⟩ = 0. Hence, T ∗ g ∈ V ⊥ , so πV T ∗ f = πV T ∗ πTV f + πV T ∗ g = πV T ∗ πTV f .

68 | 3 Basic notations and results

3.4 p-subordinate operator Definition 3.4.1. Let G and S be two linear operators on a Banach space X. The operator S is called subordinate to G with order p ∈ [0, 1] (or p-subordinate) if 𝒟(G) ⊂ 𝒟(S) and there is a constant c > 0 such that ‖Sφ‖ ≤ c‖Gφ‖p ‖φ‖1−p

for all φ ∈ 𝒟(G).

If p = 1, then S is said to be subordinate to G. We give a much simpler way by using the following lemma. Lemma 3.4.1. Let G and S be two linear operators on a Hilbert space X such that 𝒟(G) ⊂ 𝒟(S). Suppose that G is normal with compact resolvent and S is G-bounded with order p, (p ∈ [0, 1]). Then we have the following: (i) if N(G) = {0}, then S is G-subordinate with order p; (ii) if N(G) ≠ {0}, then S − P is (G + P)-subordinate with order p, where P denotes the orthogonal projection onto N(G). Proof. (i) The operator G has a compact resolvent. Then Theorem 2.13.1 implies that 0 ∈ ρ(G) and G is invertible, so we estimate 󵄩 󵄩p 󵄩 󵄩p a‖φ‖ = a󵄩󵄩󵄩G−1 Gφ󵄩󵄩󵄩 ‖φ‖1−p ≤ a󵄩󵄩󵄩G−1 󵄩󵄩󵄩 ‖Gφ‖p ‖φ‖1−p . Hence, 󵄩 󵄩p ‖Sφ‖ ≤ (b + a󵄩󵄩󵄩G−1 󵄩󵄩󵄩 )‖Gφ‖p ‖φ‖1−p ,

for every φ ∈ 𝒟(G).

Then, in accordance with Definition 2.5.2, we deduce that S is G-subordinate with order p. This proves the first item. (ii) Assume that N(G) ≠ {0} and let P be the orthogonal projection onto N(G). We define the operator G0 := G + P. Hence, we get 󵄩󵄩 󵄩 p 1−p 󵄩󵄩(S − P)φ󵄩󵄩󵄩 ≤ b‖Gφ‖ ‖φ‖ + (a + 1)‖φ‖ p ≤ b(‖G0 φ‖ + ‖Pφ‖) ‖φ‖1−p + (a + 1)‖φ‖

≤ b(‖G0 φ‖p + ‖Pφ‖p )‖φ‖1−p + (a + 1)‖φ‖ ≤ b‖G0 φ‖p ‖φ‖1−p + (a + b + 1)‖φ‖.

This shows that S−P is G0 -bounded with order p. Since 0∈ρ(G0 ), S−P is G0 -subordinate with order p. This ends the proof of the lemma. We denote sectors in the complex plane by Ω(φ− , φ+ ) := {reiθ such that r ≥ 0, φ− < θ < φ+ } and

Ω(φ) := Ω(−φ, φ)

for the sectors lying between the rays with arguments φ− , φ+ and −φ, φ, respectively.

3.4 p-subordinate operator

| 69

Lemma 3.4.2 (A. S. Markus [155, Lemma 6.8]). Suppose that L is a normal operator with compact resolvent, σ(L) ∩ Ω(2θ) ⊂ ℝ, and the operator T − L is p-subordinate to L (0 ≤ p < 1). Assume that there exists a sequence {rk }∞ 1 of positive numbers tending monotonically to ∞ such that, for some q > b, sup n+ (rk − qrkp , rk + qrkp , L) = m < ∞, k

(3.4.1)

where n+ (rk − qrkp , rk + qrkp , L) denotes the sum of multiplicities of all eigenvalues of L contained in the interval (rk − qrkp , rk + qrkp ). Then the numbers xk ∈ (rk − (q − b)rkp /8, rk + (q − b)rkp /8) can be chosen in such a way that ∞ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨∫⟨(λ − A)−1 f , g⟩dλ󵄨󵄨󵄨 < ∞ 󵄨󵄨 󵄨󵄨 k=1 Γk

holds for any f , g ∈ X. Condition (3.4.1) can be rewritten in the following form: lim inf n+ (r − qr p , r + qr p , L) < ∞. r→∞

(3.4.2)

Lemma 3.4.3 (A. S. Markus [155, Lemma 6.11]). If lim inf r p−1 n+ (r − qr p , r + qr p , L) < ∞, r→∞

(3.4.3)

then (3.4.2) holds for any q > 0, where n+ (r − qr p , r + qr p , L) denotes the sum of multiplicities of all eigenvalues of L contained in the interval (r − qr p , r + qr p ). Proof. Let us first consider the case where p = 0. Assume that (3.4.2) fails to hold. Then lim n+ (0, r + q, L) − n+ (0, r, L) = ∞, so, for any a > 0, there exists an r0 > 0 such that n+ (0, r + q, L) − n+ (0, r, L) > a

(r ≥ r0 ).

Consequently, n+ (0, r, L) − n+ (0, r0 , L) >

r − r0 a (r ≥ r0 ) q

and, hence, lim inf r −1 n+ (0, r, L) ≥

a , q

70 | 3 Basic notations and results which contradicts (3.4.3), because a is arbitrary. If p > 0, then it follows from (3.4.3) that 1

n (0, r 1−p , L) lim inf + 0, by what has been proved. This means that 1

lim inf(n+ (0, (r 1−p + q) 1−p , L) − n+ (0, r, L)) < ∞. r→∞

(3.4.4)

Since 1

1

(r 1−p + q) 1−p = r(1 + qr p−1 ) 1−p

≥ r + (1 − p)−1 qr p > r + qr p ,

we have, by (3.4.4), for any q > 0, lim inf(n+ (0, r + qr p , L) − n+ (0, r, L)) < ∞. r→∞

The lemma is proved.

3.5 Finite-order and sine-type functions 3.5.1 Finite-order function Definition 3.5.1. (i) An entire function f (z) is said to be of finite order if there exist M > 0 and p > 0 such that Mf (r) ≤ Mer

p

for r sufficiently large, where 󵄨 󵄨 Mf (r) := max󵄨󵄨󵄨f (z)󵄨󵄨󵄨. |z|=r

The greatest lower bound ρ of such numbers p is called the order of f , that is, ρ = lim

r→∞

log log Mf (r) log r

.

3.5 Finite-order and sine-type functions | 71

Similarly, we define the type σ of an entire function f (z) of order ρ to be the greatest lower bound of the positive numbers L such that Mf (r) ≤ MeLr

p

for r sufficiently large, that is, σ = lim

r→∞

log Mf (r) rp

.

(ii) An entire function of finite order p is said to be of the minimum type if, for any ε > 0, there exists M > 0 such that p

Mf (r) ≤ Meεr . Theorem 3.5.1 (E. Hille and R. S. Phillips [106]). Suppose that f (z) is an entire function of finite order which does not exceed ρ > 0. Suppose f (z) is bounded on a set of rays and arg z = θj , j = 1, 2, . . . , n, such that the angles between consecutive rays are less than πρ . Then f (z) is constant.

3.5.2 A function of the sine type Definition 3.5.2. An entire function f (z) is said to be of the exponential type if the inequality 󵄨󵄨 󵄨 B|z| 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ Ae ,

∀z ∈ ℂ,

(3.5.1)

holds for some positive constants A and B. The smallest of constants B such that (3.5.1) holds is said to be of the exponential type of f . Remark 3.5.1. An entire function of the exponential type is of finite order at most 1. By f (x) = O(ϕ(x)), we mean generally that 󵄨󵄨 󵄨 󵄨󵄨f (x)󵄨󵄨󵄨 < Aϕ(x) if x is sufficiently near some given limit. In particular, O(1) means a bounded function. By f (x) = o(ϕ(x)), we mean that f (x) →0 ϕ(x) as x tends to a given limit. In particular, o(1) means a function that tends to zero.

72 | 3 Basic notations and results Theorem 3.5.2 (R. M. Young [194, Theorem 5, p. 64]). If f (z) is an entire function of finite order ρ, then n(r) = O(r ρ+ξ ) for every positive number ξ , where n(r) denotes the number of zeros of f (z) contained in the disk 𝔹(0, r), where 𝔹(0, r) is the closure of the open ball 𝔹(0, r). We present the following important result. Theorem 3.5.3 (R. M. Young [194, Theorem 6, p. 64]). If f (z) is an entire function of finite order ρ and if z1 , z2 , z3 , . . . are its zeros, other than z = 0, then the series ∞

1

∑

|zn |α

n=1

is convergent whenever α > ρ. Proof. We suppose without loss of generality that the zeros of f (z) have been numbered so that 0 < |z1 | ≤ |z2 | ≤ ⋅ ⋅ ⋅ . Given α > ρ, choose β so that ρ < β < α. Since f (z) is of order ρ, we deduce from Theorem 3.5.2 that n(r) ≤ Ar β for some constant A and all values of r. Take r = |zn |. Then n(r) = n and, hence, n ≤ A|zn |β

for n = 1, 2, 3, . . . .

The result follows at once, since ∞

1

n=1

nβ

∑

α

< ∞.

Definition 3.5.3. The growth indicator of an exponential-type function f is a 2π-periodic function on ℝ, defined by the following equality: 1 󵄨 󵄨 hf (ϕ) = lim sup log󵄨󵄨󵄨f (reiϕ )󵄨󵄨󵄨, r→∞ r

ϕ ∈ [−π, π].

The indicator diagram of f is a convex set Gf such that hf (ϕ) = sup Re(ke−iϕ ), k∈Gf

ϕ ∈ [−π, π].

3.5 Finite-order and sine-type functions | 73

Definition 3.5.4. An entire function f of the exponential type is said to be a function of the Cartwright class if ∫ ℝ

max(log |f (x)|, 0) dx < +∞. 1 + x2

In particular, the function f of the exponential type satisfying the condition ∫ ℝ

|f (x)|2 dx < ∞ 1 + x2

belongs to the Cartwright class. Remark 3.5.2. The indicator diagram of a Cartwright class function is an interval [iα, iβ], α ≤ β of the imaginary axis. Its length is the width of the indicator diagram (see [35, pp. 59–60]). Definition 3.5.5. An entire function of the exponential type with simple zeros {λn }∞ 1 and with the width of the indicator diagram T is called a generating function of expo2 nential family {eiλn t }∞ 1 in L (0, T), T > 0. Definition 3.5.6. An entire function f of the exponential type is said to be of the sine type if: (i) the zeros of f lie in a strip {z ∈ ℂ such that |Im z| ≤ H} for some H > 0; and (ii) there exist h ∈ ℝ and positive constants c1 , c2 such that 󵄨 󵄨 c1 ≤ 󵄨󵄨󵄨f (x + ih)󵄨󵄨󵄨 ≤ c2 , for all x ∈ ℝ. For the sake of completeness, we first recall the following notions, which will be used in the sequel. The spectral radius of an operator A is defined by r(A) := sup |λ|. λ∈σ(A)

We define a few concepts of the essential spectral radius. Definition 3.5.7. A complex number λ is a normal point of a bounded linear operator T in a Banach space X if either λ belongs to the resolvent set of T or λ is an isolated point in the spectrum of T and the corresponding spectral projection has a finitedimensional range.

74 | 3 Basic notations and results Definition 3.5.8. The essential spectral radius ress (T) of a bounded linear operator T is the infimum of the set of the number r > 0 such that σ(T) ∩ {λ such that |λ| > r} consists of normal points of T, in other words, ress (T) is the radius of the complement of the normal points.

3.6 Singular values 3.6.1 Singular values of a compact operator Let K be a compact operator acting on a separable Hilbert space X. Note that K ∗ K is compact positive and self-adjoint. Let λ1 (K ∗ K) ≥ λ2 (K ∗ K) ≥ λ3 (K ∗ K) ≥ ⋅ ⋅ ⋅ be the sequence of non-zero eigenvalues of K ∗ K, where every eigenvalue is repeated as many times as its multiplicity. Definition 3.6.1. We say that a sequence of singular values of compact operator K is the sequence of eigenvalues of the compact positive operators √K ∗ K, ranked in descending order, where each eigenvalue is repeated as many times as its multiplicity. The eigenvalues of the operator √K ∗ K are called the s-numbers of the operator K. Note sn (K) are the s-numbers of the operator K, so we have sn (K) = √λn (K ∗ K). Let us give a proposition which will be used later. Proposition 3.6.1 (J. M. Paoli [167]). Let A ∈ ℒ(X) and let F be a closed subspace of X invariant by A. Then sn (A|F ) ≤ sn (A) for all n ≥ 1. Proof. Consider the linear application Φ : ℒ(X) 󳨀→ ℒ(X),

A 󳨀→ P ∘ A,

where P is the orthogonal projection on F. We have ‖P‖ ≤ 1 and ‖A|F ‖ ≤ ‖A‖,

3.6 Singular values | 75

so 󵄩 󵄩󵄩 󵄩󵄩Φ(A)󵄩󵄩󵄩 ≤ ‖A‖. Thus, ‖Φ‖ ≤ 1. Moreover, Φ(Fn−1 (X)) ⊂ Fn−1 (X), where Fn (X) is a subspace of ℒ(X) of operators of rank less than or equal to n. Let A󸀠 ∈ Fn−1 (F) and let A be the extension of A󸀠 to X obtained, for example, taking A = 0 on F ⊥ . Then A ∈ Fn−1 (X) and Φ(A) = A󸀠 . Hence, Φ(Fn−1 (X)) = Fn−1 (F). Thus, dist(Φ(A), Fn−1 (F)) ≤ dist(A, Fn−1 (X)) for all A ∈ ℒ(X). Moreover, since F is invariant by A, A|F = Φ(A) and sn (A|F ) = dist(A|F , Fn−1 (F)) ≤ dist(A, Fn−1 (X)) = sn (A).

This completes the proof. Proposition 3.6.2. Let A be a compact operator on a Hilbert space X and let B be a bounded operator on X. Then: (i) sj (BA) ≤ ‖B‖sj (A) for all j = 1, 2, . . .; and (ii) sj (AB) ≤ ‖B‖sj (A) for all j = 1, 2, . . .. 3.6.2 Numerical range Let X be a complex Hilbert space and let A be a bounded linear operator in X. Then the numerical range of A is the set W(A) := {⟨Ax, x⟩ x ∈ 𝕊X },

76 | 3 Basic notations and results where 𝕊X := {x ∈ X such that ‖x‖ = 1} is the unit sphere in X. The following properties of W(A) are immediate: W(αI + βA) = α + βW(A) for α, β ∈ ℂ, W(A∗ ) = {λ, λ ∈ W(A)}, W(U ∗ AU) = W(A),

for any unitary U.

By the well-known Toeplitz–Hausdorff theorem, the numerical range is a convex subset of ℂ and it satisfies the so-called spectral inclusion property σp (A) ⊂ W(A) and

σ(A) ⊂ W(A).

Note that W(A) is closed if dim X < ∞. Further, the resolvent of A can be estimated in terms of the distance to the numerical range. We have 1 󵄩󵄩 −1 󵄩 , 󵄩󵄩(λ − A) 󵄩󵄩󵄩 ≤ dist(λ, W(A))

λ ∉ W(A),

(3.6.1)

where dist(λ, W(A)) denotes the distance between λ and the set W(A). If a point λ ∈ W(A) is a corner of the numerical range (i. e., W(A) lies in a sector with vertex λ and angle less than π), then λ ∈ σ(A). If, in addition, λ ∈ W(A), then λ ∈ σp (A). The estimate (3.6.1) implies that, if λ ∈ σp (A) is a boundary point of W(A), then there are no associated vectors at λ. For many applications it is useful to consider unbounded operators. Assume now that A ∈ ℒ(𝒟(A), X) is a closed linear operator with domain 𝒟(A) densely and continuously embedded in X. Its numerical range is then defined by W(A) := {⟨Ax, x⟩ x ∈ 𝒟(A) ∩ 𝕊X }.

3.6.3 Carleman class Cp Definition 3.6.2. Let K be a compact operator in a Hilbert space X. Then K is said to belong to the Carleman class Cp , p > 0, with order p, if the series ∞

∑ spn (K)

n=1

converges, where (sn (K))n are the eigenvalues of the operator √K ∗ K.

3.7 Fractional operators | 77

We denote by Cp (X) (p > 0) the set of operators of the Carleman class Cp . In the particular case where p = 2, C2 (X) is exactly the space of Hilbert–Schmidt operators and, for p = 1, C1 (X) is called the space of nuclear operators or the space of trace class operators on X. For a systematic treatment of the operators of the Carleman class, we refer the reader to Gohberg and Krein’s book [94]. Theorem 3.6.1 (V. B. Lidskii [153]). If K belongs to the Carleman class Cp for some p > 0, then (I − λK)−1 can be represented as the ratio of two entire functions. We have (I − λK)−1 =

Dλ , d(λ)

(3.6.2)

for which the inequalities 󵄨󵄨 󵄨 c |λ|p 󵄨󵄨d(λ)󵄨󵄨󵄨 ≤ e 1

p

and ‖Dλ ‖ ≤ ec2 |λ|

(3.6.3)

are valid for all λ ∈ ℂ (c1 and c2 being positive constants). Theorem 3.6.2 (V. B. Lidskii [153]). Under the conditions of Theorem 3.6.1, the resolvent (I − λK)−1 can be represented as a ratio of entire functions (I − λK)−1 =

DK (λ) Δ K (λ)

such that 󵄨󵄨 󵄨 ε(λ)|λ|p 󵄨󵄨DK (λ)󵄨󵄨󵄨 ≤ e

p 󵄨 󵄨 and 󵄨󵄨󵄨Δ K (λ)󵄨󵄨󵄨 ≤ eε(λ)|λ| ,

(3.6.4)

where ε(λ) → 0 as λ → ∞.

3.7 Fractional operators Let X be a Hilbert space and let K be a compact operator on X. If the resolvent of K satisfies the estimate M 󵄩󵄩 −1 󵄩 , 󵄩󵄩(K − λ) 󵄩󵄩󵄩 ≤ |λ|

(3.7.1)

for λ lying in the sector S = {λ ∈ ℂ such that π −

δ δ ≤ arg(λ) ≤ π + }, 2 2

where δ > 0, then we define, for α ∈ ]0, 1[, the power K α as follows: ∞

sin(απ) 1 K = ∫ λα [(K + λ)−1 − ]dλ. π λ α

0

(3.7.2)

78 | 3 Basic notations and results Theorem 3.7.1 (A. Intissar [112]). Let K be a compact operator such that its resolvent satisfies the estimate (3.7.1) for λ lying in the sector S, given in (3.7.2). If K belongs to the Carleman class Cp , then we know its fractional powers K α belong to the Carleman class C p for every 0 < α < 1. α

Corollary 3.7.1 (V. I. Macaev and Ju. A. Palant [154]). Let K be a compact operator acting on a Hilbert space X. We suppose that: (i) |arg⟨Kφ, φ⟩| ≤ π2 for any φ ∈ X; and (ii) K ∈ Cp (X) for some p > 0. Then √K ∈ C2p (X).

3.8 Phragmén–Lindelöf theorems Theorem 3.8.1 (Phragmén–Lindelöf, R. M. Young [194, Theorem 10 p. 80]). Let f (z) be continuous on a closed sector of opening πα and analytic in the open sector. Suppose that, on the bounding rays of the sector, 󵄨󵄨 󵄨 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ M and that, for some β < α, 󵄨󵄨 󵄨 rβ 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ e , whenever z lies inside the sector and |z| = r is sufficiently large. Then 󵄨󵄨 󵄨 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ M throughout the sector. Proof. We suppose without loss of generality that the given sector is symmetric with respect to the positive real axis and that it has its vertex at the origin. We introduce the auxiliary function γ

g(z) = e−εz f (z), where β < γ < α and ε > 0. Here z γ denotes that single-valued analytic branch of the multiple-valued function z γ = exp(γ log z) that takes positive values for positive real z. Setting z = reiθ , we have 󵄨󵄨 󵄨 󵄨 −εr γ cos(γθ) 󵄨󵄨 󵄨󵄨g(z)󵄨󵄨󵄨 = e 󵄨󵄨f (z)󵄨󵄨󵄨.

3.9 Fredholm determinant | 79

On the bounding rays of the sector, cos(γθ) is positive and, hence, 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨g(z)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ M. On the arc |θ| ≤

π 2α

of the circle |z| = r, we have π 󵄨󵄨 󵄨 󵄨 −εr γ cos(γ 2α ) 󵄨󵄨 󵄨󵄨g(z)󵄨󵄨󵄨 ≤ e 󵄨󵄨f (z)󵄨󵄨󵄨

< er

β

π ) −εr γ cos(γ 2α

as soon as r is sufficiently large. However, this last expression approaches zero as r → ∞, so, if r is sufficiently large, 󵄨󵄨 󵄨 󵄨󵄨g(z)󵄨󵄨󵄨 ≤ M on this arc as well. By the maximum modulus principle, 󵄨󵄨 󵄨 󵄨󵄨g(z)󵄨󵄨󵄨 ≤ M throughout that part of the sector for which |z| ≤ r and hence throughout the entire sector, since r can be made arbitrarily large. Therefore, we conclude that 󵄨󵄨 󵄨 ε|z|γ 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ Me everywhere within the sector. Since ε is arbitrary, the result follows. The theorem is sharp in the sense that the conclusion is no longer valid when β = α. Indeed, we need only consider the function f (z) = ez

α

for |arg z| ≤

π ; 2α

f (z) is bounded on the rays of the sector but certainly not in the interior. Corollary 3.8.1. An entire function of order less than one that is bounded on a line must reduce to a constant.

3.9 Fredholm determinant Definition 3.9.1. Let K be an operator of the trace class on a Hilbert space X and (μn )n be the sequence of non-zero eigenvalues of K, counted according to their algebraic multiplicities. Then the Fredholm determinant d(λ) associated to K is given by ∞

d(λ) = ∏(1 − λμn ). n=1

80 | 3 Basic notations and results For an operator K belonging to the Carleman class Cp for some p > 1, it is then necessary to go back to the definition of d(λ) since the product is no longer always convergent. One sets ∞

1

m−1

d(λ) = ∏(1 − λμk )eλμk +⋅⋅⋅+ m−1 (λμk ) k=1

,

where m is an integer not less than 1 such that 1 < p ≤ m. We recall that the Fredholm determinant of K −1 is the function defined by ∞

d(ξ ) = ∏(1 − k=1

ν

ξ k ) , ξk

where the numbers ξk , k = 1, 2, . . ., are the unrepeated eigenvalues of K and the integers νk are the algebraic multiplicities of ξk . Let us also recall that the geometric multiplicity of ξk is the dimension of the eigenspace N(K − ξk ) as opposed to the algebraic multiplicity which is the dimension νk of the generalized eigenspace Hk = ⋃ N(K − ξk )m . m≥1

3.10 Denseness of generalized eigenvectors (0) Let 𝒞p,∞ (X) be the set defined by 1

(0)

𝒞p,∞ (X) = {A ∈ ℒ(X) such that lim n p sn (A) = 0} n→∞

= {A ∈ ℒ(X) such that sn (A) = o(n

− p1

) as n → ∞}.

Definition 3.10.1. A sequence of vectors {f1 , f2 , f3 , . . .} in a Hilbert space X is said to be complete if the zero vector alone is perpendicular to every fn . The main result given by M. T. Aimar, A. Intissar, and J. M. Paoli in [13, 14, 167] can be formulated as follows. Theorem 3.10.1 (M. T. Aimar, A. Intissar, and J. M. Paoli [13, 14, 167]). Let A be a closed and linear operator with a dense domain 𝒟(A) on a Hilbert space X. We assume that: (i) there exists λ ∈ ρ(A) such that (λ − A)−1 is a compact operator; (ii) there exists ξ0 ∈ ρ(A) such that (0) (ξ0 − A)−1 ∈ 𝒞p,∞ (X),

for some p > 0; and

3.11 Inequalities | 81

(iii) there are γ1 , γ2 , . . . , γm , m half-lines from the origin of the complex plane such that: (a) the angle between any two adjacent half-lines is less than or equal to πp (which implies that m ≥ [2p] + 1, where [2p] designates the entire part of 2p, except for around 2p strictly positive or we have m = 2p); (b) ‖(λ − A)−1 ‖ is bounded for |λ| sufficiently large on each of these half-lines, i. e., there is M > 0 and R > 0 such that 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − A) 󵄩󵄩󵄩 ≤ M, for |λ| ≥ R and λ ∈ ⋃1≤i≤m γi . Then the system of generalized eigenvectors of A is complete in X. Theorem 3.10.2 (A. S. Markus [155, Theorem 4.3]). Let L be a normal operator whose resolvent is an operator of finite order and whose spectrum lies on a finite number of rays arg λ = αk (k = 1, . . . , n). If T − L is compact relative to L, then the operator T has a compact resolvent and the system of its root vectors is complete in X.

3.11 Inequalities In this section, we need to recall some classical inequalities that we will later make use of. Lemma 3.11.1 (Young’s inequality). Given a, b, and β being positive numbers such that β ∈ [0, 1]. Then aβ b1−β ≤ βa + (1 − β)b. Proof. The function f : x 󳨀→ ex is convex on ℝ (since f (x) = ex > 0 on ℝ). Hence, for x, y ∈ ℝ and β ∈ [0, 1], we have 󸀠󸀠

f (βx + (1 − β)y) ≤ βf (x) + (1 − β)f (y). This means that eβx+(1−β)y ≤ βex + (1 − β)ey , or β

1−β

(ex ) (ey )

≤ βex + (1 − β)ey .

By putting a = ex and b = ey , we obtain aβ b1−β ≤ βa + (1 − β)b.

82 | 3 Basic notations and results Lemma 3.11.2 (Hölder’s inequality). For all p, q > 1 such that (a1 , . . . , an ), (b1 , . . . , bn ) ∈ ℝn , we have n

n

p

1 p

n

q

1 p

+

1 q

= 1 and for all

1 q

∑ |ak ||bk | ≤ ( ∑ |ak | ) ( ∑ |bk | ) .

k=1

Proof. Taking a = implies

p

|ak | , ∑nk=1 |ak |p

|ak |

(∑nk=1 |ak |p )

1 p

k=1

b=

q

|bk | , ∑nk=1 |bk |q

|bk |

k=1

and β = p1 , Young’s inequality (Lemma 3.11.1)

≤

1 q

(∑nk=1 |bk |q )

1 |bk |q 1 |ak |p + . n p p ∑k=1 |ak | q ∑nk=1 |bk |q

Hence, after summation, we have n

∑

k=1

|ak |

(∑nk=1 |ak |p )

1 p

|bk |

(∑nk=1 |bk |q )

≤

1 q

|a |p |b |q 1 n 1 n ∑ n k p + ∑ n k q p k=1 ∑k=1 |ak | q k=1 ∑k=1 |bk |

1 1 + p q = 1. =

Hence, 1 p

n

n

k=1

k=1

1 q

n

∑ |ak ||bk | ≤ ( ∑ |ak |p ) ( ∑ |bk |q ) . k=1

Lemma 3.11.3 (Minkowski’s inequality). For all p ≥ 1 and for all (a1 , . . . , an ) ∈ ℝn , (b1 , . . . , bn ) ∈ ℝn , we have 1 p

n

1 p

n

1 p

n

( ∑ |ak + bk |p ) ≤ ( ∑ |ak |p ) + ( ∑ |bk |p ) . k=1

k=1

k=1

Proof. For p = 1, the inequality is trivial. For p > 1, we have |ak + bk |p = |ak + bk ||ak + bk |p−1

≤ |ak ||ak + bk |p−1 + |bk ||ak + bk |p−1 .

Hence, after summation, we have n

n

n

k=1

k=1

k=1

∑ |ak + bk |p ≤ ∑ |ak ||ak + bk |p−1 + ∑ |bk ||ak + bk |p−1 .

Applying Hölder’s inequality, with q = n

p−1

∑ |ak ||ak + bk |

k=1

p , p−1 n

(p > 1), we obtain p

1 p

n

≤ ( ∑ |ak | ) ( ∑ |ak + bk | k=1

k=1

(p−1)q

1 q

)

(3.11.1)

3.11 Inequalities | 83 1 p

n

1− p1

n

≤ ( ∑ |ak |p ) ( ∑ |ak + bk |p ) k=1

.

k=1

Similarly, we have n

p−1

∑ |bk ||ak + bk |

k=1

n

p

1 p

n

1− p1

p

≤ ( ∑ |bk | ) ( ∑ |ak + bk | ) k=1

k=1

.

Hence, (3.11.1) implies n

n

k=1

k=1

1− p1

∑ |ak + bk |p ≤ ( ∑ |ak + bk |p )

1 p

n

1 p

n

[( ∑ |ak |p ) + ( ∑ |bk |p ) ]. k=1

k=1

Thus, n

1 p

1 p

n

n

1 p

( ∑ |ak + bk |p ) ≤ ( ∑ |ak |p ) + ( ∑ |bk |p ) . k=1

k=1

k=1

Lemma 3.11.4 (Interpolation inequality). Given a, b, θ, and δ being positive numbers such that 0 ≤ δ ≤ θ. Then aδ bθ−δ ≤ aθ + bθ . Proof. We discuss three cases: – If a < b, then ba > 1. Thus, θ−δ

b ( ) a

θ

b 1, then δ

θ

a a ( ) 0. Then there exists an integer N0 > 0 such that, for all M, N ≥ N0 , 󵄩󵄩 󵄩󵄩 L 󵄩󵄩 󵄩󵄩 ‖AM − AN ‖Y = sup󵄩󵄩󵄩 ∑ (cM (n) − cN (n))xn 󵄩󵄩󵄩 < ε. 󵄩󵄩 󵄩󵄩n=1 L 󵄩 󵄩 Fix L > 0 and define L

yM,N = ∑ (cM (n) − cN (n))xn . n=1

We have ‖yM,N ‖ < ε for each M, N ≥ N0 . Furthermore, for each fixed n ≥ 2 and M, N ∈ ℕ∗ , we have 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨cM (n) − cN (n)󵄨󵄨󵄨‖xn ‖ = 󵄩󵄩󵄩(cM (n) − cN (n))xn 󵄩󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩n−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ (cM (k) − cN (k))xk 󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∑ (cM (k) − cN (k))xk 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩k=1 󵄩󵄩󵄩 󵄩󵄩󵄩k=1 󵄩 ≤ 2‖AM − AN ‖Y . Also, for n = 1, we have 󵄨󵄨 󵄨 󵄨󵄨cM (n) − cN (n)󵄨󵄨󵄨‖xn ‖ ≤ ‖AM − AN ‖Y . Since (AN )N is Cauchy and xn ≠ 0, we conclude that (cN (n))N is a Cauchy sequence of scalars and, therefore, must converge to some scalar c(n) as N → ∞. Let L

yN = ∑ (c(n) − cN (n))xn . n=1

Also, keeping L fixed, we have yM,N → yN

as M → ∞.

Indeed, 󵄩󵄩 L 󵄩󵄩 󵄩󵄩 󵄩󵄩 ‖yM,N − yN ‖ = 󵄩󵄩󵄩 ∑ (c(n) − cM (n))xn 󵄩󵄩󵄩 󵄩󵄩n=1 󵄩󵄩 󵄩 󵄩 L

󵄨 󵄨 ≤ ∑ 󵄨󵄨󵄨c(n) − cM (n)󵄨󵄨󵄨‖xn ‖ n=1

→0

as M → ∞.

90 | 4 Bases Consequently, for all N ≥ N0 , we have ‖yN ‖ = lim ‖yM,N ‖ ≤ ε. M→∞

Our goal is to show that AN → A = (c(n))n in the norm of Y as N → ∞. Substituting the definition of yN and taking the supremum over L, we obtain, for all N ≥ N0 , 󵄩󵄩 󵄩󵄩 L 󵄩󵄩 󵄩󵄩 󵄩 sup󵄩󵄩 ∑ (c(n) − cN (n))xn 󵄩󵄩󵄩 ≤ ε. 󵄩󵄩 󵄩 L 󵄩 󵄩 󵄩n=1

(4.4.6)

Now, (cN0 (n))n ∈ Y, so the series ∑ cN0 (n)xn n

converges by the definition of Y. Hence, there is an M0 > 0 such that, for all N > M ≥ M0 , 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ cN (n)xn 󵄩󵄩󵄩 < ε. 󵄩󵄩 󵄩󵄩 0 󵄩󵄩n=M+1 󵄩󵄩

(4.4.7)

Therefore, for N > M ≥ max(M0 , N0 ) and by using equations (4.4.6) and (4.4.7), we have 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 M N 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ c(n)xn 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ (c(n) − cN (n))xn − ∑ (c(n) − cN (n))xn + ∑ cN (n)xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 0 0 󵄩󵄩n=M+1 󵄩󵄩 󵄩󵄩n=1 󵄩󵄩 n=1 n=M+1 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 M 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ (c(n) − cN0 (n))xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∑ (c(n) − cN0 (n))xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∑ cN0 (n)xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩n=1 󵄩󵄩󵄩 󵄩󵄩󵄩n=1 󵄩 󵄩n=M+1 󵄩 ≤ ε + ε + ε = 3ε. Therefore, ∞

∑ c(n)xn

n=1

converges in X. Thus, A = (c(n)) ∈ Y. Finally, equation (4.4.6) leads to AN → A in the norm of Y. Hence, Y is complete. Lemma 4.4.2. Let (xn )n be a sequence in a Banach space X and assume that xn ≠ 0 for every n. Let Y be the vector space defined in (4.4.4) with the norm ‖ ⋅ ‖Y given in (4.4.5). If (xn )n is a basis for X, then the linear operator T defined from Y into X by ∞

T : (cn )n 󳨀→ ∑ cn xn n=1

is bounded.

(4.4.8)

4.4 Natural projections associated with a basis | 91

Proof. Using Theorem 4.4.1, we know Y is a Banach space. Evidently, T is linear. If (cn ) ∈ Y, then 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩T(cn )󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩n=1 󵄩 󵄩 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩󵄩󵄩 = lim 󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 󵄩󵄩 N→∞󵄩 󵄩 󵄩󵄩n=1 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ sup󵄩󵄩󵄩 ∑ cn xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩n=1 N 󵄩 󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩(cn )n 󵄩󵄩Y . Hence, T is bounded. Theorem 4.4.2. Let (xn )n be a sequence in a Banach space X and assume that xn ≠ 0 for every n. Let Y be the vector space defined in (4.4.4) with the norm ‖ ⋅ ‖Y given in (4.4.5). If (xn )n is a basis for X, then Y is topologically isomorphic to X via the synthesis mapping T, given in (4.4.8). Proof. Let (xn )n be a basis for X. Then ∞

T(cn ) = ∑ cn xn n=1

maps Y into X by the definition of Y. Using Theorem 4.4.1, we know Y is a Banach space. Evidently, T is bijective because (xn )n is a basis and by using Lemma 4.4.2 we know T is bounded. The inverse mapping theorem therefore implies that T is a topological isomorphism of Y onto X. Theorem 4.4.3. Let (xn )n be a basis for a Banach space X, with coefficient functionals (an )n . Let Y be the vector space defined in (4.4.4), so ∞

T(cn ) = ∑ cn xn n=1

is a topological isomorphism of Y onto X. Then the partial sum operators SN , given in (4.4.1), are bounded. Moreover, 󵄩 󵄩 ‖SN ‖ ≤ 󵄩󵄩󵄩T −1 󵄩󵄩󵄩 for each N ∈ ℕ∗ . Proof. Let x ∈ X. Then ∞

x = ∑ an (x)xn . n=1

92 | 4 Bases Since the scalars an (x) are unique, T −1 is given as follows: T −1 x = (an (x)). Thus, 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 sup ‖SN x‖ = sup󵄩󵄩󵄩 ∑ an (x)xn 󵄩󵄩󵄩 󵄩󵄩 󵄩 N N 󵄩 󵄩 󵄩n=1 󵄩 󵄩 = 󵄩󵄩󵄩(an (x))n 󵄩󵄩󵄩Y 󵄩 󵄩 = 󵄩󵄩󵄩T −1 x 󵄩󵄩󵄩Y 󵄩 󵄩 ≤ 󵄩󵄩󵄩T −1 󵄩󵄩󵄩‖x‖.

(4.4.9)

Therefore, by using both equations (2.0.1) and (4.4.9), we know SN is bounded and its operator norm satisfies 󵄩 󵄩 ‖SN ‖ ≤ 󵄩󵄩󵄩T −1 󵄩󵄩󵄩. Theorem 4.4.4 (C. Heil [103, Theorem 4.13]). Let (xn )n be a basis for a Banach space X, with coefficient functionals (an )n . Let Y be the vector space defined in (4.4.4), so ∞

T(cn ) = ∑ cn xn n=1

is a topological isomorphism of Y onto X. Then sup ‖SN ‖ < ∞. N

Proof. From Theorem 4.4.3, we have 󵄩 󵄩 sup ‖SN ‖ ≤ 󵄩󵄩󵄩T −1 󵄩󵄩󵄩 < ∞. N

Theorem 4.4.5. Let (xn )n be a basis for a Banach space X, with coefficient functionals (an )n . Let Y be the vector space defined in (4.4.4), so ∞

T(cn ) = ∑ cn xn n=1

is a topological isomorphism of Y onto X. Then |‖x‖| := sup ‖SN x‖ N

forms a norm on X that is equivalent to the initial norm ‖ ⋅ ‖ and we have ‖ ⋅ ‖ ≤ |‖ ⋅ ‖| ≤ sup ‖SN ‖‖ ⋅ ‖. N

4.4 Natural projections associated with a basis | 93

Proof. It is easy to see that |‖ ⋅ ‖| has at least the properties of a semi-norm. Let x ∈ X. Then we have |‖x‖| = sup ‖SN x‖ N

≤ sup(‖SN ‖‖x‖) N

= sup ‖SN ‖‖x‖.

(4.4.10)

N

Since SN x → x in the norm of X, we have ‖x‖ = lim ‖SN x‖ N→∞

≤ sup ‖SN x‖ N

(4.4.11)

= |‖x‖|.

It follows from these two estimate equations (4.4.10) and (4.4.11) that |‖ ⋅ ‖| is a norm and that it is equivalent to ‖ ⋅ ‖. Theorem 4.4.6. Let (xn )n be a basis for a Banach space X, with coefficient functionals (an )n . Let Y be the vector space defined in (4.4.4), so ∞

T(cn ) = ∑ cn xn n=1

is a topological isomorphism of Y onto X. Then the coefficient functionals an are continuous linear functionals on X that satisfy 1 ≤ ‖an ‖‖xn ‖ ≤ 2 sup ‖SN ‖, N

n ∈ ℕ∗ .

Proof. Since (xn )n is a basis, each xn is non-zero, so (an )n satisfies the estimates (4.4.3) for n ≥ 2. Since a1 (x)x1 = S1 x, the same estimate (4.4.3) is also valid for n = 1. Consequently, each an is bounded and ‖an ‖‖xn ‖ ≤ 2 sup ‖SN ‖ N

for each n. As in the discussion following equation (4.2.1), by the uniqueness we must have am (xn ) = δmn , so (xn )n and (an )n are biorthogonal and, therefore, 1 = an (xn ) ≤ ‖an ‖‖xn ‖, which completes the proof.

94 | 4 Bases Theorem 4.4.7. Let (xn )n be a basis for a Banach space X, with coefficient functionals (an )n . Let Y be the vector space defined in (4.4.4), so ∞

T(cn ) = ∑ cn xn n=1

is a topological isomorphism of Y onto X. Then (xn )n is a Schauder basis for X and (an )n is the unique sequence in X ∗ that is biorthogonal to (xn )n . Proof. Since the coefficient functionals are continuous, (xn )n is a Schauder basis. We have observed that (an )n is a biorthogonal sequence in X ∗ . This biorthogonal system is unique because of the fact that (xn )n is complete.

4.5 A characterization of bases Lemma 4.5.1. Let {ek }∞ k=1 be a basis of X and let f ∈ X, i. e., ∞

f = ∑ ck ek k=1

for some coefficients {ck }∞ k=1 . Consider 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 |‖f ‖| := sup󵄩󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 < ∞. 󵄩󵄩 n 󵄩 󵄩󵄩k=1 󵄩 Then |‖ ⋅ ‖| is a norm on X, so X is a Banach space with respect to this norm. Proof. Let {ek }∞ k=1 be a basis of X. Then each f ∈ X has a unique expansion ∞

f = ∑ ck ek k=1

and 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 |‖f ‖| := sup󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 < ∞. 󵄩󵄩 n 󵄩 󵄩󵄩k=1 󵄩 It is easy to see that |‖ ⋅ ‖| has at least the properties of a semi-norm. If |‖f ‖| = 0, then 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 = 0 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩k=1 for all n ∈ ℕ∗ . Hence, ck = 0 for all k ∈ ℕ∗ and, therefore, f = 0, so |‖ ⋅ ‖| is a norm on X and that X is a Banach space with respect to this norm.

4.5 A characterization of bases | 95

Let {ek }∞ k=1 be a family of non-zero vectors in X and let 𝒜 be the vector space consisting of all f ∈ X which can be expanded as ∞

f = ∑ ck ek k=1

∞ for some coefficients {ck }∞ k=1 . Let {fj }j=1 be a sequence of 𝒜. Then we may write fj in the following form: ∞

fj = ∑ ck ek (j)

k=1

for appropriate coefficients {ck }∞ k=1 . If there exists a constant K such that, for all m, ∗ n ∈ ℕ with m ≤ n, (j)

󵄩󵄩 m 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 ≤ K 󵄩󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩k=1 󵄩 󵄩k=1 󵄩

(4.5.1)

∗ for all scalar-valued sequences {ck }∞ k=1 , then, by (4.5.1), for each i∈ℕ and all n ≥ m ≥ i, ∗ we have, for all j, l ∈ ℕ ,

󵄩󵄩 m 󵄩󵄩 󵄩󵄩 󵄨󵄨 (j) 󵄩󵄩󵄩 (j) (l) 󵄨󵄨 (l) 󵄩 c − c ‖e ‖ ≤ K (c − c )e ∑ 󵄨󵄨 i 󵄩󵄩 k󵄩 󵄩󵄩 i 󵄨󵄨 i k k 󵄩󵄩 󵄩󵄩 󵄩k=1 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 (j) ≤ K 2 󵄩󵄩󵄩 ∑ (ck − ck(l) )ek 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=1 󵄩 󵄩󵄩 n 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 (j) ≤ K 2 (󵄩󵄩󵄩 ∑ ck ek − fj 󵄩󵄩󵄩 + ‖fj − f ‖ 󵄩󵄩󵄩k=1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 + ‖fl − f ‖ + 󵄩󵄩󵄩 fl − ∑ ck(l) ek 󵄩󵄩󵄩). 󵄩󵄩 󵄩󵄩 k=1 󵄩 󵄩

(4.5.2)

(4.5.3)

The results obtained by O. Christensen, appearing in the remaining part of this section, can be found in [66]. Lemma 4.5.2. If {ek }∞ k=1 is a basis, then equation (4.5.1) holds for all scalar-valued sequences {ck }∞ . k=1 Proof. If {ek }∞ k=1 is a basis, then each f ∈ X has a unique expansion ∞

f = ∑ ck ek k=1

and 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 |‖f ‖| := sup󵄩󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 < ∞. 󵄩󵄩 n 󵄩 󵄩󵄩k=1 󵄩

96 | 4 Bases Using Lemma 4.5.1, |‖ ⋅ ‖| is a norm on X and that X is a Banach space with respect to this norm. By the definition of |‖ ⋅ ‖|, we have ‖f ‖ ≤ |‖f ‖|, for all f ∈ X, meaning that the identity operator is a continuous and injective mapping of (X, |‖ ⋅ ‖|) onto (X, ‖ ⋅ ‖). From Theorem 2.10.1, it follows that this operator has a continuous inverse, i. e., that there exists a constant K > 0 such that |‖f ‖| ≤ K‖f ‖ for all f ∈ X. In particular, fixing an arbitrary n ∈ ℕ∗ and considering n

f = ∑ ck ek , k=1

we obtain (4.5.1). ∞ Lemma 4.5.3. Let {ek }∞ k=1 be a family of non-zero vectors in X. If the family {ek }k=1 is ∗ complete in X and there exists a constant K such that, for all m, n ∈ ℕ with m ≤ n, equation (4.5.1) holds for all scalar-valued sequences {ck }∞ k=1 , then the vector space 𝒜 is closed in X.

Proof. Let f ∈ X and let {fj }∞ j=1 be a sequence in 𝒜 such that fj → f as j → ∞. Write ∞

fj = ∑ ck ek (j)

k=1

∗ for appropriate coefficients {ck }∞ k=1 . Given ε > 0, choose N ∈ ℕ such that (j)

‖f − fj ‖ ≤

ε 2K 2

for j ≥ N,

(4.5.4)

where K is a constant given in (4.5.1). By letting n → ∞, it follows from the estimates (4.5.3) and (4.5.4) that 󵄨󵄨 (j) (l) 󵄨 󵄨󵄨ci − ci 󵄨󵄨󵄨‖ei ‖ ≤ ε

for all i ∈ ℕ∗ , j, l ≥ N,

(4.5.5)

and, via the intermediate step (4.5.2), 󵄩󵄩 m 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ (c(j) − c(l) )ek 󵄩󵄩󵄩 ≤ ε 󵄩󵄩 󵄩󵄩 k k 󵄩󵄩k=1 󵄩󵄩

for all m ∈ ℕ∗ , j, l ≥ N.

(4.5.6)

(l) For each i ∈ ℕ∗ , the sequence {ci(l) }∞ l=1 is convergent by (4.5.5), say, ci → ci as l → ∞. By letting l → ∞ in both (4.5.5) and (4.5.6), we obtain

󵄨󵄨 (j) 󵄨 󵄨󵄨ci − ci 󵄨󵄨󵄨‖ei ‖ ≤ ε

for all i ∈ ℕ∗ , j ≥ N,

4.5 A characterization of bases | 97

and 󵄩󵄩 󵄩󵄩 m 󵄩 󵄩󵄩 󵄩󵄩 ∑ (c(j) − ck )ek 󵄩󵄩󵄩 ≤ ε 󵄩󵄩 󵄩󵄩 k 󵄩󵄩 󵄩󵄩k=1

for all m ∈ ℕ∗ , j ≥ N.

(4.5.7)

Now, for given m ∈ ℕ∗ and all j ∈ ℕ∗ , we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 󵄩󵄩 󵄩󵄩 m m 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 f − ∑ ck ek 󵄩󵄩󵄩 ≤ ‖fj − f ‖ + 󵄩󵄩󵄩 fj − ∑ c(j) ek 󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∑ (c(j) − ck )ek 󵄩󵄩󵄩. 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k k 󵄩󵄩 󵄩󵄩 󵄩󵄩k=1 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=1 k=1 For a given ε > 0, we choose N ∈ ℕ∗ so that (4.5.7) holds. By fixing a sufficiently large value for j > N, we obtain ‖f − fj ‖ ≤ ε. After that, we obtain 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 󵄩 󵄩󵄩 fj − ∑ c(j) ej 󵄩󵄩󵄩 ≤ ε 󵄩󵄩 󵄩󵄩 k 󵄩󵄩 󵄩󵄩 k=1 by choosing m ∈ ℕ∗ sufficiently large. Thus, 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 󵄩 󵄩󵄩 f − ∑ ck ek 󵄩󵄩󵄩 ≤ 3ε 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=1 for m sufficiently large. We conclude that ∞

f = ∑ ck ek , k=1

i. e., the series ∞

∑ ck ek

k=1

converges to f and f ∈ 𝒜 as desired. ∞ Lemma 4.5.4. Let {ek }∞ k=1 be a family of non-zero vectors in X. If the family {ek }k=1 is ∗ complete in X and there exists a constant K such that, for all m, n ∈ ℕ with m ≤ n, equation (4.5.1) holds for all scalar-valued sequences {ck }∞ k=1 , then the vector space 𝒜 is the whole space X.

Proof. Since {ek }∞ k=1 is assumed to be complete, we know that 𝒜 is dense in X, so the result follows from Lemma 4.5.3.

98 | 4 Bases It is clear that a basis for X is complete and consists of non-zero vectors. Adding an extra condition leads to a characterization of bases. Theorem 4.5.1 (O. Christensen [66, Theorem 3.1.4]). A complete family of non-zero vectors {ek }∞ k=1 in X is a basis for X if and only if there exists a constant K such that, for all m, n ∈ ℕ∗ with m ≤ n, equation (4.5.1) holds for all scalar-valued sequences {ck }∞ k=1 . Proof. The first implication follows from Lemma 4.5.2. For the other implication, assume that a complete family {ek }∞ k=1 of non-zero vectors satisfies (4.5.1). By using Lemma 4.5.4, we infer that 𝒜 = X. To prove that {ek }∞ k=1 is a basis, we only need to show that, if ∞

∑ ck ek = 0,

k=1

then ck = 0 for all k ∈ ℕ∗ . This again follows from (4.5.1). If ∞

∑ ck ek = 0,

k=1

then, for each i ∈ ℕ∗ and all n ≥ i, 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 |ci |‖ei ‖ ≤ K 󵄩󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩. 󵄩󵄩 󵄩󵄩 󵄩k=1 󵄩 From this we obtain the result by letting n → ∞. ∞ Corollary 4.5.1. The coefficient functionals {ck }∞ k=1 associated with a basis {ek }k=1 for X ∗ are continuous and can thus be considered as elements in the dual X . If there exists a constant C > 0 such that

‖ek ‖ ≥ C

(4.5.8)

for all k ∈ ℕ∗ , then the norms of {ck }∞ k=1 are uniformly bounded. Proof. We use Theorem 4.5.1 and the notation introduced there. Let f ∈ X and write ∞

f = ∑ ck (f )ek . k=1

Then, for any j ∈ ℕ and all n ≥ j, we have ∗

󵄩󵄩 n 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩󵄩 󵄩󵄩 󵄨󵄨cj (f )󵄨󵄨‖ej ‖ ≤ K 󵄩󵄩 ∑ ck (f )ek 󵄩󵄩󵄩. 󵄩󵄩 󵄩󵄩 󵄩k=1 󵄩 Letting n → ∞ in the least equation and using equation (4.5.8), we deduce K 󵄨󵄨 󵄨 ‖f ‖ 󵄨󵄨cj (f )󵄨󵄨󵄨 ≤ ‖ej ‖

4.6 Biorthogonal systems | 99

≤

K ‖f ‖, C

which completes the proof. ∞ Theorem 4.5.2. Let {ek }∞ k=1 be a basis for X and let {ck }k=1 be the associated coefficient functionals. Then: ∗ (i) {ck }∞ k=1 is a basis for its closed span in X and its associated biorthogonal system is ∞ ∗∗ {ek }k=1 (considered as elements in X ); and ∗ (ii) if X is reflexive, then {ck }∞ k=1 is a basis for X .

4.6 Biorthogonal systems Theorem 4.6.1 (C. Heil [103, Theorem 3.3.2]). Assume that {ek }∞ k=1 is a basis for the ∞ Hilbert space X. Then there exists a unique family {gk }k=1 in X for which ∞

f = ∑ ⟨f , gk ⟩ek , k=1

for all f ∈ X.

(4.6.1)

∞ ∞ {gk }∞ k=1 is a basis for X and {ek }k=1 and {gk }k=1 are biorthogonal. ∞ Proof. By Corollary 4.5.1, the coefficient functionals {ck }∞ k=1 associated with {ek }k=1 are continuous. The Riesz representation theorem (see Theorem 3.2.6) implies that there exists a unique family {gk }∞ k=1 such that

ck (f ) = ⟨f , gk ⟩,

for all f ∈ X.

Hence, ∞

f = ∑ ⟨f , gk ⟩ek , k=1

for all f ∈ X.

We leave it to the reader to verify that no other family {gk }∞ k=1 can satisfy (4.6.1) and ∞ ∞ ∞ that {ek }k=1 and {gk }k=1 are biorthogonal. The fact that {gk }k=1 is a basis for X follows from Theorem 4.5.2.

4.7 Orthonormal bases Definition 4.7.1. A sequence {ek }∞ k=1 in X is an orthonormal system if ⟨ek , ej ⟩ = δkj . An orthonormal basis is an orthonormal system {ek }∞ k=1 which is a basis for X.

100 | 4 Bases ∞ Note that an orthonormal system {ek }∞ k=1 is a Bessel sequence. In fact, if {ck }k=1 ∈ ∗ l (ℕ) and m, n ∈ ℕ , n > m, then 2

2 2 󵄩󵄩 n m 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 n 󵄩󵄩 󵄩󵄩 ∑ ck ek − ∑ ck ek 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=m+1 󵄩󵄩 k=1 k=1 n

= ∑ |ck |2 , k=m+1

as in the reasoning given before Theorem 3.1.1. This implies that ∞

∑ ck ek

k=1

is convergent and that 󵄩󵄩 ∞ 󵄩󵄩2 ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∑ ck ek 󵄩󵄩󵄩 = ∑ |ck |2 . 󵄩󵄩 󵄩󵄩 󵄩󵄩k=1 󵄩󵄩 k=1 Theorem 4.7.1 (Hilbert–Schmidt theorem). Let K be a compact operator and let it be self-adjoint on a separable Hilbert space X. Then there is an orthonormal basis {ϕn }∞ 1 of X such that Kϕn = λn ϕn

and

lim λ n→∞ n

= 0.

The next theorem gives equivalent conditions for an orthonormal system {ek }∞ k=1 to be an orthonormal basis. Theorem 4.7.2 (O. Christensen [66, Theorem 3.4.2]). For an orthonormal system {ek }∞ k=1 , the following are equivalent: (i) {ek }∞ k=1 is an orthonormal basis; (ii) f = ∑∞ k=1 ⟨f , ek ⟩ek for all f ∈ X; (iii) ⟨f , g⟩ = ∑∞ k=1 ⟨f , ek ⟩⟨ek , g⟩ for all f , g ∈ X; 2 2 (iv) ∑∞ |⟨f , e k ⟩| = ‖f ‖ for all f ∈ X; k=1 ∞ (v) span{ek }k=1 = X; and (vi) if ⟨f , ek ⟩ = 0 for all k ∈ ℕ∗ , then f = 0. Proof. For the proof of (i) 󳨐⇒ (ii), let f ∈ X. If {ek }∞ k=1 is an orthonormal basis, there exist coefficients {ck }∞ such that k=1 ∞

f = ∑ ck ek . k=1

Given any j ∈ ℕ , we have ∗

∞

⟨f , ej ⟩ = ∑ ck δkj = cj k=1

4.7 Orthonormal bases | 101

and (ii) follows. Statement (iii) is an obvious consequence of (ii) and (iv) is a special case of (iii). The implications (iv) 󳨐⇒ (v) 󳨐⇒ (vi) are clear. For the proof of (vi) 󳨐⇒ (i), let f ∈ X. Since {ek }∞ k=1 is a Bessel sequence, we know that ∞

g := ∑ ⟨f , ek ⟩ek k=1

is well defined. Furthermore, ⟨f − g, ej ⟩ = 0 for all j ∈ ℕ∗ , so, by (vi), ∞

f = g = ∑ ⟨f , ek ⟩ek . k=1

To prove that {ek }∞ k=1 is a basis, we only need to show that no other linear combination of {ek }∞ can be equal to f . This follows from the argument we used to prove that (ii) k=1 follows from (i). The equality in Theorem 4.7.2 (iv) is called Parseval’s equation. Via Corollary 3.1.2, we obtain the following important consequence of Theorem 4.7.2. Corollary 4.7.1. If {ek }∞ k=1 is an orthonormal basis, then each f ∈X has an unconditionally convergent expansion ∞

f = ∑ ⟨f , ek ⟩ek . k=1

In particular, the dual basis equals the basis itself. Proposition 4.7.1 (O. Christensen [66, Proposition 3.4.8]). Assume that {ek }∞ k=1 is a sequence of normalized vectors in X and that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , ek ⟩󵄨󵄨󵄨 = ‖f ‖2 ,

k=1

for all f ∈ X.

Then {ek }∞ k=1 is an orthonormal basis for X. Proof. In view of Theorem 4.7.2, we only have to prove that {ek }∞ k=1 is an orthonormal system. For each j ∈ ℕ∗ , we have ∞

󵄨 󵄨2 󵄨 󵄨2 1 = ‖ej ‖2 = ∑ 󵄨󵄨󵄨⟨ej , ek ⟩󵄨󵄨󵄨 = 1 + ∑ 󵄨󵄨󵄨⟨ej , ek ⟩󵄨󵄨󵄨 , k=1

which shows that ⟨ej , ek ⟩ = 0 for k ≠ j.

k =j̸

102 | 4 Bases Let X be a separable Hilbert space. We choose a sequence {fk }∞ k=1 in X such that span{fk }∞ k=1 = X. By extracting a subsequence if necessary, we assume that, for each n ∈ ℕ∗ , fn+1 ∈ ̸ span{fk }nk=1 . By applying the Gram–Schmidt process to {fk }∞ k=1 , we obtain an orthonormal system {ek }∞ in X for which k=1 ∞ span{ek }∞ k=1 = span{fk }k=1 = X,

so we have the following theorem. Theorem 4.7.3. Every separable Hilbert space X has an orthonormal basis. ∞ Lemma 4.7.1. Let {ek }∞ k=1 be an orthonormal basis for X and let {δk }k=1 be the canonical 2 orthonormal basis for l (ℕ). Then the operator

U : X 󳨀→ l2 (ℕ),

∞

∞

k=1

k=1

U( ∑ ck ek ) = ∑ ck δk ,

2 {ck }∞ k=1 ∈ l (ℕ),

(4.7.1)

is an isometry. Proof. Let {ek }∞ k=1 be an orthonormal basis for X. Then each f ∈ X has a unique expansion ∞

f = ∑ ⟨f , ek ⟩ek . k=1

2 Since Uf ∈ l2 (ℕ) and {δk }∞ k=1 is the canonical orthonormal basis for l (ℕ), we have

󵄩󵄩 ∞ 󵄩󵄩2 󵄩󵄩 󵄩󵄩 ‖Uf ‖2 = 󵄩󵄩󵄩 ∑ ⟨f , ek ⟩δk 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩k=1 󵄩 ∞

󵄨 󵄨2 = ∑ 󵄨󵄨󵄨⟨f , ek ⟩󵄨󵄨󵄨 k=1

= ‖f ‖2 , so U is an isometry, which completes the proof. Based on Theorem 4.7.3, we can prove that every separable Hilbert space can be identified with l2 (ℕ). Theorem 4.7.4. Every separable infinite-dimensional Hilbert space X is isometrically isomorphic to l2 (ℕ).

4.7 Orthonormal bases | 103

Proof. Suppose that {ek }∞ k=1 is an orthonormal basis for X. Note that ∞

∑ ck ek

k=1

2 is convergent for all {ck }∞ k=1 ∈ l (ℕ). Also, each f ∈ X has a unique expansion with 2 l -coefficients, namely, ∞

f = ∑ ⟨f , ek ⟩ek . k=1

2 Let {δk }∞ k=1 be the canonical orthonormal basis for l (ℕ). Then we can define the operator U by equation (4.7.1). Then, using Lemma 4.7.1, we know U is an isometry, so U maps X bijectively onto l2 (ℕ). This completes the proof. ∞ Lemma 4.7.2. Let {ek }∞ k=1 and {fk }k=1 be two orthonormal bases for X. Define the operator

U : X 󳨀→ X,

∞

∞

k=1

k=1

U( ∑ ck ek ) = ∑ ck fk ,

2 {ck }∞ k=1 ∈ l (ℕ).

(4.7.2)

Then U maps X boundedly and bijectively onto X. Proof. The proof is left to the reader. Lemma 4.7.3. The operator U, given in (4.7.2), is unitary. Proof. Given f , g ∈ X. We write ∞

f = ∑ ⟨f , ek ⟩ek k=1

and ∞

g = ∑ ⟨g, ek ⟩ek . k=1

If {fk }∞ k=1 is an orthonormal basis for X and if U is the operator given in (4.7.2), then, via both the definition of U and Theorem 4.7.2, we have ⟨U ∗ Uf , g⟩ = ⟨Uf , Ug⟩ ∞

∞

k=1

k=1

= ⟨ ∑ ⟨f , ek ⟩fk , ∑ ⟨g, ek ⟩fk ⟩ ∞

= ∑ ⟨f , ek ⟩⟨g, ek ⟩ k=1

= ⟨f , g⟩.

104 | 4 Bases Hence, U ∗ U = I. Since U is onto (see Lemma 4.7.2), it follows that U is unitary. This completes the proof. The following theorem characterizes all orthonormal bases for X, starting with one orthonormal basis. Theorem 4.7.5. Let {ek }∞ k=1 be an orthonormal basis for X. Then the orthonormal bases for X are precisely the sets {Uek }∞ k=1 , where U : X 󳨀→ X is a unitary operator. Proof. Using Lemma 4.7.3, the operator U, given in (4.7.2), is unitary. Then ⟨Uek , Uej ⟩ = ⟨U ∗ Uek , ej ⟩ = ⟨ek , ej ⟩ = δkj ,

i. e., {Uek }∞ k=1 is an orthonormal system. The conclusion that it is a basis follows from the fact that U is onto (see Lemma 4.7.2).

4.8 The Gram matrix ∗ Let {fk }∞ k=1 be a Bessel sequence. We can compose the bounded operators T and T; hereby we obtain the bounded operator

T ∗ T : l2 (ℕ) 󳨀→ l2 (ℕ),

∞

∞

T ∗ T{ck }∞ k=1 = {⟨∑ cl fl , fk ⟩} l=1

k=1

.

2 Letting {ek }∞ k=1 be the canonical orthonormal basis for l (ℕ), the jkth entry in the ma∗ trix representation for T T is

⟨T ∗ Tek , ej ⟩ = ⟨Tek , Tej ⟩ = ⟨fk , fj ⟩. Identifying T ∗ T with its matrix representation, we write T ∗ T = {⟨fk , fj ⟩}j,k=1 . ∞

∞ The matrix {⟨fk , fj ⟩}∞ j,k=1 is called the Gram matrix associated with {fk }k=1 and the above argument shows that it defines a bounded operator on l2 (ℕ) when {fk }∞ k=1 is a Bessel sequence.

4.9 Hilbert–Schmidt operator | 105

4.9 Hilbert–Schmidt operator Definition 4.9.1. A Hilbert–Schmidt operator on a separable Hilbert space X is a bounded operator A on X for which ∞

‖A‖22 := ∑ ‖Aen ‖2 < ∞, n=1

where

{en }∞ 1

is an orthonormal basis of X.

Theorem 4.9.1. Let X be a separable Hilbert space. Then all Hilbert–Schmidt operators on X are compact. Theorem 4.9.2 (N. Dunford and J. T. Schwartz [77, XI, Theorem 6.26]). Let A be a Hilbert–Schmidt operator with non-zero eigenvalues λ1 , λ2 , . . . repeated in accordance with their multiplicities. Then the infinite product ∞

φλ (A) = ∏(1 − i=1

λi λλi )e λ

converges and defines an analytic function for λ ≠ 0. For each fixed λ ≠ 0, φλ (A) is a continuous complex-valued function on the Banach space of all Hilbert–Schmidt operators. Theorem 4.9.3 (N. Dunford and J. T. Schwartz [77, XI, Theorem 6.27]). If λ is in the resolvent set of the Hilbert–Schmidt operator A, then 1 ‖A‖2 󵄩󵄩 −1 󵄩 )}. 󵄩󵄩φλ (A)(λ − A) 󵄩󵄩󵄩 ≤ |λ| exp{ (1 + 2 |λ|2 Corollary 4.9.1 (N. Dunford and J. T. Schwartz [77, XI, Corollary 6.28]). Let A be a quasi-nilpotent (i. e., σ(A) = {0}) Hilbert–Schmidt operator. Then, for every λ ≠ 0, we have ‖A‖2 1 󵄩󵄩 −1 󵄩 )}. 󵄩󵄩(λ − A) 󵄩󵄩󵄩 ≤ |λ| exp{ (1 + 2 |λ|2

4.10 The trace class Definition 4.10.1. Let X be a separable Hilbert space and let {φn }∞ n=1 be an orthonormal basis. Then, for any positive operator A ∈ ℒ(X), we define ∞

tr(A) = ∑ ⟨φn , Aφn ⟩. n=1

The number tr(A) is called the trace of A and is independent of the orthonormal basis chosen. Indeed, given an orthonormal basis {φn }∞ n=1 , we define ∞

trφ (A) = ∑ ⟨φn , Aφn ⟩. n=1

106 | 4 Bases If {ψn }∞ n=1 is another orthonormal basis, then ∞

trφ (A) = ∑ ⟨φn , Aφn ⟩ n=1 ∞

󵄩 1 󵄩2 = ∑ 󵄩󵄩󵄩A 2 φn 󵄩󵄩󵄩 n=1 ∞

∞

1 󵄨 󵄨2 = ∑ ( ∑ 󵄨󵄨󵄨⟨ψm , A 2 φn ⟩󵄨󵄨󵄨 )

n=1 m=1 ∞

∞

󵄨 1 󵄨2 = ∑ ( ∑ 󵄨󵄨󵄨⟨A 2 ψm , φn ⟩󵄨󵄨󵄨 ) m=1 n=1 ∞ 1

󵄩 󵄩2 = ∑ 󵄩󵄩󵄩A 2 ψm 󵄩󵄩󵄩 m=1 ∞

= ∑ ⟨ψm , Aψm ⟩ m=1

= trψ (A). Since all the terms are positive, interchanging the sums is allowed. Theorem 4.10.1. Let X be a separable Hilbert space and let {φn }∞ n=1 be an orthonormal basis. Let A ∈ ℒ(X) be a positive operator. Then (i) tr(A + B) = tr(A) + tr(B); (ii) tr(λA) = λ tr(A) for all λ ≥ 0; (iii) tr(UAU −1 ) = tr(A) for any unitary operator U; and (iv) if 0 ≤ A ≤ B, then tr(A) ≤ tr(B). Proof. The assertions (i), (ii), and (iv) are obvious. To prove (iii) we note that, if {φn }∞ n=1 is an orthonormal basis, then so is {Uφn }∞ . Hence, n=1 tr(UAU −1 ) = trUφ (UAU −1 ) = trφ (A)

= tr(A), which completes the proof.

4.11 Equivalent bases In this section we prove several results related to the invariance of bases under topological isomorphisms. We begin with the easy fact that bases are preserved by topological isomorphisms.

4.11 Equivalent bases | 107

Lemma 4.11.1. Let X, Y be Banach spaces. If (xn )∞ n=1 is a basis for X and T : X 󳨀→ Y is a topological isomorphism, then (Txn )∞ is a basis for Y. n=1 Proof. Let y be any element of Y. Then T −1 y ∈ X, so, since (xn )∞ n=1 is a basis for X, there are unique scalars (cn )∞ such that n=1 ∞

T −1 y = ∑ cn xn . n=1

Since T is continuous, we have y = T(T −1 y) ∞

= ∑ cn Txn . n=1

If ∞

y = ∑ bn Txn n=1

is another representation of y, then, since T −1 is continuous, we have ∞

T −1 y = ∑ bn xn . n=1

Hence, bn = cn ∞ for each n, since (xn )∞ n=1 is a basis for X, so (Txn )n=1 is a basis for Y. This completes the proof. ∞ Definition 4.11.1. Two Schauder bases, {bn }∞ n=1 in V and {cn }n=1 in W, are said to be equivalent if there exist two positive constants c and C, such that, for every integer N and all sequences {αn }∞ n=1 of scalars,

󵄩󵄩 N 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 c󵄩󵄩󵄩 ∑ αn bn 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ αn cn 󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩 ∑ αn bn 󵄩󵄩󵄩 . 󵄩󵄩n=1 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩n=1 󵄩󵄩 󵄩 󵄩V 󵄩󵄩n=1 󵄩W 󵄩 󵄩V The simplest and perhaps the most obvious way to construct new bases from old ones is through an isomorphism of the underlying space. Thus, if {vn }∞ n=1 is a fixed but arbitrary basis for a Banach space X and if the bounded invertible operator T trans∞ ∞ forms {vn }∞ n=1 into {wn }n=1 , that is, if Tvn = wn , for n = 1, 2, 3, . . . , then {wn }n=1 is also ∞ ∞ a basis for X. In fact, it is easy to see that {vn }n=1 and {wn }n=1 are equivalent in the following sense.

108 | 4 Bases ∞ Definition 4.11.2. Let X be a Banach space. Two bases {vn }∞ n=1 and {wn }n=1 of X are called equivalent if ∞

∑ cn vn

n=1

is convergent if and only if ∞

∑ cn wn

n=1

is convergent. Theorem 4.11.1 (Paley–Wiener). Let {φn }∞ n=1 be a basis for a Banach space X and sup∞ pose that {fn }n=1 is a sequence of elements of X such that 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∑ ci (φi − fi )󵄩󵄩󵄩 ≤ λ󵄩󵄩󵄩∑ ci φi 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩 for some constant λ, 0 ≤ λ < 1, and all choices of the scalars c1 , . . . , cn , (n = 1, 2, 3, . . .). ∞ Then {fn }∞ n=1 is a basis for X equivalent to {φn }n=1 . Proof. It follows, by assumption, that the series ∞

∑ cn (φn − fn )

n=1

is convergent whenever the series ∞

∑ cn φn

n=1

is convergent. Define a mapping T : X 󳨀→ X by setting ∞

∞

n=1

n=1

T( ∑ cn φn ) = ∑ cn (φn − fn ). The operator T is clearly linear and bounded and ‖T‖ ≤ λ < 1. Thus, the operator I − T is one-to-one and onto (invertible). Further, (I − T)φn = fn , Now, the result follows.

for every n.

4.11 Equivalent bases | 109

Corollary 4.11.1 (R. M. Young [194, Corollary p. 38]). Let {φn }∞ n=1 be a basis for a Banach space X and let {fn }∞ be the associated sequence of coefficient functionals. If {ψn }∞ n=1 n=1 is a sequence of vectors in X for which ∞

∑ ‖fn ‖‖φn − ψn ‖ < 1,

n=1

∞ then {ψn }∞ n=1 is a basis for X equivalent to {φn }n=1 .

Proof. Let x = ∑ ci φi be an arbitrary finite sum. Then 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∑ ci (φi − ψi )󵄩󵄩󵄩 = 󵄩󵄩󵄩∑ fi (x)(φi − ψi )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≤ ∑ 󵄩󵄩fi (x)(φi − ψi )󵄩󵄩󵄩 ∞

≤ ( ∑ ‖fn ‖‖φn − ψn ‖)‖x‖ n=1

∞ 󵄩󵄩 󵄩󵄩 = ( ∑ ‖fn ‖‖φn − ψn ‖)󵄩󵄩󵄩∑ ci φi 󵄩󵄩󵄩. 󵄩 󵄩 n=1

The result follows from Theorem 4.11.1, since ∞

0 ≤ ∑ ‖fn ‖‖φn − ψn ‖ < 1. n=1

This completes the proof. ∞ ∞ Let (xn )∞ n=1 and (yn )n=1 be equivalent bases for a Hilbert space X and let (fn )n=1 and be their respective biorthogonal sequences. Observe to begin with that a se∞ quence biorthogonal to a basis is also a basis, so that (fn )∞ n=1 and (gn )n=1 are themselves bases for X. Let T be a bounded invertible operator on X such that Txn = yn (n = 1, 2, 3, . . .). Then the adjoint operator T ∗ is also bounded and invertible. We have the following assertion:

(gn )∞ n=1

T ∗ gn = fn

(n = 1, 2, 3, . . .).

Indeed, for fixed n and all values of m, we have ⟨T ∗ gn , xm ⟩ = ⟨gn , Txm ⟩ = ⟨gn , ym ⟩

110 | 4 Bases = δnm

= ⟨fn , xm ⟩. ∞ ∞ Since (xm )∞ m=1 is complete, the assertion follows. This proves that (fn )n=1 and (gn )n=1 are equivalent bases for X, so we have the following theorem.

Theorem 4.11.2 (R. M. Young [194, Theorem 8]). In a Hilbert space equivalent bases have equivalent biorthogonal sequences.

4.12 Riesz basis 4.12.1 Definition In Theorem 4.7.5, we characterized all orthonormal bases in terms of unitary operators acting on a single orthonormal basis. Formally, the definition of a Riesz basis appears by weakening the condition on the operator. Definition 4.12.1. A basis ℛ := {fk }∞ k=1 in a separable Hilbert space X is said to be a Riesz basis for X, if it is equivalent to an orthonormal basis, i. e., fk = Uek

for all k ∈ ℕ∗ ,

where {ek }∞ k=1 is an orthonormal basis for X and U is a bounded invertible (one-to-one and onto) operator on X. Lemma 4.12.1. A Riesz basis is a Bessel sequence. Proof. Assume that {fk }∞ k=1 is a Riesz basis for X. In accordance with the definition of a Riesz basis, we write ∞ {fk }∞ k=1 = {Uek }k=1 ,

where U is a one-to-one and onto bounded operator and {ek }∞ k=1 is an orthonormal basis. Consider f ∈ X. Then we have ∞

∞

k=1

k=1

󵄨 󵄨2 󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨⟨f , Uek ⟩󵄨󵄨󵄨 󵄩 󵄩2 = 󵄩󵄩󵄩U ∗ f 󵄩󵄩󵄩 󵄩 󵄩2 ≤ 󵄩󵄩󵄩U ∗ 󵄩󵄩󵄩 ‖f ‖2 ≤ ‖U‖2 ‖f ‖2 .

(4.12.1)

This proves that a Riesz basis is a Bessel sequence, which completes the proof.

4.12 Riesz basis | 111

4.12.2 Completeness results The results of this section, given by R. M. Young, can be found in [194]. Lemma 4.12.2 (R. M. Young [194, Theorem 9]). If the sequence {φn }∞ n=1 forms a Riesz basis for a separable Hilbert space X, then the sequence {φn }∞ is complete in X and n=1 ∞ possesses a complete biorthogonal sequence {gn }n=1 such that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨φ, φn ⟩󵄨󵄨󵄨 < ∞

n=1

(4.12.2)

and ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨φ, gn ⟩󵄨󵄨󵄨 < ∞

n=1

(4.12.3)

for every φ in X. ∞ Proof. Let {gn }∞ n=1 be the unique sequence in X biorthogonal to {φn }n=1 . By Theo∞ rem 4.11.2, {gn }n=1 is also a Riesz basis for X. Since every vector φ in X has the two biorthogonal expansions ∞

φ = ∑ ⟨φ, gn ⟩φn n=1

and ∞

φ = ∑ ⟨φ, φn ⟩gn , n=1

the result follows immediately from the definition of a Riesz basis. Let {fk }∞ k=1 be a Riesz basis for X. In accordance with the definition of a Riesz basis, we write ∞ {fk }∞ k=1 = {Uek }k=1 ,

where U is a one-to-one and onto bounded operator and {ek }∞ k=1 is an orthonormal basis. Define a new inner product ⟨f , g⟩1 on X by setting ⟨f , g⟩1 := ⟨Uf , Ug⟩.

(4.12.4)

Let ‖ ⋅ ‖1 be the norm generated by this inner product. Then we have ‖f ‖ ≤ ‖f ‖1 ≤ ‖U‖‖f ‖ ‖U −1 ‖

(4.12.5)

for every f in X. It is easy to see a Riesz basis is actually a basis. In fact, one can characterize Riesz bases in terms of bases satisfying extra conditions as follows.

112 | 4 Bases Lemma 4.12.3 (R. M. Young [194, Theorem 9]). If there is an equivalent inner product on a separable Hilbert space X, with respect to which the sequence {φn }∞ n=1 becomes an orthonormal basis for X, then the sequence {φn }∞ is complete in X and there exist n=1 positive constants A and B such that, for an arbitrary positive integer n and arbitrary scalars c1 , c2 , . . . , cn , we have 󵄩󵄩2 󵄩󵄩 n n 󵄩󵄩 󵄩󵄩 󵄩 A ∑ |ci | ≤ 󵄩󵄩∑ ci φi 󵄩󵄩󵄩 ≤ B ∑ |ci |2 . 󵄩󵄩 󵄩󵄩 i=1 i=1 󵄩 󵄩i=1 n

2

(4.12.6)

Proof. Suppose that ⟨f , g⟩1 , given in (4.12.4), is an equivalent inner product on X relative to which the sequence {φn }∞ n=1 forms an orthonormal basis. From the relations m‖f ‖ ≤ ‖f ‖1 ≤ M‖f ‖ for every f , where m and M are positive constants not depending on f , it follows that, for arbitrary scalars c1 , . . . , cn , we have 󵄩󵄩 n 󵄩󵄩2 n 󵄩󵄩 󵄩󵄩 1 n 2 󵄩 󵄩󵄩 ≤ 1 ∑ |ci |2 . |c | ≤ c φ ∑ ∑ 󵄩 i i i 󵄩 󵄩 2 2 󵄩󵄩 󵄩󵄩 M i=1 m i=1 󵄩i=1 󵄩 Clearly, the sequence {φn }∞ n=1 is complete in X. Lemma 4.12.4 (R. M. Young [194, Theorem 9]). If the sequence {φn }∞ n=1 is complete in a separable Hilbert space X and there exist positive constants A and B such that, for an arbitrary positive integer n and arbitrary scalars c1 , c2 , . . . , cn , one has (4.12.6), then the sequence {φn }∞ n=1 forms a Riesz basis for X. Proof. Let {en }∞ n=1 be an arbitrary orthonormal basis for X. It follows, by assumption, that there exist bounded linear operators T and S such that Ten = φn and Sφn = en , (n = 1, 2, 3, . . .). Certainly, ST = I. Since {φn }∞ n=1 is complete, we also have TS = I. Hence, T is invertible and {φn }∞ n=1 is a Riesz basis for X. Lemma 4.12.5 (R. M. Young [194, Theorem 9]). If the sequence {φn }∞ n=1 forms a Riesz basis for a separable Hilbert space X, then the sequence {φn }∞ is complete in X and its n=1 Gram matrix (⟨φi , φj ⟩)i,j=1 ∞

generates a bounded invertible operator on l2 (ℕ).

4.12 Riesz basis | 113

Proof. Let T be a bounded invertible operator on X that carries some orthonormal ba∞ sis {en }∞ n=1 into the basis {φn }n=1 . If A = (aij ) denotes the matrix of the (invertible) op∗ ∞ erator T T relative to {en }n=1 , then aij = ⟨T ∗ Tej , ei ⟩ = ⟨Tej , Tei ⟩ = ⟨φj , φi ⟩.

Therefore, the Gram matrix of {φn }∞ n=1 is the conjugate of A. 4.12.3 Dual basis associated to a Riesz basis The dual basis associated to a Riesz basis is also a Riesz basis. Theorem 4.12.1 (O. Christensen [66, Theorem 3.6.3]). Let {fk }∞ k=1 be a Riesz basis for X. Then there exists a unique sequence {gk }∞ in X such that k=1 ∞

f = ∑ ⟨f , gk ⟩fk , k=1

for all f ∈ X.

(4.12.7)

∞ ∞ Further {gk }∞ k=1 is also a Riesz basis and {fk }k=1 and {gk }k=1 are biorthogonal. Moreover, the series (4.12.7) converges unconditionally for all f ∈ X.

Proof. In accordance with the definition of a Riesz basis, we write ∞ {fk }∞ k=1 = {Uek }k=1 ,

where U is a one-to-one and onto bounded operator and {ek }∞ k=1 is an orthonormal −1 basis. Consider f ∈ X. By expanding U f in the orthonormal basis {ek }∞ k=1 , we have ∞

U −1 f = ∑ ⟨U −1 f , ek ⟩ek k=1 ∞

= ∑ ⟨f , (U −1 )∗ ek ⟩ek . k=1

Therefore, f = UU −1 f ∞

= ∑ ⟨f , (U −1 ) ek ⟩Uek k=1 ∞

= ∑ ⟨f , gk ⟩fk , k=1

∗

114 | 4 Bases where gk := (U −1 )∗ ek . Since (U −1 )∗ is one-to-one and onto bounded, {gk }∞ k=1 is a Riesz basis by definition. Using Lemma 4.12.1, {fk }∞ is a Bessel sequence. Hence, by usk=1 ing Corollary 3.1.2, the series (4.12.7) converges unconditionally. The rest of the proof follows immediately from Theorem 4.6.1. ∞ ∞ ∞ We call {gk }∞ k=1 the dual Riesz basis of {fk }k=1 . Therefore, {fk }k=1 and {gk }k=1 are duals of each other and ∞

∞

k=1

k=1

f = ∑ ⟨f , gk ⟩fk = ∑ ⟨f , fk ⟩gk ,

for all f ∈ X.

4.12.4 Characterization of Riesz basis ∞ Proposition 4.12.1 (O. Christensen [66, Proposition 3.6.4]). If {fk }∞ k=1 = {Uek }k=1 is a Riesz basis for X, then there exist constants A, B > 0 such that ∞

󵄨 󵄨2 A‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 ≤ B‖f ‖2 k=1

The largest possible value for the constant A is 2

B is ‖U‖ .

1 ‖U −1 ‖2

for all f ∈ X. and the smallest possible value for

Proof. By using Lemma 4.12.1, we know every Riesz basis {Uek }∞ k=1 is a Bessel sequence with optimal upper bound ‖U‖, which follows already from the estimate in (4.12.1). The result about the lower bound follows from 󵄩 󵄩 ‖f ‖ = 󵄩󵄩󵄩(U ∗ )−1 U ∗ f 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩(U ∗ )−1 󵄩󵄩󵄩󵄩󵄩󵄩U ∗ f 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 = 󵄩󵄩󵄩U −1 󵄩󵄩󵄩󵄩󵄩󵄩U ∗ f 󵄩󵄩󵄩, which completes the proof. Proposition 4.12.2. In a separable Hilbert space X, the sequence {φn }∞ n=1 is a Riesz basis if and only if it satisfies the following property that ⟨φn , u⟩ obey: ∞

󵄨 󵄨2 A‖u‖2 ≤ ∑ 󵄨󵄨󵄨⟨φn , u⟩󵄨󵄨󵄨 ≤ B‖u‖2 , n=1

(4.12.8)

for strictly positive constants A and B independent of u. The results obtained by R. M. Young, appearing in the remaining part of this section, can be found in [194]. Lemma 4.12.6. If the sequence {φn }∞ n=1 forms a Riesz basis for a separable Hilbert space X, then there is an equivalent inner product on X, with respect to which the sequence {φn }∞ n=1 becomes an orthonormal basis for X.

4.12 Riesz basis | 115

Proof. Since {φn }∞ n=1 is a Riesz basis for X, there exists a bounded invertible operator ∞ U that transforms {φn }∞ n=1 into some orthonormal basis {en }n=1 , i. e., Uφn = en

for n = 1, 2, 3, . . . .

Define a new inner product ⟨f , g⟩1 on X by equation (4.12.4) and let ‖⋅‖1 be the norm generated by this inner product. Then we have equation (4.12.5), so the new inner product is equivalent to the original one. Clearly, ⟨φi , φj ⟩1 = ⟨Uφi , Uφj ⟩ = ⟨ei , ej ⟩ = δij for every i and j, which completes the proof. Lemma 4.12.7. If the sequence {φn }∞ n=1 is complete in a separable Hilbert space X and its Gram matrix (⟨φi , φj ⟩)i,j=1 ∞

generates a bounded invertible operator on l2 (ℕ), then the sequence {φn }∞ n=1 is complete in X and there exist positive constants A and B such that, for an arbitrary positive integer n and arbitrary scalars c1 , c2 , . . . , cn , one has (4.12.6). Proof. Suppose that the Gram matrix of {φn }∞ n=1 generates a bounded invertible operator on l2 (ℕ). Let {en }∞ be an arbitrary orthonormal basis for X. Consider the transn=1 formation T : X 󳨀→ X, defined by ∞

∞

∞

i=1

i=1

j=1

T(∑ ci ei ) = ∑ ei (∑ ⟨φi , φj ⟩cj ) 2 whenever {ci }∞ i=1 ∈l (ℕ). It is easy to see that T is linear, bounded, and invertible. A simple calculation shows that

󵄩󵄩 ∞ 󵄩󵄩2 ∞ ∞ 󵄩󵄩 󵄩 󵄩󵄩∑ ci φi 󵄩󵄩󵄩 = ⟨T(∑ ci ei ), ∑ ci ei ⟩. 󵄩󵄩 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩 i=1 i=1

(4.12.9)

Since T is a positive operator, it has a (unique) positive square root (see F. Riesz and B. Sz.-Nagy [173, p. 2651]). We call this √T. Equation (4.12.9) may then be put in the following form: 󵄩󵄩 ∞ 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 ∞ 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∑ ci φi 󵄩󵄩󵄩 = 󵄩󵄩󵄩√T(∑ ci ei )󵄩󵄩󵄩 , 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1

116 | 4 Bases from which it follows at once that 2

󵄩󵄩 󵄩󵄩 ∞ ∞ 󵄩󵄩 1 ∞ 󵄩󵄩󵄩 2 󵄩󵄩 ≤ ‖T‖ ∑ |ci |2 , c φ |c | ≤ ∑ ∑ 󵄩 i i i 󵄩 󵄩 󵄩󵄩 󵄩󵄩 ‖T −1 ‖ i=1 i=1 󵄩 󵄩i=1 which completes the proof. Lemma 4.12.8. If the sequence {φn }∞ n=1 is complete in a separable Hilbert space X and possesses a complete biorthogonal sequence {gn }∞ n=1 such that (4.12.2) and (4.12.3) hold ∞ for every φ in X, then the sequence {φn }n=1 forms a Riesz basis for X. Proof. Consider the linear transformation from X into l2 (ℕ) defined by φ 󳨀→ {⟨φ, φn ⟩}. The reader will verify without great difficulty that this mapping is closed. By the closed graph theorem it is continuous and hence there exists a positive constant C for which ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨φ, φn ⟩󵄨󵄨󵄨 ≤ C 2 ‖φ‖2

n=1

for all φ.

(4.12.10)

Similarly, there exists a positive constant D for which ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨φ, gn ⟩󵄨󵄨󵄨 ≤ D2 ‖φ‖2

n=1

for all φ.

(4.12.11)

Fix an arbitrary orthonormal basis {en }∞ n=1 for X and define operators S and T on the ∞ linear subspaces spanned by the sequences {φn }∞ n=1 and {gn }n=1 , respectively, by setting ∞

∞

i=1

i=1

∞

∞

i=1

i=1

S(∑ ci φi ) = ∑ ci ei and T(∑ ci gi ) = ∑ ci ei . By virtue of the two inequalities (4.12.10) and (4.12.11), we have 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩S(∑ ci φi )󵄩󵄩󵄩 ≤ D󵄩󵄩󵄩∑ ci φi 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩 and 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩T(∑ ci gi )󵄩󵄩󵄩 ≤ C 󵄩󵄩󵄩∑ ci gi 󵄩󵄩󵄩. 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 i=1 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩

4.12 Riesz basis | 117

∞ Since both sequences {φn }∞ n=1 and {gn }n=1 are complete, each of the operators S and T can be extended by continuity to a bounded linear operator on the entire space. Let

φ = ∑ ai φi and g = ∑ bi gi be finite sums. A straightforward calculation shows that ⟨Sφ, Tg⟩ = ⟨φ, g⟩. By continuity, this holds for every pair of vectors φ and g. Hence, ⟨φ, S∗ Tg⟩ = ⟨φ, g⟩, so S∗ T = I. The existence of a right-inverse for S∗ implies that S∗ is onto and hence that S is bounded from below (see A. E. Taylor [184, p. 2341]). Since the range of S is dense in X, we conclude that S is invertible. Thus, the sequence {φn }∞ n=1 forms a Riesz basis for X. This completes the proof. The following equivalent characterizations of Riesz bases can be found in the book of R. M. Young [194]. Theorem 4.12.2 (R. M. Young [194, Theorem 9, p. 32]). Let X be a separable Hilbert space. Then the following statements are equivalent: (i) The sequence {φn }∞ n=1 forms a Riesz basis for X. (ii) There is an equivalent inner product on X, with respect to which the sequence {φn }∞ n=1 becomes an orthonormal basis for X. (iii) The sequence {φn }∞ n=1 is complete in X and there exist positive constants A and B such that, for an arbitrary positive integer n and arbitrary scalars c1 , c2 , . . . , cn , one has (4.12.6). (iv) The sequence {φn }∞ n=1 is complete in X and its Gram matrix (⟨φi , φj ⟩)i,j=1 ∞

generates a bounded invertible operator on l2 (ℕ). (v) The sequence {φn }∞ n=1 is complete in X and possesses a complete biorthogonal se∞ quence {gn }n=1 such that (4.12.2) and (4.12.3) hold for every φ in X. Proof. The results follow from Lemmas 4.12.2–4.12.8.

118 | 4 Bases 4.12.5 Collection of vectors near a Riesz basis Let {φn }∞ n=1 be a Riesz basis on X. There is a bounded invertible operator G and an orthonormal basis {en }∞ n=1 of X such that φn = Gen

for all n.

Consider the operators ∞

Su = ∑ ⟨φn , u⟩ϕn ,

(4.12.12)

Vu = ∑ ⟨φn , u⟩φn ,

(4.12.13)

n=1 ∞

n=1

and ∞

Wu = ∑ ⟨ϕn , u⟩φn ,

(4.12.14)

n=1

where {ϕn }∞ n=1 is a sequence of vectors of X. The space N(S) is trivial if and only if the 2 ϕn are linearly independent since any sequence (cn )∞ n=1 ∈ l (ℕ) is the set of Fourier coefficients of some vector u ∈ X. The space N(W) is trivial if and only if the ϕn are a basis since any vector orthogonal to all the ϕn lies in N(W) and the reverse inclusion follows from the linear independence of the vectors φn . Moreover, V is self-adjoint. Since the (φn )∞ n=1 form a Riesz basis, σ(V) ⊂ [A, B], where A and B are given in (4.12.8). Since A > 0, V −1 exists and we have 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 −1 󵄩󵄩 1 󵄩󵄩󵄩 󵄩 ⟨φ , u⟩(ϕ − φ ) V (S − V)u = ∑ 󵄩 󵄩󵄩 󵄩󵄩 n n 󵄩 󵄩󵄩 A 󵄩󵄩󵄩󵄩n=1 n 󵄩󵄩 1 2

+∞ 1 ∞󵄨 󵄨 ≤ ( ∑ 󵄨󵄨󵄨⟨φn , u⟩󵄨󵄨󵄨) ( ∑ ‖ϕn − φn ‖2 ) A n=1 n=1 1

1 2

1 2

B 2 +∞ < ( ∑ ‖ϕ − φn ‖2 ) ‖u‖ A n=1 n < ‖u‖

(4.12.15)

and, similarly, 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 −1 󵄩󵄩 1 󵄩󵄩󵄩 󵄩 V (W − V)u ≤ ⟨ϕ − φ , u⟩φ ∑ 󵄩󵄩 󵄩󵄩 󵄩 n n󵄩 󵄩󵄩 A 󵄩󵄩󵄩󵄩n=1 n 󵄩󵄩 < ‖u‖.

(4.12.16)

4.12 Riesz basis | 119

Theorem 4.12.3 (A. Schueller [181, Lemma 1.1]). Let (φn )∞ n=1 be a Riesz basis of a separable Hilbert space X. Let (ϕn )∞ be a collection of vectors in X such that n=1 +∞

∑ ‖ϕn − φn ‖2
0, V −1 exists and 󵄩󵄩 −1 󵄩󵄩 1 󵄩󵄩V 󵄩󵄩 < . A It suffices to show that 󵄩󵄩 −1 󵄩 󵄩󵄩V (S − V)󵄩󵄩󵄩 < 1 and 󵄩󵄩 −1 󵄩 󵄩󵄩V (W − V)󵄩󵄩󵄩 < 1. Using equations (4.12.15) and (4.12.16), we have 󵄩󵄩 −1 󵄩 󵄩󵄩V (S − V)u󵄩󵄩󵄩 < ‖u‖ and 󵄩󵄩 −1 󵄩 󵄩󵄩V (W − V)u󵄩󵄩󵄩 < ‖u‖. Thus, W −1 and S−1 both exist and are bounded, so (ϕn )∞ n=1 is a Riesz basis of X. 4.12.6 Sequence quadratically close ∞ Definition 4.12.2. Two sequences of vectors {fn }∞ n=1 and {gn }n=1 in a normed vector space are said to be quadratically close if ∞

∑ ‖fn − gn ‖2 < ∞.

n=1

120 | 4 Bases Let X be a separable Hilbert space, let {en }∞ n=1 be an orthonormal basis for X, and let {fn }∞ be an ω-independent sequence that is quadratically close to {en }∞ n=1 n=1 . Define an operator T : X 󳨀→ X by setting ∞

Tf = ∑ ⟨f , en ⟩(en − fn ). n=1

(4.12.17)

Lemma 4.12.9. Let X be a separable Hilbert space and let {en }∞ n=1 be an orthonormal ba∞ sis for X. If {fn }∞ is quadratically close to {e } , then the operator T, given in (4.12.17), n n=1 n=1 is compact. Proof. It is clear that T is linear and that ∞

‖T‖2 ≤ ∑ ‖en − fn ‖2 . n=1

Furthermore, since Ten = en − fn , it follows that ∞

∞

n=1

n=1

∑ ‖Ten ‖2 ≤ ∑ ‖en − fn ‖2 < ∞.

This shows that T is a Hilbert–Schmidt operator and hence compact (see Theorem 4.9.1). This completes the proof. Theorem 4.12.4 (Bari). Let X be a separable Hilbert space and let {en }∞ n=1 be an orthonormal basis for X. If {fn }∞ is an ω-independent sequence that is quadratically n=1 ∞ close to {en }∞ n=1 , then {fn }n=1 is a Riesz basis for X. Proof. Let T : X 󳨀→ X be the operator defined in (4.12.17). Using Lemma 4.12.9, we infer that T is compact. We complete the proof by showing that N(I − T) = {0}. If (I − T)f = 0, then, from the equations ∞

∞

n=1 ∞

n=1

0 = (I − T)f = ∑ ⟨f , en ⟩en − ∑ ⟨f , en ⟩(en − fn ) = ∑ ⟨f , en ⟩fn n=1

and the fact that

{fn }∞ n=1

is ω-independent, it follows that f = 0. Hence, N(I − T) = {0}

and the Fredholm alternative shows that I − T is invertible. Clearly, (I − T)en = fn for every n. This completes the proof.

4.12 Riesz basis | 121

4.12.7 Sum direct of subspaces The results obtained by W. S. Tang, appearing in the remaining part of this section, can be found in [183]. Theorem 4.12.5. Let V and W be closed linear subspaces of X. The following conditions are equivalent: (i) W ⊕ V ⊥ = X; and ∞ (ii) there exist Riesz bases {vn }∞ n=1 and {wn }n=1 for V and W, respectively, such that ∞ ∞ {vn }n=1 is biorthogonal to {wn }n=1 . ∞ Proof. (ii) 󳨐⇒ (i) Let {vn }∞ n=1 and {wn }n=1 be two Riesz bases for V and W, respectively, such that

⟨vn , wk ⟩ = δnk ,

n, k ∈ ℕ∗ .

(4.12.18)

Then ∞

∞

n=1

n=1

∞

∞

n=1

n=1

V = {f ∈ X such that f = ∑ an vn , ∑ |an |2 < ∞} and W = {g ∈ X such that g = ∑ bn wn , ∑ |bn |2 < ∞}.

(4.12.19)

∞ Let f ∈ X. Then, by the Riesz basis property of {vn }∞ n=1 and {wn }n=1 , we know

πW f = ∑⟨f , vn ⟩wn is a well-defined vector in W. By using both (4.12.18) and (4.12.20), we get ⟨f − πW f , vk ⟩ = 0,

k ∈ ℕ∗ .

Hence, f − πW f ∈ V ⊥ , so f = πW f + (f − πW f ) ∈ W + V ⊥ . Therefore, W + V ⊥ = X. Let g ∈ W ∩ V ⊥ . By using both (4.12.18) and (4.12.19), we have g = ∑⟨g, vn ⟩wn = 0.

(4.12.20)

122 | 4 Bases Hence, W ∩ V ⊥ = {0}, so W ⊕ V ⊥ = X. ∞ (i) 󳨐⇒ (ii) Suppose that W ⊕ V ⊥ = X. Let {vn }∞ n=1 and {un }n=1 be dual Riesz bases ∞ ∞ for V, i. e., {vn }n=1 and {un }n=1 are both Riesz bases for V and

⟨vn , uk ⟩ = δnk ,

for all n, k.

Recall that, if F : V 󳨀→ V is the frame operator defined by F(v) = ∑⟨v, vn ⟩vn ,

v ∈ V,

then un = F −1 (vn ),

for all n.

Consider PV the orthogonal projection of X on V by PV : X 󳨀→ X defined by ∞

PV f := ∑ ⟨f , vn ⟩un , n=1

f ∈ X.

Let G := PV |W be the restriction of PV to W. If f ∈ W ∩ N(G), then f ∈ W ∩ V ⊥ = {0}, so G is one-to-one and G(W) = PV (W)

= PV (W + V ⊥ ) = PV (X) = V.

Thus, G is an invertible bounded operator from W onto V. Let wn = G−1 (un ),

for all n.

Then {wn }∞ n=1 is a Riesz basis of W and, for every n, un = G(wn ) ∞

= ∑ ⟨wn , vk ⟩uk . k=1

Hence, ⟨wn , vk ⟩ = δnk Therefore, (i) if and only if (ii).

for all n, k.

4.12 Riesz basis | 123

Corollary 4.12.1. Let V and W be closed linear subspaces of X. The following conditions are equivalent: (i) V ⊕ W ⊥ = X; (ii) W ⊕ V ⊥ = X; and ∞ (iii) there exist Riesz bases {vn }∞ n=1 and {wn }n=1 for V and W, respectively, such that ∞ ∞ {vn }n=1 is biorthogonal to {wn }n=1 . Proof. Interchanging the roles of V and W in Theorem 4.12.5, we also have (i) if and only if (iii). The result follows from Theorem 4.12.5. Let us first state a probably folk result on the vector sum of two closed linear subspaces of a Hilbert space. We leave its proof to the reader. Theorem 4.12.6. Let V and W be closed linear subspaces of a complex Hilbert space X. The following conditions are equivalent: (i) sup{|⟨v, w⟩| such that v ∈ V, w ∈ W, and ‖v‖ = ‖w‖ = 1} < 1; (ii) there exists a positive constant C such that ‖v + w‖2 ≥ C(‖v‖2 + ‖w‖2 ),

v ∈ V, w ∈ W;

and (iii) V + W is closed in X and V ∩ W = {0}. ̃ and W ̃ are Riesz bases for V and W, respectively, then Moreover, if V ∩ W = {0} and V (i), (ii), and (iii) are each equivalent to: ̃∪W ̃ is a Riesz basis for V + W. (iv) V Corollary 4.12.2. Let U1 and U2 be closed linear subspaces of X and U1 ∩ U2 = {0}. If U1 + U2 is closed, then V1 + V2 is closed for any closed linear subspace Vi of Ui , i = 1, 2. 4.12.8 Concept of the angle between two closed linear subspaces The expression in condition (i) of Theorem 4.12.6 is closely related to the concept of the angle θ(V, W) (0 ≤ θ(V, W) ≤ π2 ) between two closed linear subspaces V and W of a Hilbert space X, which is defined by cos(θ(V, W)) :=

inf

v∈V, ‖v‖=1

‖πW v‖,

where πW (⋅) is the orthogonal projection of X on W. It is easy to see that cos2 (θ(V, W)) = 1 −

v∈V,

sup

w∈W ⊥ ,

‖v‖=‖w‖=1

󵄨󵄨 󵄨2 󵄨󵄨⟨v, w⟩󵄨󵄨󵄨 .

(4.12.21)

124 | 4 Bases Note that, in general, θ(V, W) and θ(W, V) are not necessarily equal, but by (4.12.21) we always have θ(V, W) = θ(W ⊥ , V ⊥ ). If V ⊕ W ⊥ = X, then θ(V, W) = θ(W, V). The results obtained by W. S. Tang, appearing in the remaining part of this section, can be found in [183]. Theorem 4.12.7. Let V and W be closed linear subspaces of X. The following conditions are equivalent: (i) V ⊕ W ⊥ = X; and (ii) cos(θ(V, W)) > 0 and cos(θ(W, V)) > 0. Proof. (i) 󳨐⇒ (ii) Suppose that (i) holds. Since V + W ⊥ is closed and V ∩ W ⊥ = {0}, by Theorem 4.12.6, 󵄨 󵄨 sup{󵄨󵄨󵄨⟨v, w⟩󵄨󵄨󵄨 such that v ∈ V, w ∈ W ⊥ , ‖v‖ = ‖w‖ = 1} < 1. Hence, by (4.12.21), cos(θ(V, W)) > 0. Since (ii) of Corollary 4.12.1 also holds, interchanging the roles of V and W in the above argument, cos(θ(W, V)) > 0 holds as well. (ii) 󳨐⇒ (i) Suppose that (ii) holds. Since cos(θ(V, W)) > 0, 󵄨 󵄨 sup{󵄨󵄨󵄨⟨v, w⟩󵄨󵄨󵄨 such that v ∈ V, w ∈ W ⊥ , ‖v‖ = ‖w‖ = 1} < 1. In view of Theorem 4.12.6, V + W ⊥ is closed and V ∩ W ⊥ = {0}. Since cos(θ(W, V)) > 0 as well, by the above argument, W + V ⊥ is closed and W ∩ V ⊥ = {0}. Hence, V + W ⊥ = (V + W ⊥ )

⊥⊥

= (V ⊥ ∩ W) = X. ⊥

As a consequence of Theorems 4.12.5 and 4.12.7 and Corollary 4.12.1, we have the following. Corollary 4.12.3. Let V and W be closed linear subspaces of X. The following conditions are equivalent: (i) V ⊕ W ⊥ = X; (ii) W ⊕ V ⊥ = X; (iii) there exist Riesz bases {vn } and {wn } for V and W, respectively, such that {vn } is biorthogonal to {wn }; and (iv) cos(θ(V, W)) > 0 and cos(θ(W, V)) > 0.

4.12 Riesz basis | 125

4.12.9 Paley–Wiener criterion Theorem 4.12.8 (Paley–Wiener). Let {en }∞ n=1 be an orthonormal basis for a Hilbert ∞ space X and let {fn }∞ n=1 be “close” to {en }n=1 in the sense that 󵄩󵄩 󵄩󵄩 ∞ ∞ 󵄩 󵄩󵄩 󵄩󵄩∑ ci (ei − fi )󵄩󵄩󵄩 ≤ λ√∑ |ci |2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩i=1 i=1 for some constant λ, 0 ≤ λ < 1, and arbitrary scalars c1 , . . . , cn (n = 1, 2, 3, . . .). Then {fn }∞ n=1 is a Riesz basis for X. Theorem 4.12.9 (Kadec’s 41 -theorem, R. M. Young [194, Theorem 14]). If {λn } is a sequence of real numbers for which |λn − n| ≤ L
0 be numbers such that the angles Ωk = {λ ∈ ℂ such that |φ − αk | < θk }

130 | 4 Bases (k = 1, . . . , n) are disjoint. By Lemma 3.4.3, the assertion of Lemma 3.4.2 is valid for each of these angles. This means that the eigenvalues of T in Ωk (except perhaps for a finite number) can be broken up into finite groups Λ kj , j = 1, 2, . . . in the increasing order of Re(λe−iαk ) in such way that the Riesz projections Pkj corresponding to these groups satisfy the condition ∞

󵄨 󵄨 ∑ 󵄨󵄨󵄨⟨Pkj f , g⟩󵄨󵄨󵄨 < ∞

(4.14.2)

j=1

for any f , g ∈ X. The set n ∞

σ(T)\ ⋃ ⋃ Λ kj k=1 j=1

is finite in view of Corollary 2.18.1. If it contains s (≥ 0) numbers and {Pi }si=1 are the corresponding Riesz projections, then it follows from (4.14.2) that, for any f , g ∈ X, s

n ∞

i=1

k=1 j=1

󵄨 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨⟨Pi f , g⟩󵄨󵄨󵄨 ∑ ∑ 󵄨󵄨󵄨⟨Pkj f , g⟩󵄨󵄨󵄨 < ∞. By Theorem 3.10.2, the system of subspaces s

{R(Pkj )}k=1,...,n; j=1,2,... {R(Pi )}i=1 is complete in X and in view of Lemma 4.14.2 it forms an unconditional basis for X. The theorem is proved.

4.15 Riesz basis of finite-dimensional A-invariant subspaces Definition 4.15.1. A subspace U ⊂ X is called A-invariant if x ∈U ∩ 𝒟(A) implies Ax ∈U. If we speak of a finite-dimensional A-invariant subspace U, we additionally assume that dim U < ∞ and U ⊂ 𝒟(A). Let us now recall the following result due to C. Wyss [189, Theorem 6.1]. Theorem 4.15.1 (C. Wyss [189, Theorem 6.1]). Let G be a normal operator with compact resolvent whose spectrum lies on a finite number of rays from the origin. Let S be p-subordinate to G with 0 ≤ p < 1. If lim inf r→∞

n(𝔹(0, r), G) < ∞, r 1−p

then T = G + S admits a Riesz basis of finite-dimensional T-invariant subspaces. Let us now recall the following result due to S. Charfi, A. Damargi, and A. Jeribi [59, Theorem 1.2].

4.15 Riesz basis of finite-dimensional A-invariant subspaces |

131

Theorem 4.15.2 (S. Charfi, A. Damargi, and A. Jeribi [59, Theorem 1.2]). Let G be a normal operator on a Hilbert space X, having compact resolvent. Assume that the spectrum of G lies on a finite number of rays from the origin and, for some p ∈ [0, 1[, lim inf r→∞

n(𝔹(0, r), G) < ∞. r 1−p

If the operator S := A − G is G-bounded with order p, then A admits a Riesz basis of finite-dimensional A-invariant subspaces. Proof. To prove this theorem, we distinguish the following two cases: – N(G) = {0}. According to the first item of Lemma 3.4.1, S is G-subordinate with order p. Hence, as a consequence of Theorem 4.15.1, we conclude that T admits a Riesz basis of finite-dimensional, T-invariant subspaces. – N(G) ≠ {0}. In this case, we consider the orthogonal projection onto N(G) denoted by P and we let G0 = G + P. It follows from the second item of Lemma 3.4.1 that S − P is p-subordinate to G0 . Hence, Theorem 4.15.1 guarantees that T admits a Riesz basis of finite-dimensional, T-invariant subspaces. This achieves the proof.

5 Semi-groups 5.1 C0 -semi-group 5.1.1 Definitions Definition 5.1.1. Let X be a Banach space. A strongly continuous semi-group is an operator-valued function T(t) from ℝ+ into ℒ(X) that satisfies the following properties: T(t + s) = T(t)T(s)

for t, s ≥ 0,

T(0) = I, and 󵄩 󵄩󵄩 + 󵄩󵄩T(t)z0 − z0 󵄩󵄩󵄩 → 0 as t → 0

for all z0 ∈ X.

We shall subsequently use the standard abbreviation C0 -semi-group for a strongly continuous semi-group. Definition 5.1.2. The infinitesimal generator A of a C0 -semi-group on a Banach space X is defined by 1 Az = lim+ (T(t) − I)z, t→0 t whenever the limit exists, the domain of A, 𝒟(A), being the set of elements in X for which the limit exists.

5.1.2 Hille–Yosida theorem We recall the Hille–Yosida theorem. Theorem 5.1.1 (Hille–Yosida theorem). A necessary and sufficient condition for a closed, densely defined, linear operator A on a Hilbert space X to be the infinitesimal generator of a C0 -semi-group is that there exist real numbers M, w such that, for all real α > w, α ∈ ρ(A), we have M 󵄩󵄩 −r 󵄩 󵄩󵄩(α − A) 󵄩󵄩󵄩 ≤ (α − w)r for all r ≥ 1. In this case, 󵄩󵄩 󵄩 wt 󵄩󵄩T(t)󵄩󵄩󵄩 ≤ Me . https://doi.org/10.1515/9783110493863-005

134 | 5 Semi-groups 5.1.3 Dissipative operator Let X be a Hilbert space. An operator −A is said to be dissipative if Re⟨Au, u⟩ ≥ 0 for every u ∈ 𝒟(A) and −A is maximal dissipative if it has no proper dissipative extension. It is known that a closed, maximal dissipative operator is densely defined and that −A is closed and maximal dissipative if and only if −A∗ is and, also, if and only if −A is the infinitesimal generator of a contraction semi-group {exp(−tA)}0 0, { dt {u(0) = ϕ0 ∈ 𝒟(A)

(5.2.1)

and 2

{ d v (t) = Av(t), for t > 0, { dt 2 {v(0) = φ ∈ 𝒟(A).

(5.2.2)

If A generates a C0 -semi-group, then the solution u(t) (respectively v(t)) of (5.2.1) (respectively (5.2.2)) can be written as u(t) = e−tA ϕ0

1 2

(respectively v(t) = e−tA φ).

5.2.1 Diagonalization of the semi-group e−tA Our concern with the problem (5.2.1) is the series expansion of the solution u(t) in terms of the generalized eigenvectors of the operator A for every t > 0 or at least for t large enough. For this reason, V. B. Lidskii [153] and G. Geymonat and P. Grisvard [92] established that the solution u(t) of the problem (5.2.1) can be expanded into a series of the generalized eigenvectors of A.

136 | 5 Semi-groups Theorem 5.2.1 (V. B. Lidskii [153]). Let A be a closed, densely defined, and linear operator on a Hilbert space X satisfying: (i) A generates an analytic semi-group and A−1 is compact; and (ii) there exists p < 1 such that A−1 ∈ Cp . Then the solution u(t) of the problem (5.2.1) can be expanded in X into the following series: ∞

∞

k=1

k=1

u(t) = ∑ Pk e−tA ϕ0 = ∑ e−tA Pk ϕ0

for every t > 0,

where Pk is the projector operator corresponding to λk (an eigenvalue of A) on X and given by Pk =

1 ∫ (A − λ)−1 dλ 2πi Ck

and Ck is a positively oriented circle with center λk (an eigenvalue of A) such that no other eigenvalue lies inside or on the circle. Theorem 5.2.2 (G. Geymonat and P. Grisvard [92]). Let A be a closed, densely defined, and linear operator on a Hilbert space X satisfying the following: (i) there exists δ > 0 such that the resolvent set ρ(A) of A includes in the double sector 󵄨󵄨 π 󵄨󵄨󵄨 󵄨 S = {λ ∈ ℂ such that 󵄨󵄨󵄨arg λ ± 󵄨󵄨󵄨 ≤ δ} 󵄨󵄨 2 󵄨󵄨 and that there exists an integer n ≥ −1 such that n 󵄩󵄩 −1 󵄩 󵄩󵄩(A − λ) 󵄩󵄩󵄩 ≤ C(1 + |λ|)

for every λ ∈ S; (ii) A−1 ∈ Cp for every p > 1; and (iii) there exists c such that d(λ) = O(ec|λ| ) and there exists a sequence of real numbers rj → ∞ and numbers a > 0 and b ≥ 0 such that 󵄨󵄨 󵄨 −br 󵄨󵄨d(λ)󵄨󵄨󵄨 ≥ ae j for Re λ = −rj , where d(λ) is the Fredholm determinant given in Definition 3.9.1.

5.2 Diagonalization of semi-groups | 137

Then there exists t0 > 0 such that the solution u(t) of the problem (5.2.1) can be expanded in X into the series ∞

∞

k=1

k=1

u(t) = ∑ Pk e−tA ϕ0 = ∑ e−tA Pk ϕ0

for every t > t0 ,

where Pk is the projector operator corresponding to λk (an eigenvalue of A) on X and given by Pk =

1 ∫ (A − λ)−1 dλ 2πi Ck

and Ck is a positively oriented circle with center λk (an eigenvalue of A) such that no other eigenvalue lies inside or on the circle.

5.2.2 Diagonalization of the semi-group e−tA

1 2

Our concern with the problem (5.2.2) is the series expansion of the solution u(t) in terms of the generalized eigenvectors of the operator A for every t > 0, or at least for t large enough. For this reason, we will put some conditions on the resolvent of the operator A that allow us to use the previous theorems of V. B. Lidskii (Theorem 5.2.1) or that of G. Geymonat and P. Grisvard (Theorem 5.2.2) to establish that the solution v(t) of the problem (5.2.2) can be expanded into a series of the generalized eigenvectors of A. Theorem 5.2.3 (A. Intissar [112, Theorem 3.1]). Let A be a closed, densely defined, and linear operator on a Hilbert space X satisfying the following: (i) A − λ is invertible for every λ ≤ 0 and there exists a constant M > 0 such that M 󵄩󵄩 −1 󵄩 󵄩󵄩(A − λ) 󵄩󵄩󵄩 ≤ |λ| + 1 for every λ ≤ 0; and (ii) the operator A−1 is in the Carleman class Cp for some 0 < p < 21 . Then the solution v(t) of the elliptic problem of the Dirichlet type (5.2.2) can be expanded in X into the following series: ∞

1 2

∞

1 2

v(t) = ∑ Pk e−tA φ = ∑ e−tA Pk φ k=1

k=1

for every t > 0,

where Pk is the projector operator corresponding to λk (an eigenvalue of A) on X and given by Pk =

1 ∫ (A − λ)−1 dλ 2πi Ck

138 | 5 Semi-groups and Ck is a positively oriented circle with center λk (an eigenvalue of A) such that no other eigenvalue lies inside or on the circle. Proof. Under the assumption (i), it is well known that, for every α such that 0 < α < 1, one can define the negative fractional power of the operator A by the following formula: ∞

A

−α

sin(απ) = ∫ λ−α (A + λ)−1 dλ. π 0

1

Hence, the square root of A exists. We only need to prove that A− 2 is in the desired Carleman class. This is the crucial point of the proof. In fact, by applying Theorem 3.7.1 1 for A−1 and by using assumption (ii), we infer that A− 2 belongs to the Carleman class C2p with 0 < 2p < 1. Hence, the result follows from the fact that −√A generates an analytic semi-group, together with Lidskii’s theorem (Theorem 5.2.1). In the following, we will handle the case where A−1 belongs to the Carleman class Cp for every p > 21 . For this purpose, we will use Von Koch’s approach on the Fredholm determinants [187, 188]. In fact, let T ∈ C2 , i. e., T is a Hilbert–Schmidt operator, and let (fj )j be an arbitrary orthonormal basis of the Hilbert space X. Consider the matrices (⟨Tfj , fk ⟩)j,k=1,...,n for n = 1, 2, . . ., associated with the operator T, and let us define the Von Koch determinant of I − T by n

det(I − T) = lim det(δjk − ⟨Tfj , fk ⟩)j,k=1,...,n exp(∑⟨Tfj , fj ⟩), n→∞

j=1

(5.2.3)

where δjk = 1 if j = k and 0 if j ≠ k. Theorem 5.2.4 (A. Intissar [112, Theorem 3.2]). Let A be a densely defined closed linear operator with a non-empty resolvent set ρ(A) on a Hilbert space X. We assume that the following conditions hold: (i) the inverse of A belongs to the Carleman class Cp for every p > 21 ; (ii) A − λ is invertible for every λ ≤ 0 and there exists a constant M > 0 such that M 󵄩󵄩 −1 󵄩 󵄩󵄩(A − λ) 󵄩󵄩󵄩 ≤ |λ| + 1 for every λ ≤ 0; and (iii) there exists a constant C such that 1 2

d(λ) = O(eC|λ| ), where d(λ) is the Fredholm determinant given in Definition 3.9.1.

5.2 Diagonalization of semi-groups | 139

Then there exists t0 > 0 such that the solution v(t) of the elliptic problem of the Dirichlet type (5.2.2) can be expanded in X into the series 1 2

∞

1 2

∞

v(t) = ∑ Pk e−tA φ = ∑ e−tA Pk φ k=1

k=1

for every t > t0 ,

where Pk is the projector operator corresponding to λk (an eigenvalue of A) on X and given by Pk =

1 ∫ (A − λ)−1 dλ 2πi Ck

and Ck is a positively oriented circle with center λk (an eigenvalue of A) such that no other eigenvalue lies inside or on the circle. Proof. Since the inverse K = A−1 belongs to the Carleman class Cp for every p > 21 , in accordance with Theorem 3.7.1, we have √K ∈ C1+ε for every ε > 0 and, in particular, √K ∈ C2 . We consider the operator n

√Kn = ∑⟨⋅, fj ⟩√Kfj j=1

(n = 1, 2, . . .), where (fj )∞ j=1 is an arbitrary orthonormal basis of the Hilbert space X. Then the sequence of operators √Kn converges uniformly to the operator √K in the C2 -norm. By using equation (5.2.3), we have p

det(I − √Kn ) = lim det(δjk − ⟨√Kn fj , fk ⟩)j,k=1,...,p exp(∑⟨√Kn fj , fj ⟩). p→∞

j=1

Then, by passing to the limit as n → ∞, we obtain lim det(I − √Kn ) = det(I − √K).

n→∞

Let Pn be the orthogonal projection onto the linear hull of the vectors (fj )j (1 ≤ j ≤ n). We have √Kn = Pn √KPn and, consequently, det(I − √Kn ) det(I + √Kn ) = det[(I − √Kn )(I + √Kn )] exp[tr(√Kn − √Kn )]

140 | 5 Semi-groups = det[(I − √Kn )(I + √Kn )]

= det(I − Kn ), where Kn = Pn KPn . Consequently,

det(I − √K) det(I + √K) = det(I − K) and det(I − λ√K) det(I + λ√K) = det(I − λ2 K) = d(λ2 ).

(5.2.4)

Hence, the result follows from equation (5.2.4), assumption (iii), and the fact that √A satisfies the conditions of the Geymonat–Grisvard theorem (Theorem 5.2.2).

5.2.3 Riesz basis formed by eigenvectors of A In this section, we consider on a Hilbert space X the linear system (5.2.1), where A is an unbounded linear operator on X and ϕ0 ∈ X. If the operator A generates a C0 -semigroup T(t) on X, then the solution of (5.2.1) can be written in L2 ([0, T], X) as ϕ(t) := T(t)ϕ0 . In this section, we show that, under some conditions on the eigenvalues and eigenvectors of A, the eigenvectors of the system (5.2.1) form a Riesz basis in Hilbert space X. Theorem 5.2.5. Let A be a linear unbounded densely defined operator on a separable Hilbert space X satisfying: (i) A generates a C0 -semi-group; (ii) the resolvent of A is compact; (iii) the eigenvalues {λn }∞ n=1 of A are simple; 2 (iv) the family {eλn t }∞ n=1 forms a Riesz basis of L (0, T), T > 0; and (v) the system of eigenvectors of A is complete in X. Then the eigenvectors associated with the operator A form a Riesz basis in X. Proof. Since the resolvent of A is compact, A has a discrete spectrum, that is, σ(A) = σp (A) = {λn , n ≥ 1}. Let sp(A) be the {T(t)}-invariant spectral subspace of X defined by m

sp(A) := { ∑ E(λk , A)ϕ, ∀ϕ ∈ X, ∀m ∈ ℕ∗ }, k=1

5.2 Diagonalization of semi-groups | 141

where E(λk , A) is the Riesz projection corresponding to λk on X. Since the system of the generalized eigenvectors of A is complete in X, we have X = sp(A). For any f ∈ X and ϕ ∈ sp(A), the function ⟨T(t)ϕ, f ⟩ is in the subspace E 1 := L2 [0, T], where E1 is the span generated by the family {eλn t }∞ n=1 , i. e., N

E1 := span{ ∑ αk eλk t , ∀N ≥ 1}. k=1

Then, for any ϕ ∈ X, ⟨T(t)ϕ, f ⟩ ∈ L2 [0, T] can be written as the sum ∞

⟨T(t)ϕ, f ⟩ = ∑ ⟨E(λn , A)ϕ, f ⟩eλn t . n=1

Moreover, by using the Riesz property of the exponential family {eλn t }∞ n=1 in L [0, T], we know that there exist two positive constants K1 and K2 such that 2

∞

T

∞

0

n=1

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 K1 ∑ 󵄨󵄨󵄨⟨E(λn , A)ϕ, f ⟩󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨⟨T(t)ϕ, f ⟩󵄨󵄨󵄨 dt ≤ K2 ∑ 󵄨󵄨󵄨⟨E(λn , A)ϕ, f ⟩󵄨󵄨󵄨 . n=1

(5.2.5)

In accordance with both assumption (i) and Theorem 5.1.1, we have 󵄩󵄩 󵄩 wt 󵄩󵄩T(t)󵄩󵄩󵄩 ≤ Me ,

(5.2.6)

for M ≥ 1 and w ≥ 0. Using equations (5.2.5) and (5.2.6), we have e 󵄨 󵄨2 K1 ∑ 󵄨󵄨󵄨⟨E(λn , A)ϕ, f ⟩󵄨󵄨󵄨 ≤ M 2 [ ∞

2wT

n=1

−1 ]‖ϕ‖2 ‖f ‖2 . 2w

Consequently, e 󵄩 󵄩2 K1 ∑ 󵄩󵄩󵄩E(λn , A)ϕ󵄩󵄩󵄩 ≤ M 2 [ ∞

n=1

2wT

−1 ]‖ϕ‖2 . 2w

Since A is the generator of the C0 -semi-group T(t) in a Hilbert space X, the adjoint of A, A∗ , has the same property (see [78, p. 2354]). In other words, A∗ is the generator of the C0 -semi-group T ∗ (t) with the same conjugate eigenvalues λn . Furthermore, each λn is an isolated eigenvalue of A∗ with E ∗ (λn , A) = E(λn , A∗ ). For any f , ϕ ∈ X, we have ⟨T ∗ (t)ϕ, f ⟩ = ⟨ϕ, T(t)f ⟩ ∈ L2 [0, T]

142 | 5 Semi-groups and ∞

⟨T ∗ (t)ϕ, f ⟩ = ∑ ⟨E(λn , A∗ )ϕ, f ⟩eλn t . n=1

By using a similar reasoning as before, we show that ∞

[e 󵄩2 󵄩 K1 ∑ 󵄩󵄩󵄩E ∗ (λn , A)ϕ󵄩󵄩󵄩 ≤ M 2

2wT

n=1

− 1] ‖ϕ‖2 . 2w

(5.2.7)

Let f ∈ sp(A). Then we have m

f = ∑ E(λk , A)f . k=1

Hence, using equation (5.2.7), we infer that ‖f ‖2 := ⟨f , f ⟩ m

= ⟨ ∑ E(λk , A)f , f ⟩ m

k=1

= ∑ ⟨E(λk , A)f , E ∗ (λk , A)f ⟩ k=1

1 2

m

m

󵄩 󵄩 󵄩2 󵄩2 ≤ ( ∑ 󵄩󵄩󵄩E(λk , A)f 󵄩󵄩󵄩 ) ( ∑ 󵄩󵄩󵄩E ∗ (λk , A)f 󵄩󵄩󵄩 ) k=1

1 2

m

1 2

k=1

1

[e2wT − 1] 2 2 󵄩 󵄩2 ‖f ‖ ) . ≤ ( ∑ 󵄩󵄩󵄩E(λk , A)f 󵄩󵄩󵄩 ) (M 2 2wK 1

k=1

Thus, m 2wT − 1] 󵄩 󵄩2 [e ‖f ‖2 ≤ ∑ 󵄩󵄩󵄩E(λk , A)f 󵄩󵄩󵄩 M 2 . 2wK 1

k=1

Taking the limit, we get ‖f ‖2 ≤ M 2

[e2wT − 1] ∞ 󵄩󵄩 󵄩2 ∑ 󵄩E(λ , A)f 󵄩󵄩󵄩 , 2wK1 n=1󵄩 n

∀f ∈ X.

Hence, we deduce ∞ 2wT 2wK1 − 1] ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩2 󵄩2 2 2 [e E(λ , A)f ≤ ‖f ‖ ≤ M ∑ ∑ 󵄩E(λ , A)f 󵄩󵄩󵄩 , 󵄩 󵄩 n 󵄩 󵄩 2wK1 n=1󵄩 n M 2 [e2wT − 1] n=1

This makes us conclude that the series ∞

∑ E(λn , A)f

n=1

∀f ∈ X.

5.2 Diagonalization of semi-groups | 143

converges in X with a sum equal to f and causes {E(λn , A)f }n=1 ∞

to form a Riesz basis in X. As a consequence of the foregoing result from the Riesz basis property of eigenvectors, we have the following result. Corollary 5.2.1. System (5.2.1) satisfies the spectrum determined growth assumption, i. e., S(A) = w(A), where S(A) := sup Re λ λ∈σ(A)

and 󵄩 󵄩 w(A) := inf{w such that there is C ≥ 1 ; 󵄩󵄩󵄩eAt 󵄩󵄩󵄩 ≤ Cewt ∀ t ≥ 0}.

6 Discrete operator and denseness of the generalized eigenvectors 6.1 Hilbert–Schmidt discrete operator Let X be a complex Hilbert space with an inner product ⟨⋅, ⋅⟩ and a norm ‖ ⋅ ‖ and let A be a Fredholm operator in X. Let Φ denote the component of the Fredholm set of A which contains the origin. Clearly, Φ is a connected open set in ℂ, 0 ∈ Φ, and, for each λ ∈ Φ, the linear operator λ − A is a Fredholm operator in X. Let Φ = ℂ. If σ(A) = {λi }∞ i=1 , mi (0 < mi < ∞) is the ascent of the operator λi − A for i = 1, 2, . . . and Pi , i = 1, 2, . . ., is the projection of X onto the generalized eigenspace N([λi − A]mi ) along R([λi − A]mi ). Then we proceed to study the subspaces ∞

M∞ := ⋂ R([λi − A]mi ) = {x ∈ X such that Pi x = 0, i = 1, 2, . . .} i=1

and ∞

S∞ := {x ∈ X such that x = ∑ Pi x}. i=1

A basic property satisfied by the generalized eigenspaces is the fact that N([λj − A]mj ) ⊂ R([λi − A]mi ) for all i ≠ j. As an immediate consequence we obtain the following properties: N([λi − A]mi ) ∩ N([λj − A]mj ) = {0}

for all i ≠ j

and Pi Pj = δij Pi

for all i, j = 1, 2, . . . ,

where δij is the Kronecker delta. 6.1.1 Elementary properties Some additional elementary properties are stated in the following lemma. Lemma 6.1.1. For all λ ∈ ℂ and for i = 1, 2, . . ., we have: (i) (λ − A)x ∈ N([λi − A]mi ) for all x ∈ N([λi − A]mi ); and (ii) (λ − A)y ∈ R([λi − A]mi ) for all y ∈ 𝒟(A) ∩ R([λi − A]mi ). https://doi.org/10.1515/9783110493863-006

(6.1.1)

146 | 6 Discrete operator and denseness of the generalized eigenvectors In addition, for all λ ∈ ρ(A) and for i = 1, 2, . . ., (iii) (λ − A)−1 z ∈ 𝒟(A) ∩ R([λi − A]mi ) for all z ∈ R([λi − A]mi ). The next two lemmas are simple consequences of the above properties. Lemma 6.1.2. For i = 1, 2, . . ., the operator λi −A from 𝒟(A)∩R([λi −A]mi ) into R([λi −A]mi ) is one-to-one and onto. Lemma 6.1.3. For positive integers with i ≠ j, the operator λi − A from m

m

𝒟(A) ∩ R([λi − A] i ) ∩ R([λj − A] j )

into R([λi − A]mi ) ∩ R([λj − A]mj ) is one-to-one and onto. Theorem 6.1.1 (P. Lang and J. Locker [146]). Let A be a Fredholm operator in X with Φ = ℂ and assume that there exists a point ξ0 ∈ ρ(A). Then, for each λ ∈ ℂ, the operator λ − A : 𝒟(A) ∩ M∞ 󳨀→ M∞ is one-to-one and onto. Proof. Let us fix any λ ∈ ℂ. From Lemma 6.1.1 (ii), it is clear that λ − A maps 𝒟(A) ∩ M∞ into M∞ and, for λ∈ρ(A), certainly λ−A is one-to-one on 𝒟(A)∩M∞ , while, for λ∈σ(A), Lemma 6.1.2 shows that λ − A is one-to-one on 𝒟(A) ∩ M∞ . Therefore, we only need to check the onto property. First, assume λ ∈ ρ(A) and take any y ∈ M∞ . Then, by using Lemma 6.1.1 (iii), we have x = (λ − A)−1 y ∈ 𝒟(A) ∩ M∞ with (λ − A)x = y and, hence, in this case λ − A maps 𝒟(A) ∩ M∞ onto M∞ . Second, assume λ ∈ σ(A), say, λ = λi , and take any y ∈ M∞ . By using Lemma 6.1.2, there exists a unique x ∈ 𝒟(A) ∩ R([λi − A]mi ) such that (λi − A)x = y. To complete the proof, it is sufficient to show that x ∈ M∞ . Indeed, for any j ≠ i, there exists a unique z ∈ 𝒟(A) ∩ R([λi − A]mi ) ∩ R([λj − A]mj ) with (λi − A)z = y in accordance with Lemma 6.1.3. Since x and z both belong to m

𝒟(A) ∩ R([λi − A] i ),

6.1 Hilbert–Schmidt discrete operator

| 147

where λi − A is one-to-one, we must have x = z ∈ R([λj − A]mj ). We conclude that x ∈ R([λj − A]mj ) for j = 1, 2, . . ., so x ∈ M∞ . An immediate corollary of Theorem 6.1.1 is the following. Corollary 6.1.1. The subspace M∞ is either zero-dimensional or infinite-dimensional. Let X be a complex Hilbert space and A ∈ 𝒞 (X). Let Φ be the component containing the origin. We divide Φ into three disjoint sets U, S, T according to the following definition: (i) λ ∈ U if limn→∞ α[(A − λ)n ] = 0; (ii) λ ∈ S if 0 < limn→∞ α[(A − λ)n ] < ∞; and (iii) λ ∈ T if limn→∞ α[(A − λ)n ] = ∞. Theorem 6.1.2 (S. Kaniel and M. Schechter [138, Theorem 2.1]). Let X be a Hilbert space and let A be a densely defined linear operator on X. If A is a Fredholm operator, then one of the following cases occurs: (i) the set U contains all points of Φ with the possible exception of a denumerable set of points in S having no finite limit point; or (ii) Φ = T. 6.1.2 Direct sum Let X be a Hilbert space and A ∈ 𝒞 (X). For each λ ∈ ℂ, define ν(λ) = lim dim N([λ − A]k ). k→∞

We note that λ is an eigenvalue of A if and only if ν(λ) > 0, in which case ν(λ) is the dimension of the generalized eigenspace corresponding to λ, i.e., ν(λ) is the algebraic multiplicity of λ. The following theorem, given in [146], presents some of the basic spectral properties of the Fredholm operator A relative to the set Φ, where Φ denotes the component of the Fredholm set of A which contains the origin. Theorem 6.1.3 (P. Lang and J. Locker [146]). Let A be a Fredholm operator in X and assume there exists a point ξ0 ∈ Φ ∩ ρ(A). Then: (i) We have 0 ≤ ν(λ) < ∞ for all λ ∈ Φ, with λ ∈ ρ(A) if and only if ν(λ) = 0 and λ ∈ σ(A) if and only if ν(λ) > 0.

148 | 6 Discrete operator and denseness of the generalized eigenvectors (ii) The set Φ ∩ σ(A) is a denumerable set having no finite limit points in Φ. (iii) For each point λ0 ∈Φ, there exists a δ0 > 0 such that the punctured disk 0 < |λ−λ0 | < δ0 lies in Φ ∩ ρ(A). (iv) For each λ0 ∈ Φ, the ascent and descent of the operator λ0 − A are finite and equal. In addition, if m0 denotes the ascent of λ0 − A, then X = N([λ0 − A]m0 ) ⊕ R([λ0 − A]m0 )

(6.1.2)

as a topological direct sum. Proof. Since ν(ξ0 ) = 0, it follows from Theorem 6.1.2 that 0 ≤ ν(λ) < ∞ for all λ ∈ Φ and the set of all λ ∈ Φ, where 0 < ν(λ) < ∞, is a denumerable set having no finite limit points in Φ. Also, i(ξ0 − A) = 0 and, hence, i(λ − A) = 0

for all λ ∈ Φ,

(6.1.3)

or ⊥

dim N(λ − A) = dim N((λ − A)∗ ) = dim[R(λ − A)]

for all λ ∈ Φ.

(6.1.4)

It is immediate from (6.1.4) that λ ∈ Φ belongs to ρ(A) if and only if ν(λ) = 0. This establishes the first three parts of the theorem. Next, fix any point λ0 ∈ Φ and assume λ0 − A has ascent m0 , which is finite because ν(λ0 ) < ∞. From (6.1.3), we see that the products [λ0 − A]k , k = 0, 1, 2, . . ., are all Fredholm operators of index 0, so k

dim N([λ0 − A]k ) = dim N([(λ0 − A)∗ ] ) = dim R([λ0 − A]k )

⊥

(6.1.5)

for k = 0, 1, 2, . . .. From (6.1.5) we conclude that the ascent and descent of λ0 − A are equal. The decomposition (6.1.2) follows from both Theorem 2.14.1 and the fact that R((λ0 − ξ0 ) − λ0 + A) = R(ξ0 − A) = X. 6.1.3 Laurent expansion for the resolvent Now, we state the Laurent expansion for the resolvent (λ − A)−1 about any point λ0 ∈ Φ. Theorem 6.1.4. Let A be a Fredholm operator in X and assume there exists a point ξ0 ∈ Φ ∩ ρ(A). Then:

6.1 Hilbert–Schmidt discrete operator

| 149

(i) If λ0 ∈Φ∩ρ(A), then there exists δ0 > 0 such that the disk |λ −λ0 | < δ0 lies in Φ∩ρ(A) and ∞

(λ − A)−1 = ∑ (λ − λ0 )n An n=0

for all |λ − λ0 | < δ0 , where n+1

An = (−1)n [(λ0 − A)−1 ]

and where the series converges in the uniform operator norm. (ii) If λ0 ∈ Φ ∩ σ(A) and if m0 (0 < m0 < ∞) is the ascent of λ0 − A, then there exists δ0 > 0 such that the punctured disk 0 < |λ − λ0 | < δ0 lies in Φ ∩ ρ(A) and m0

Bn (λ − λ0 )n n=1

∞

(λ − A)−1 = ∑ (λ − λ0 )n An + ∑ n=0

for all 0 < |λ − λ0 | < δ0 , where the coefficients An and Bn belong to ℒ(X) and where the series converges in the uniform operator norm. In addition, Bn ≠ 0 for 1 ≤ n ≤ m0 , B1 is the projection of X onto the generalized eigenspace N((λ0 − A)m0 ) along R((λ0 − A)m0 ), and n

(λ0 − A)A0 = I − B1 ,

(6.1.6)

n

(−1) (λ0 − A) An = A0 , n = 0, 1, 2, . . . , n

n

Bn+1 = (−1) (λ0 − A) B1 , n = 0, 1, . . . , m0 − 1.

(6.1.7) (6.1.8)

6.1.4 Spectral theory of the inverse of an HS discrete operator Definition 6.1.1. A closed and densely defined linear operator A in X is a Hilbert– Schmidt (HS) discrete operator if there exists a point ξ0 ∈ ρ(A) such that (ξ0 − A)−1 is an HS operator. Let A be an HS discrete operator in X. According to Theorem 6.1.1, the operator λ − A maps 𝒟(A) ∩ M∞ into M∞ is one-to-one and onto for each λ ∈ ℂ, where M∞ is given in (6.1.1). Let S = A|𝒟(A)∩M∞ , so the operator λ − S maps 𝒟(A) ∩ M∞ into M∞ is one-to-one and onto. From this we know the inverse operator (λ − S)−1 : M∞ 󳨀→ 𝒟(A) ∩ M∞ is one-to-one and onto for each λ ∈ ℂ. It follows, from the closed graph theorem, that (λ − S)−1 ∈ ℒ(M∞ )

150 | 6 Discrete operator and denseness of the generalized eigenvectors and, clearly, (λ − S)−1 = (λ − A)−1 |M∞ for each λ ∈ ρ(A). We are going to examine these inverses. Let L = ξ0 − S : 𝒟(A) ∩ M∞ 󳨀→ M∞ . Then let us set K = L−1 : M∞ 󳨀→ 𝒟(A) ∩ M∞ ,

(6.1.9)

where K ∈ ℒ(M∞ ) and K = (ξ0 − A)−1 |M∞ . Let (uα )α∈I and (vβ )β∈J be two orthonormal ⊥ bases for M∞ and M∞ , respectively, where I and J are countable index sets. Clearly, the vectors (uα )α∈I ∪ (vβ )β∈J form an orthonormal basis for X and 󵄩 󵄩2 ∑ ‖Kuα ‖2 = ∑ 󵄩󵄩󵄩(ξ0 − A)−1 uα 󵄩󵄩󵄩

α∈I

α∈I

󵄩 󵄩2 󵄩 󵄩2 ≤ ∑ 󵄩󵄩󵄩(ξ0 − A)−1 uα 󵄩󵄩󵄩 + ∑ 󵄩󵄩󵄩(ξ0 − A)−1 vβ 󵄩󵄩󵄩 < ∞. α∈I

β∈J

Hence, K is an HS operator on M∞ with 󵄨󵄩 󵄩󵄨 |‖K‖| ≤ 󵄨󵄨󵄨󵄩󵄩󵄩(ξ0 − A)−1 󵄩󵄩󵄩󵄨󵄨󵄨, where |‖ ⋅ ‖| denotes the HS norm of an operator. We have the following theorem. Theorem 6.1.5. The operator K, given in (6.1.9), is an HS operator on M∞ with 󵄨󵄩 󵄩󵄨 |‖K‖| ≤ 󵄨󵄨󵄨󵄩󵄩󵄩(ξ0 − A)−1 󵄩󵄩󵄩󵄨󵄨󵄨, where |‖ ⋅ ‖| denotes the HS norm of an operator. The spectrum of K is given by σ(K) = {0}. In fact, let λ ≠ 0 in σ(K). Since K is compact (see Theorem 4.9.1), λ is an eigenvalue of K and there exists u ≠ 0 in M∞ with Ku = λu. This implies that u ∈ 𝒟(A) ∩ M∞ and u = λ(ξ0 − A)u,

6.1 Hilbert–Schmidt discrete operator

| 151

or 1 Au = (ξ0 − )u λ and, hence, ξ0 − for some i and

1 = λi λ

u ∈ N(λi − A) ∩ M∞ ⊂ N([λi − A]mi ) ∩ R([λi − A]mi ) = {0}, which is a contradiction. Thus, we have the following theorem. Theorem 6.1.6. Let K be the operator given in (6.1.9). Then σ(K) = {0} and K is a quasinilpotent operator on M∞ . First, a straightforward calculation shows that −1

(λ − S)(ξ0 − λ)−1 K((ξ0 − λ)−1 − K)

=I

on M∞

and −1

(ξ0 − λ)−1 K((ξ0 − λ)−1 − K) (λ − S) = I

on 𝒟(A) ∩ M∞ ,

so −1

(λ − S)−1 = (ξ0 − λ)−1 K((ξ0 − λ)−1 − K)

(6.1.10)

for all λ ≠ ξ0 in ℂ. This is a companion result for (2.13.2) and it follows formally from (2.13.2) by restriction and inversion. Second, by a corollary of Carleman’s inequality (see Corollary 4.9.1), we have 1 −1 󵄩 󵄩󵄩 −1 −1 2 2 󵄩󵄩((ξ0 − λ) − K) 󵄩󵄩󵄩 ≤ |ξ0 − λ| exp{ (1 + |‖K‖| |ξ0 − λ| )} 2

(6.1.11)

for all λ ≠ ξ0 in ℂ. The combination of both (6.1.10) and (6.1.11) leads to the following estimate: 1

1 e 2 ‖K‖ 󵄩󵄩 −1 󵄩 exp{ |‖K‖|2 |ξ0 − λ|2 } 󵄩󵄩(λ − S) 󵄩󵄩󵄩 ≤ 2 |ξ0 − λ|2

(6.1.12)

for all λ ≠ ξ0 in ℂ. In addition, if |λ| > 2|ξ0 |, then 1 |λ| < |ξ0 − λ| < 2|λ| 2 and from (6.1.12) we obtain 1

4e 2 ‖K‖ 󵄩󵄩 −1 󵄩 exp(2|‖K‖|2 |λ|2 ) 󵄩󵄩(λ − S) 󵄩󵄩󵄩 ≤ |λ|2 for all λ ∈ ℂ with |λ| > 2|ξ0 |.

(6.1.13)

152 | 6 Discrete operator and denseness of the generalized eigenvectors 6.1.5 Singular values of the resolvent Theorem 6.1.7 (M. T. Aimar, A. Intissar, and J. M. Paoli [13]). Let A be a closed linear operator with a dense domain 𝒟(A) on a Hilbert space X. We suppose that: (i) there exists λ ∈ ρ(A) such that (λ − A)−1 is a compact operator; and (0) (ii) there exists ξ0 ∈ ρ(A) such that (ξ0 − A)−1 ∈ 𝒞p,∞ (X) for some p > 0. Then sn ((λ − A)−1 |M∞ ) = o(n

− p1

) as n → ∞.

Proof. Let ξ0 ∈ ρ(A). We have −1 (ξ0 − A)−1 |M∞ = (ξ0 − A|M∞ ) .

By combining both hypothesis (ii) and Proposition 3.6.1, we have sn ((ξ0 − A)−1 |M∞ ) = o(n

− p1

)

as n → ∞. Let K1 = (ξ0 − A)−1 |M∞ and let λ ∈ ℂ\{ξ0 }. We have (λ − A)|M∞ = λ − A|M∞

= λ − ξ0 + ξ0 − A|M∞ .

This implies that (λ − A)|M∞ K1 = [(λ − ξ0 ) + (ξ0 − A|M∞ )]K1 = (λ − ξ0 )K1 + I.

Hence, 1 1 (λ − A)|M∞ K1 = −K1 + ξ0 − λ ξ0 − λ 1 = − K1 , ξ0 − λ so −1

1 1 (λ − A)|M∞ K1 ( − K1 ) ξ0 − λ ξ0 − λ

=I

6.2 Denseness of the generalized eigenvectors of an HS discrete operator

| 153

on M∞ . We similarly show −1

1 1 K1 ( − K1 ) (λ − A)|M∞ = I ξ0 − λ ξ0 − λ on 𝒟(A) ∩ M∞ . Thus, −1

((λ − A)|M∞ )

−1

=

1 1 K( − K1 ) , ξ0 − λ 1 ξ0 − λ

which is closed and all defined on M∞ and therefore bounded by the closed graph theorem. By referring to Proposition 3.6.2 (i), we deduce that − p1

sn ((λ − A)−1 |M∞ ) = o(n

)

as n → ∞.

6.2 Denseness of the generalized eigenvectors of an HS discrete operator All results of this section can be found in [146].

6.2.1 The subspaces M∞ and S∞ for a Fredholm operator Let X be a complex Hilbert space with an inner product ⟨⋅, ⋅⟩ and a norm ‖ ⋅ ‖. Suppose that A is a Fredholm operator in X with Φ = ℂ and assume that there exists a point ξ0 ∈ ρ(A). We know that λ − A is a Fredholm operator in X for each λ ∈ ℂ and, according to Theorem 6.1.3, the spectrum σ(A) is a denumerable set having no finite limit points (it may be finite or empty). Let σ(A) = {λi }∞ i=1 , let mi (0 < mi < ∞) denote the ascent of the operator λi − A for i = 1, 2, . . ., and let Pi , i = 1, 2, . . ., denote the projection of X onto the generalized eigenspace N([λi − A]mi ) along R([λi − A]mi ). We know that the adjoint operators A∗ and (λ − A)∗ = λ − A∗ are Fredholm operators in X for all λ ∈ ℂ, ξ0 ∈ ρ(A∗ ), ∗ and σ(A∗ ) = {λi }∞ i=1 . Also, the ascent and descent of the operator λi − A are both equal to mi for i = 1, 2, . . . (see equation (6.1.5)) and m

⊥

m

⊥

N([λi − A∗ ] i ) = R([λi − A]mi ) and R([λi − A∗ ] i ) = N([λi − A]mi )

154 | 6 Discrete operator and denseness of the generalized eigenvectors for i = 1, 2, . . .. It is well known that Pi∗ , i = 1, 2, . . ., is the projection of X onto the generalized eigenspace N([λi − A∗ ]mi ) along R([λi − A∗ ]mi ). Now, we are ready to introduce the closed subspaces ∞

M∞ = ⋂ R([λi − A]mi ) = {x ∈ X such that Pi x = 0, i = 1, 2, . . .} i=1

and ∞

m

∗ M∞ = ⋂ R([λi − A∗ ] i ) = {x ∈ X such that Pi∗ x = 0, i = 1, 2, . . .}, i=1

together with the subspaces ∞

S∞ = {x ∈ X such that x = ∑ Pi x} i=1

and ∞

∗ S∞ = {x ∈ X such that x = ∑ Pi∗ x}. i=1

∗ We observe that x ∈ M∞ if and only if Pi∗ x = 0 for i = 1, 2, . . . if and only if

⟨Pi u, x⟩ = ⟨u, Pi∗ x⟩ = 0 ⊥ ⊥ . Summing up, we have for all u ∈ X and for i = 1, 2, . . . if and only if x ∈ S∞ = S∞ completed the proof.

Theorem 6.2.1 (P. Lang and J. Locker [146, Theorem 3.1]). The Hilbert space X has the orthogonal direct sum decompositions ∗ ∗ ⊕M . X = S∞ ⊕ M∞ = S∞ ∞

6.2.2 Sufficient conditions S∞ = X and M∞ = {0} Let us fix any element x ∈ M∞ . Take any y ∈ X and define f : ℂ 󳨀→ ℂ by f (λ) = ⟨(λ − S)−1 x, y⟩ for all λ ∈ ℂ, where S = A|𝒟(A)∩M∞ .

6.2 Denseness of the generalized eigenvectors of an HS discrete operator |

155

Lemma 6.2.1. Let λ0 ∈ ρ(A), x ∈ M∞ , and y ∈ X. Then f is analytic on a neighborhood of λ0 . Proof. Take any point λ0 ∈ρ(A). According to Theorem 6.1.4 (i), there exists δ0 > 0 such that the disk |λ − λ0 | < δ0 lies in ρ(A) and ∞

f (λ) = ∑ (λ − λ0 )n ⟨An x, y⟩ n=0

for all |λ − λ0 | < δ0 . Consequently, f is analytic on a neighborhood of each point λ0 ∈ ρ(A). Let λ0 ∈ σ(A), x ∈ M∞ , and y ∈ X. Then, by Theorem 6.1.4 (ii), there exists δ0 > 0 such that the punctured disk 0 < |λ − λ0 | < δ0 lies in ρ(A) and m0

⟨Bn x, y⟩ (λ − λ0 )n n=1

∞

f (λ) = ∑ (λ − λ0 )n ⟨An x, y⟩ + ∑ n=0

(6.2.1)

for all 0 < |λ − λ0 | < δ0 , where m0 denotes the ascent of λ0 − A, An and Bn satisfy equations (6.1.6)–(6.1.8), and B1 is the projection of X onto N([λ0 − A]m0 ) along R([λ0 − A]m0 ). However, x ∈ M∞ implies B1 x = 0 and, by equation (6.1.8), Bn x = 0 for n = 1, . . . , m0 . Hence, (6.2.1) simplifies to ∞

f (λ) = ∑ (λ − λ0 )n ⟨An x, y⟩ n=0

for all 0 < |λ − λ0 | < δ0 . Thus, we have the following lemma. Lemma 6.2.2. Let λ0 ∈ σ(A), x ∈ M∞ , and y ∈ X. Then there exists δ0 > 0 such that the punctured disk 0 < |λ − λ0 | < δ0 lies in ρ(A) and ∞

f (λ) = ∑ (λ − λ0 )n ⟨An x, y⟩, n=0

(6.2.2)

where An satisfies equations (6.1.6)–(6.1.8). We can show that (6.2.2) remains valid for λ = λ0 . Indeed, from (6.1.6) and (6.1.7), we see that A0 maps X into 𝒟(A) and An maps X into 𝒟(An ) with A0 x = (−1)n (λ0 − A)n An x ∈ R([λ0 − A]n ) for n = 0, 1, 2, . . .. Thus, A0 x ∈ 𝒟(A) ∩ R([λ0 − A]m0 ), where m0 denotes the ascent of λ0 − A and, by (6.1.6), (λ0 − A)A0 x = (I − B1 )x = x.

156 | 6 Discrete operator and denseness of the generalized eigenvectors Moreover, (λ0 − S)−1 x belongs to 𝒟(A) ∩ R([λ0 − A]m0 ) with (λ0 − A)(λ0 − S)−1 x = x, since λ0 − A is one-to-one on 𝒟(A) ∩ R([λ0 − A]m0 ) (see Lemma 6.1.2). We must have A0 x = (λ0 − S)−1 x and, hence, ⟨A0 x, y⟩ = ⟨(λ0 − S)−1 x, y⟩ = f (λ0 ). We conclude that (6.2.2) is valid for all |λ − λ0 | < δ0 , which proves that f is analytic on a neighborhood of each point λ0 ∈ σ(A). Hence, we have the following lemma. Lemma 6.2.3. f is analytic on a neighborhood of each point λ0 ∈ σ(A). Theorem 6.2.2 (P. Lang and J. Locker [146, Theorem 4.3]). Let A be an HS discrete linear operator in X with a spectrum σ(A) = {λi }∞ i=1 . Assume that there exists a set of five rays, arg λ = θj , j = 1, . . . , 5, such that: (i) the angles between the adjacent rays are less than π2 ; (ii) for |λ| sufficiently large, all the points on the five rays belong to ρ(A) and ‖(λ − A)−1 ‖ is bounded for these λ; and (iii) on at least one of the rays, ‖(λ − A)−1 ‖ → 0 as λ → ∞. Then ∗ =X S∞ = S∞

and

∗ M∞ = M∞ = {0}.

∗ = X. Proof. It is sufficient to prove that M∞ = {0} for Theorem 6.2.1, which gives S∞ ∗ By applying the same argument to A , we get ∗ M∞ = {0}

and S∞ = X. Let us fix any element x ∈ M∞ . Take any y ∈ X and define f : ℂ 󳨀→ ℂ by f (λ) = ⟨(λ − S)−1 x, y⟩ for all λ ∈ ℂ, where S = A|𝒟(A)∩M∞ . We assert that f is an entire function of order ≤2. First, according to Lemma 6.2.1, f is analytic on a neighborhood of each point λ0 ∈ ρ(A).

6.2 Denseness of the generalized eigenvectors of an HS discrete operator |

157

Second, choose any point λ0 ∈ σ(A). Then, by Lemma 6.2.2, there exists δ0 > 0 such that the punctured disk 0 < |λ − λ0 | < δ0 lies in ρ(A) and f satisfies (6.2.2), for all 0 < |λ − λ0 | < δ0 . Third, by Lemma 6.2.3, f is analytic on a neighborhood of each point λ0 ∈ σ(A). Fourth, the above demonstrates that f is an entire function (see the proof of Lemma 6.2.3). By applying (6.1.13), we obtain 1

󵄨󵄨 󵄨 4e 2 ‖K‖ ‖x‖‖y‖ exp(2|‖K‖|2 |λ|2 ) 󵄨󵄨f (λ)󵄨󵄨󵄨 ≤ r02

(6.2.3)

for all |λ| > r0 , where r0 = max(2|ξ0 |, 1) > 0. From (6.2.3), it is immediate that f has an order ≤ 2. This establishes the assertion. To complete the proof, we observe that f is bounded on each of the five rays arg λ = θj , j = 1, . . . , 5. By condition (ii), we have 󵄨󵄨 󵄨 󵄩 −1 󵄩 󵄨󵄨f (λ)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩(λ − A) 󵄩󵄩󵄩‖x‖‖y‖ ≤ γ‖x‖‖y‖

for |λ| sufficiently large and for λ on any of the five rays. Then the Phragmén–Lindelöf theorem (see Theorem 3.5.1) implies that f is constant on ℂ and hence, in view of condition (iii), f (λ) = ⟨(λ − S)−1 x, y⟩ = 0 that

for all λ ∈ ℂ.

(6.2.4)

Now, (6.2.4) is valid for an arbitrary y ∈ X, so (λ − S)−1 x = 0 or x = 0. This proves M∞ = {0}.

The proof is complete.

7 Frames in Hilbert spaces The main feature of a basis {fk }∞ k=1 in a Hilbert space X is that every f ∈ X can be represented as an (infinite) linear combination of the elements fk in the basis ∞

f = ∑ ck (f )fk .

(7.0.1)

k=1

The coefficients ck (f ) are unique. We now introduce the concept of frames. A frame is also a sequence of elements {fk }∞ k=1 in X, which allows every f ∈ X to be written as in (7.0.1). However, the corresponding coefficients are not necessarily unique. Thus, a frame might not be a basis; arguments for generalizing the basis concept were given in Chapter 4.

7.1 Frame in Hilbert space Throughout this part, X denotes a separable Hilbert space and I is a countable index set. Each orthonormal basis {φn }n for a Hilbert space X satisfies the Plancherel equality, which states that 󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨φ, φn ⟩󵄨󵄨󵄨 = ‖φ‖2

n∈I

for all φ ∈ X. However, a sequence can satisfy the Plancherel equality without being orthonormal or a basis. Here we present an (elementary) finite-dimensional example. Let X = ℝ2 and set φ1 = (1, 0), φ2 = (0, 1), φ3 = ( √12 , √12 ), and φ4 = (− √12 , √12 ). Each of {φ1 , φ2 } and {φ3 , φ4 } is an orthonormal basis for ℝ2 , so 4

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨φ, φn ⟩󵄨󵄨󵄨 = 2‖φ‖2 ,

n=1

x ∈ ℝ2 .

− 21

Therefore, the family {2 φn }4n=1 satisfies the Plancherel equality, but it is not orthogonal and it is not a basis for ℝ2 . 7.1.1 Frame In this section, we will briefly recall the definitions and some basic properties of the notion of frames. Definition 7.1.1. A family {φn }n∈I is said to be a frame for X if there exist numbers A, B > 0 such that 󵄨 󵄨2 A‖φ‖2 ≤ ∑ 󵄨󵄨󵄨⟨φ, φn ⟩󵄨󵄨󵄨 ≤ B‖φ‖2 , n∈I

https://doi.org/10.1515/9783110493863-007

for all φ ∈ X.

(7.1.1)

160 | 7 Frames in Hilbert spaces The constants A and B are called a lower and upper frame bound, respectively. The largest possible lower frame bound is called the optimal lower frame bound and the smallest possible upper frame bound is the optimal upper frame bound. Remark 7.1.1. (i) It should be noted here that frames are a natural generalization of the concept of Riesz bases. Indeed, a family {φn }n∈I is a Riesz basis for X if and only if {φn }n∈I is a frame and is ω-linearly independent. (ii) It is clear that a frame is a complete set of vectors since the relations ⟨φ, φn ⟩ = 0, n = 1, 2, 3, . . ., imply that φ = 0. (iii) If {φn }n is a complete orthonormal sequence in X, then Parseval’s identity shows that (7.1.1) holds with A = B = 1. (iv) Recall that the complex exponential functions { √12π eikx }+∞ k=−∞ constitute an or2 thonormal basis for L2 (−π, π). Thus, {eikx }+∞ k=−∞ is a frame for L (−π, π) with bounds A = B = 2π.

Now, we give some examples for which the frames are not necessarily Riesz bases. Let {φn }∞ n=1 be an orthonormal basis in X. (i) Define F1 := {fn }∞ n=1 given by 0 fn = { φn−1

if n = 1, if n ≠ 1.

We have ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , fn ⟩󵄨󵄨󵄨 = ‖f ‖2 .

n=1

Then F1 is a frame for X with bounds A = B = 1. However, F1 is not a Riesz basis since it is not ω-linearly independent. In fact, there exists c1 ≠ 0 such that ∞

∞

n=1

n=2

∑ cn fn = ∑ cn φn−1 = 0.

(ii) Define F2 := {fn }∞ n=1 given by φ fn = { 1 φn−1

if n = 1, if n ≠ 1.

We have ∞

󵄨 󵄨2 ‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , fn ⟩󵄨󵄨󵄨 ≤ 2‖f ‖2 . n=1

Hence, F2 is a frame for X with bounds 1 and 2. It is easy to see that F2 is not a Riesz basis, although F2 is neither ω-linearly independent. Indeed, there exists c2 = −c1 ≠ 0

7.1 Frame in Hilbert space

| 161

such that ∞

∞

n=1

n=3

∑ cn fn = c1 φ1 + c2 φ1 + ∑ cn φn−1 = 0.

Consequently, F2 is not a Riesz basis. (iii) Let F3 := {φ1 , √12 φ2 , √12 φ2 , √13 φ3 , √13 φ3 , √13 φ3 , . . .}. It is clear that ∞

󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨f , fn ⟩󵄨󵄨󵄨 = ‖f ‖2 .

n=1

Then F3 is a frame for X with bounds A = B = 1. We easily see that F3 is not a Riesz basis since it is not ω-linearly independent. In fact, there exist cn ≠ 0, n ≥ 2, such that ∞

∑ cn fn = 0.

n=1

Proposition 7.1.1. Let {φn }n∈I be a frame for X with bounds A and B and let (an )n∈I be a sequence of scalars. If inf |an | > 0 n∈I

and sup |an | < +∞, n∈I

then (an φn )n∈I is a frame for X. Proof. Since {φn }n∈I is a frame for X, there exist numbers A, B > 0 such that 󵄨 󵄨2 A‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , φn ⟩󵄨󵄨󵄨 ≤ B‖f ‖2 .

(7.1.2)

n∈I

Further, we have 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 (inf |an |2 ) ∑ 󵄨󵄨󵄨⟨f , φn ⟩󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨⟨f , an φn ⟩󵄨󵄨󵄨 ≤ (sup |an |2 ) ∑ 󵄨󵄨󵄨⟨f , φn ⟩󵄨󵄨󵄨 . n∈I

n∈I

n∈I

n∈I

n∈I

Hence, equations (7.1.2) and (7.1.3) imply 󵄨 󵄨2 A inf |an |2 ‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , an φn ⟩󵄨󵄨󵄨 ≤ B sup |an |2 ‖f ‖2 . n∈I

n∈I

n∈I

As infn∈I |an | > 0, A inf |an |2 > 0. n∈I

Further, supn∈I |an | < ∞ implies that B sup |an |2 < ∞. n∈I

Consequently, (an φn )n∈I is a frame for X.

(7.1.3)

162 | 7 Frames in Hilbert spaces Let {φn }n∈I be a frame for X. Consider the operator S : X 󳨀→ X,

Sf = ∑⟨f , φi ⟩φi .

(7.1.4)

i∈I

Then S has a bounded inverse and f = SS−1 f = ∑⟨S−1 f , φi ⟩φi , i∈I

for all f ∈ X.

Theorem 7.1.1. {φn }n∈I is a frame if and only if S, given in (7.1.4), is a well-defined surjective operator on X. Proof. The only if part is well known. Now, suppose that S is well defined and surjective. In particular, 󵄨 󵄨2 ⟨Sf , f ⟩ = ∑ 󵄨󵄨󵄨⟨f , φi ⟩󵄨󵄨󵄨 < ∞, i∈I

for all f ∈ X.

Now, a standard argument (using closed graph) shows that the upper frame condition is satisfied, so S is bounded and bijective since ⊥

N(S) = R(S∗ ) = R(S)⊥ = {0}. Therefore, S has a bounded inverse S−1 : X 󳨀→ X. This means that 0 ∈ ρ(S). However, in view of Proposition 2.16.1, A := inf ⟨Sf , f ⟩ ∈ σ(S). ‖f ‖=1

Therefore, A > 0 and 󵄨 󵄨2 A‖f ‖2 ≤ ⟨Sf , f ⟩ = ∑ 󵄨󵄨󵄨⟨f , φi ⟩󵄨󵄨󵄨 < ∞, i∈I

for all f ∈ X.

The proof is complete. By using Cauchy–Schwarz on ‖f ‖2 = ∑ ci (f )⟨f , φi ⟩, i∈I

we obtain the following. Proposition 7.1.2. Let {φi }i∈I ⊂ X. Suppose that there exists a family {ci }i∈I of linear functionals on X such that: (i) the mapping f 󳨀→ {ci (f )}i∈I is continuous from X into l2 (I), i. e., there is k > 0 such that 󵄨 󵄨2 ∑ 󵄨󵄨󵄨ci (f )󵄨󵄨󵄨 ≤ k‖f ‖2 , i∈I

(ii) f = ∑i∈I ci (f )φi , for all f ∈ X.

for all f ∈ X;

and

7.1 Frame in Hilbert space

| 163

Then 󵄨2 󵄨 k −1 ‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , φi ⟩󵄨󵄨󵄨 , i∈I

for all f ∈ X.

+∞ iμk x +∞ Theorem 7.1.2. Let {λk }+∞ }k=−∞ is a k=−∞ , {μk }k=−∞ be real sequences. Suppose that {e 1 2 frame for L (−π, π) with bounds A, B. If there exists a constant L < 4 such that

|μk − λk | ≤ L

A and 1 − cos(πL) + sin(πL) < √ , B

2 then {eiλk x }+∞ k=−∞ is a frame for L (−π, π) with bounds 2

A A(1 − √ (1 − cos(πL) + sin(πL))) , B

2

B(2 − cos(πL) + sin(πL)) .

7.1.2 Frames and operators Since a frame {fk }∞ k=1 is a Bessel sequence, the operator T : l2 (ℕ) 󳨀→ X,

∞

T{ck }∞ k=1 = ∑ ck fk k=1

is bounded by Theorem 3.1.1; T is called the pre-frame operator or the synthesis operator. By Lemma 2.17.2, the adjoint operator of T is given by T ∗ : X 󳨀→ l2 (ℕ),

∞

T ∗ f = {⟨f , fk ⟩}k=1 .

The operator T ∗ is called the analysis operator. By composing T and T ∗ , we obtain the frame operator S : X 󳨀→ X,

∞

Sf = TT ∗ f = ∑ ⟨f , fk ⟩fk . k=1

(7.1.5)

Note that, since {fk }∞ k=1 is a Bessel sequence, the series defining S converges unconditionally for all f ∈X by Corollary 3.1.2. We now state some of the important properties of S. Lemma 7.1.1 (O. Christensen [66, Lemma 5.1.5]). Let {fk }∞ k=1 be a frame with frame operator S, given in (7.1.5), and frame bounds A, B. Then S is bounded, invertible, selfadjoint, and positive. Proof. S is bounded as a composition of two bounded operators. By Theorem 3.1.1, we have 󵄩 󵄩 ‖S‖ = 󵄩󵄩󵄩TT ∗ 󵄩󵄩󵄩

164 | 7 Frames in Hilbert spaces 󵄩 󵄩 = ‖T‖󵄩󵄩󵄩T ∗ 󵄩󵄩󵄩

= ‖T‖2 ≤ B. Since ∗

S∗ = (TT ∗ ) = TT ∗ = S, the operator S is self-adjoint. The inequality (7.1.1) yields A‖f ‖2 ≤ ⟨Sf , f ⟩ ≤ B‖f ‖2 for all f ∈ X, or AI ≤ S ≤ BI. Thus, S is positive. Furthermore, 0 ≤ I − B−1 S ≤

B−A I B

and, consequently, 󵄩󵄩 󵄨 󵄨 −1 󵄩 −1 󵄩󵄩I − B S󵄩󵄩󵄩 = sup 󵄨󵄨󵄨⟨(I − B S)f , f ⟩󵄨󵄨󵄨 ‖f ‖=1

B−A B < 1, ≤

which, by Theorem 2.12.1, shows that S is invertible. Lemma 7.1.2 (O. Christensen [66, Lemma 5.1.5]). Let {fk }∞ k=1 be a frame with frame op−1 ∞ erator S and frame bounds A, B. Then {S fk }k=1 is a frame with bounds B−1 , A−1 . If A, B −1 −1 −1 ∞ are the optimal bounds for {fk }∞ k=1 , then the bounds B , A are optimal for {S fk }k=1 . −1 ∞ −1 The frame operator for {S fk }k=1 is S . Proof. Note that, for f ∈ X, ∞

∞

k=1

k=1

󵄨 󵄨2 󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , S−1 fk ⟩󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨⟨S−1 f , fk ⟩󵄨󵄨󵄨 󵄩 󵄩2 ≤ B󵄩󵄩󵄩S−1 f 󵄩󵄩󵄩 󵄩 󵄩2 ≤ B󵄩󵄩󵄩S−1 󵄩󵄩󵄩 ‖f ‖2 .

−1 ∞ That is, {S−1 fk }∞ k=1 is a Bessel sequence. It follows that the frame operator for {S fk }k=1 is well defined. By definition, it acts on f ∈ X by ∞

∞

k=1

k=1

∑ ⟨f , S−1 fk ⟩S−1 fk = S−1 ∑ ⟨S−1 f , fk ⟩fk

7.1 Frame in Hilbert space

| 165

= S−1 SS−1 f = S−1 f .

(7.1.6)

−1 −1 This shows that the frame operator for {S−1 fk }∞ comk=1 equals S . The operator S mutes with both S and I, so, using Theorem 2.12.2, we multiply the inequality AI ≤ S ≤ BI by S−1 . This gives

B−1 I ≤ S−1 ≤ A−1 I, i. e., B−1 ‖f ‖2 ≤ ⟨S−1 f , f ⟩ ≤ A−1 ‖f ‖2 , for all f ∈ X. Via (7.1.6), we obtain ∞

󵄨 󵄨2 B−1 ‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , S−1 fk ⟩󵄨󵄨󵄨 ≤ A−1 ‖f ‖2 , k=1

−1 −1 for all f ∈ X. Thus, {S−1 fk }∞ k=1 is a frame with frame bounds B , A . To prove the op∞ timality of the bounds (in case A, B are optimal for {fk }k=1 ), let A be the optimal lower 1 −1 ∞ bound for {fk }∞ k=1 and assume that the optimal upper bound for {S fk }k=1 is C < A . By −1 ∞ applying what we already proved to the frame {S fk }k=1 having frame operator S−1 , we deduce that −1

∞

−1 −1 {fk }∞ k=1 = {(S ) S fk }k=1

has the lower bound C1 > A, but this is a contradiction. Thus, {S−1 fk }∞ k=1 has the optimal upper bound A1 . The argument for the optimal lower bound is similar. Theorem 7.1.3 (O. Christensen [66, Theorem 5.5.1]). A sequence {fk }∞ k=1 in X is a frame for X if and only if ∞

T : {ck }∞ k=1 󳨀→ ∑ ck fk k=1

is a well-defined mapping of l2 (ℕ) onto X. Proof. First, suppose that {fk }∞ k=1 is a frame. By Theorem 3.1.1, T is a well-defined bounded operator from l2 (ℕ) into X. By Lemma 7.1.1, the frame operator S = TT ∗ is surjective. Thus, T is surjective. For the opposite implication, suppose that T is a well-defined operator from l2 (ℕ) onto X. Then Lemmas 2.17.1 and 2.17.3 show that T is bounded and that {fk }∞ k=1 is a Bessel sequence. Let T † : X 󳨀→ l2 (ℕ)

166 | 7 Frames in Hilbert spaces denote the pseudo-inverse of T. For f ∈ X, we have ∞

f = TT † f = ∑ (T † f )k fk , k=1

where (T † f )k denotes the kth coordinate of T † f . Thus, 󵄨2 󵄨 ‖f ‖4 = 󵄨󵄨󵄨⟨f , f ⟩󵄨󵄨󵄨 󵄨󵄨 ∞ 󵄨󵄨2 󵄨󵄨 󵄨󵄨 † = 󵄨󵄨󵄨⟨ ∑ (T f )k fk , f ⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 k=1 󵄨 ∞

∞

󵄨 󵄨2 󵄨 󵄨2 ≤ ∑ 󵄨󵄨󵄨(T † f )k 󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 k=1

∞

k=1

󵄩 󵄩2 󵄨 󵄨2 ≤ 󵄩󵄩󵄩T † 󵄩󵄩󵄩 ‖f ‖2 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 . k=1

We conclude that ∞

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , fk ⟩󵄨󵄨󵄨 ≥

k=1

1 ‖f ‖2 . ‖T † ‖2

Theorem 7.1.4 (O. Christensen [66, Theorem 5.5.5]). Let {ek }∞ k=1 be an arbitrary orthonormal basis for X. Then the frames for X are precisely the families {Uek }∞ k=1 , where U : X 󳨀→ X is a bounded and surjective operator. 2 ∞ Proof. Let {δk }∞ k=1 be the canonical basis for l (ℕ) and let {ek }k=1 be an orthonormal basis for X. Let

Φ : X 󳨀→ l2 (ℕ) be the isometric isomorphism defined by Φek = δk . If {fk }∞ k=1 is a frame, then the pre-frame operator T is bounded and surjective and Tδk = fk . With U := TΦ, we have ∞ {fk }∞ k=1 = {Uek }k=1

and U is bounded and surjective. The notion that every family {Uek }∞ k=1 of the described type is a frame follows from Theorem 7.1.3; alternatively, we observe that ∞

󵄨 󵄨2 󵄩 󵄩2 ∑ 󵄨󵄨󵄨⟨f , Uek ⟩󵄨󵄨󵄨 = 󵄩󵄩󵄩U ∗ f 󵄩󵄩󵄩

k=1

and refer to Lemma 2.12.1.

7.1 Frame in Hilbert space

| 167

7.1.3 The projection method In this section, we study the following problem. Let {fi }∞ i=1 ⊂ X be total. Any f ∈ X can n be approximated by means of a subfamily {fi }i=1 in the sense that the orthogonal projection Pn f of f on span{fi }ni=1 can be written as n

Pn f = ∑ λi(n) (f )fi . i=1

Now, Pn f → f

for n → ∞,

so it is very natural to ask for conditions implying that each λi(n) (f ) converges to some value λi (f ) with n

f = ∑ λi (f )fi . i=1

In the case where we obtain a decomposition of any f ∈ X in this way, we express it by saying that the projection method works. Clearly, we have to be more exact. For example, the coefficient λi(n) (f ) does not need to be uniquely determined under the conditions given here. As we shall see, the problem can be completely solved in the case of a frame. In fact, let {fi }i∈I ⊂ X be a frame. By definition, the projection method works if and only if there exist functionals λj : X 󳨀→ ℂ such that lim ⟨f , Sn−1 fj ⟩ = λj (f )

n→∞

and f = ∑ λi (f )fi , i∈I

for all f ∈ X.

(7.1.7)

In the case in which the projection method works, we get a new way to calculate the frame coefficients. We have ⟨f , S−1 fi ⟩ = lim ⟨f , Sn−1 fi ⟩. n→∞

This result can be of importance for implementations; whereas the calculation of S−1 involves inversion of an infinite matrix, ⟨f , Sn−1 fi ⟩ can be calculated by linear algebra. Let I be a countable index set and {In }∞ n=1 be a family of finite subsets of I such that I1 ⊂ I2 ⊂ ⋅ ⋅ ⋅ ⊂ In ↗ I. Given a family {fi }i∈I ⊂ X, we define Xn := span{fi }i∈In . Lemma 7.1.3. {fi }i∈In is a frame for Xn .

168 | 7 Frames in Hilbert spaces The frame operator corresponding to {fi }i∈In is Sn : Xn 󳨀→ Xn ,

Sn f = ∑ ⟨f , fi ⟩fi . i∈In

Lemma 7.1.4. The orthogonal projection on Xn is Pn f = ∑ ⟨f , Sn−1 fi ⟩fi , i∈In

f ∈ X.

Theorem 7.1.5 (O. Christensen [62, Theorem 3.1]). Let {fi }i∈I ⊂ X be a frame. Then the projection method works if and only if, for any j ∈ I, there exists a constant cj such that 󵄩󵄩 −1 󵄩󵄩 󵄩󵄩Sn fj 󵄩󵄩 ≤ cj ,

(7.1.8)

for all n such that j ∈ In . Proof. Suppose (7.1.8) is satisfied. Fix j ∈ I and take N such that j ∈ In for all n ≥ N. Define ϕn := Sn−1 fj − S−1 fj ,

n ≥ N.

Then Sϕn = SSn−1 fj − fj

= Sn Sn−1 fj + ∑ ⟨Sn−1 fj , fi ⟩fi − fj i∈I\In

= ∑

i∈I\In

⟨Sn−1 fj , fi ⟩fi .

Hence, ϕn = ∑ ⟨Sn−1 fj , fi ⟩S−1 fi . i∈I\In

Therefore, for f ∈ X, we get 󵄨 󵄨2 󵄨󵄨 󵄨2 󵄨󵄨 −1 −1 󵄨󵄨 󵄨󵄨⟨f , ϕn ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ ⟨Sn fj , fi ⟩⟨f , S fi ⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 i∈I\In

󵄨 󵄨2 󵄨 󵄨2 ≤ ∑ 󵄨󵄨󵄨⟨Sn−1 fj , fi ⟩󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨⟨f , S−1 fi ⟩󵄨󵄨󵄨 i∈I\In

i∈I\In

󵄩 󵄩2 󵄨 󵄨2 ≤ B󵄩󵄩󵄩Sn−1 fj 󵄩󵄩󵄩 ∑ 󵄨󵄨󵄨⟨f , S−1 fi ⟩󵄨󵄨󵄨 i∈I\In

≤

Bcj2

󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨S−1 f , fi ⟩󵄨󵄨󵄨 → 0

i∈I\In

for n → ∞.

7.1 Frame in Hilbert space

| 169

Now, suppose (7.1.7) is satisfied. Fix arbitrarily j ∈ I. For any n with j ∈ In , we define a functional An f := ⟨f , Sn−1 fj ⟩.

An : X 󳨀→ ℂ,

Each An is continuous and the family {An }n is pointwise bounded, being pointwise convergent. Now, by the theorem of Banach–Steinhaus, the family {‖An ‖}n is bounded, i. e., there is a constant cj such that 󵄩 󵄩 ‖An ‖ = 󵄩󵄩󵄩Sn−1 fj 󵄩󵄩󵄩 ≤ cj ,

for all n.

Theorem 7.1.6 (O. Christensen [62, Theorem 3.3]). The projection method works for any Schauder basis. Proof. The basis condition says that there exist unique coefficient functionals {ci (⋅)}i on X for which ∞

f = ∑ ci (f )fi , i=1

for all f ∈ X. By Theorem 4.5.1, there exists a constant M such that 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 m 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∑ ci fi 󵄩󵄩󵄩 ≤ M 󵄩󵄩󵄩∑ ci fi 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩i=1 󵄩󵄩 󵄩󵄩i=1 󵄩󵄩󵄩 whenever n ≤ m and c1 , . . . , cm are arbitrarily numbers. The orthogonal projection of f ∈ X on span{fi }ni=1 is given by n

Pn f = ∑⟨f , Sn−1 fi ⟩fi , i=1

where Sn : span{fi }ni=1 󳨀→ span{fi }ni=1 ,

n

Sn f = ∑⟨f , fi ⟩fi . i=1

Clearly, n

f = lim ∑⟨f , Sn−1 fi ⟩fi , n→∞

i=1

for all f ∈ X.

Consequently, {⟨f , Sn−1 fi ⟩}∞ n=1 is a Cauchy sequence for each i and ∞

f = ∑ lim ⟨f , Sn−1 fi ⟩fi , i=1

n→∞

for all f ∈ X.

170 | 7 Frames in Hilbert spaces 7.1.4 Frame perturbations Definition 7.1.2. Let (φi )i∈I ⊂ X and (ci )i∈I be a sequence of linear functionals on X. If (i) there are A and B > 0 such that 󵄨2 󵄨 A‖f ‖2 ≤ ∑ 󵄨󵄨󵄨ci (f )󵄨󵄨󵄨 ≤ B‖f ‖2 i∈I

for all f ∈ X;

and (ii) f = ∑i∈I ci (f )φi for all f ∈ X; then {φi , ci }i∈I will be called a set of atoms for X. The optimal values of A and B are the atomic bounds. Proposition 7.1.3 (C. Heil [102, Proposition 6.6.1]). Let {φi , ci }i∈I be a set of atoms with bounds A, B. Then, for each set {ψi }i∈I ⊂ X with M := ∑ ‖φn − ψn ‖2 < n∈I

1 , B

there exists a set {di }i∈I ⊂ X such that {ψi , di }i∈I is a set of atoms with bounds A(1 + √MB)−2 , B(1 − √MB)−2 . Theorem 7.1.7 (O. Christensen [62, Proposition 2.4]). Let {φn }n∈I be a frame for X with bounds A and B and let {ψn }n∈I be a family of vectors in X such that M := ∑ ‖φn − ψn ‖2 < A. n∈I

Then the family {ψn }n∈I is a frame for X with bounds A(1 − √ M )2 and 2(B + M). A Proof. Denote the frame operator corresponding to {φn }n∈I by S. Then {φn ; S−1 φn } is a set of atoms with bounds B−1 , A−1 . Now, if ∑ ‖φn − ψn ‖2 < A,

n∈I

then Proposition 7.1.3 tells us that there exists a set {di }i∈I ⊂ X such that {ψi , di }i∈I is a

)−2 , A−1 (1 − √ M )−2 . In particular, any f ∈ X can be set of atoms with bounds B−1 (1 + √ M A A written f = ∑⟨f , di ⟩ψi i∈I

7.2 Frame of subspace or fusion frame

| 171

with −2

M 󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨f , di ⟩󵄨󵄨󵄨 ≤ A−1 (1 − √ ) ‖f ‖2 . A i∈I As a consequence of Proposition 7.1.2, we get A(1 − √

2

M 󵄨2 󵄨 ) ‖f ‖2 ≤ ∑ 󵄨󵄨󵄨⟨f , ψi ⟩󵄨󵄨󵄨 , A i∈I

for all f ∈ X. The upper estimate can be made as follows. For any f ∈ X, 󵄨 󵄨2 󵄨 󵄨2 ∑ 󵄨󵄨󵄨⟨f , ψi ⟩󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨⟨f , ψi − φi ⟩ + ⟨f , φi ⟩󵄨󵄨󵄨 i∈I

i∈I

󵄨 󵄨2 󵄨 󵄨2 ≤ ∑ 2[󵄨󵄨󵄨⟨f , ψi − φi ⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨f , φi ⟩󵄨󵄨󵄨 ] i∈I

≤ 2M‖f ‖2 + 2B‖f ‖2 = 2(B + M)‖f ‖2 , which completes the proof. Let {ei }∞ i=1 be an orthonormal basis of X. Consider f1 := e1 and 1 fi := ei−1 + ei i 2

√ π − 1, 3. Indeed, since for all i ≥ 2. Then the set {f1 } ∪ {fi }∞ i=2 is a frame with bounds 1 − 6 ∞ {ei }i=1 is a frame with bounds A = 1, B = 2, we have ∞

M := ‖f1 − e1 ‖2 + ∑ ‖fi+1 − ei ‖2 = i=1

π2 − 1 < 1. 6 2

√ π − 1, We apply Theorem 7.1.7 to deduce that {fi }∞ i=1 is a frame with bounds 1 − 6 2(B + M) ≤ 6. In fact, an easy direct estimate gives the better value 3 for the upper bound.

7.2 Frame of subspace or fusion frame We now introduce the concept of frames of subspaces which have been renamed, recently, as fusion frames. This notion can be considered as a generalization of frames (see [54]).

172 | 7 Frames in Hilbert spaces 7.2.1 Definitions Definition 7.2.1. Let W := {Wi }i∈I be a family of closed subspaces in X and let w := {wi }i∈I be a family of weights, i. e., wi > 0 for all i ∈ I. Then we say that {Wi }i∈I is a frame of subspaces with respect to {wi }i∈I for X (or Ww := {(Wi , wi )}i∈I is a fusion frame), if there exist constants 0 < 𝔸 ≤ 𝔹 < ∞ such that 󵄩2 󵄩 𝔸‖f ‖2 ≤ ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ≤ 𝔹‖f ‖2 , i∈I

for all f ∈ X,

(7.2.1)

where πWi (⋅) is the orthogonal projection onto the subspace Wi . The numbers 𝔸, 𝔹 are called the fusion frame bounds. The family Ww is called an 𝔸-tight fusion frame if 𝔸 = 𝔹, it is a Parseval fusion frame if 𝔸 = 𝔹 = 1, and it is w-uniform if w = wi = wj for all i, j ∈ I. If the right-handed inequality of (7.2.1) holds, then we say that Ww is a Bessel fusion sequence with Bessel fusion bound 𝔹. Moreover, we say that {Wi }i∈I is an orthonormal fusion basis for X if X = ⨁ Wi . i∈I

Before going further, we state a useful result from [54], which determines a relationship between frames of subspaces and frames. Before that, we recall the following definition. Definition 7.2.2. A sequence of vectors is called a frame sequence, if it is a frame only for its closed linear span.

7.2.2 A characterization of frame of subspaces Theorem 7.2.1 (P. G. Cassaza and G. Kutyniok [54, Theorem 3.2]). For each i ∈ I, let wi > 0 and let {fij }j∈Ji be a frame sequence in X with frame bounds Ai and Bi . Define Wi = span{fij }j∈Ji for all i ∈ I and choose an orthonormal basis {eij }j∈Ji for each subspace Wi . Suppose that 0 < A = inf Ai ≤ B = sup Bi < ∞. i∈I

i∈I

Then the following conditions are equivalent: (i) {wi fij }i∈I,j∈Ji is a frame for X; (ii) {wi eij }i∈I,j∈Ji is a frame for X; and (iii) {Wi }i∈I is a frame of subspaces with respect to {wi }i∈I for X.

7.2 Frame of subspace or fusion frame

| 173

Proof. Since, for each i ∈ I, {fij }j∈Ji is a frame for Wi with frame bounds Ai and Bi , we obtain 󵄩2 󵄩 󵄩2 󵄩 A ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ≤ ∑ Ai wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 i∈I

i∈I

󵄨2 󵄨 ≤ ∑ ∑ 󵄨󵄨󵄨⟨πWi (f ), wi fij ⟩󵄨󵄨󵄨 i∈I j∈Ji

󵄩 󵄩2 ≤ ∑ Bi wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 i∈I

󵄩 󵄩2 ≤ B ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 . i∈I

Now, we observe that 󵄨 󵄨2 󵄨 󵄨2 ∑ ∑ 󵄨󵄨󵄨⟨πWi (f ), wi fij ⟩󵄨󵄨󵄨 = ∑ ∑ 󵄨󵄨󵄨⟨f , wi fij ⟩󵄨󵄨󵄨 . i∈I j∈Ji

i∈I j∈Ji

This shows that, provided {wi fij }i∈I,j∈Ji is a frame for X with frame bounds C and D, the sets {Wi }i∈I form a frame of subspaces with respect to {wi }i∈I for X with frame bounds CB and DA . Moreover, if {Wi }i∈I is a frame of subspaces with respect to {wi }i∈I for X with frame bounds C and D, the calculation above implies that {wi fij }i∈I,j∈Ji is a frame for X with frame bounds AC and BD. Thus, (i) if and only if (iii). To prove the equivalence of (ii) and (iii), note that we can now actually calculate the orthogonal projections in the following way: 󵄩󵄩 󵄩󵄩󵄩2 󵄩 󵄩2 󵄩 wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 = wi2 󵄩󵄩󵄩 ∑ ⟨f , eij ⟩eij 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 j∈Ji 󵄨 󵄨2 = ∑ 󵄨󵄨󵄨⟨f , wi eij ⟩󵄨󵄨󵄨 . j∈Ji

From this, the claim follows immediately. Proposition 7.2.1. Let {wi }i∈I be a family of weights, i. e., wi > 0 for all i ∈ I such that ∑ wi2 < ∞ i∈I

and let {fij }j∈Ji be a frame sequence in X with frame bounds Ai and Bi . Define Wi = spanj∈Ji {fij } for all i ∈ I and choose an orthonormal basis {eij }j∈Ji for each subspace Wi . Suppose that 0 < A = inf Ai ≤ B = sup Bi < ∞. i∈I

i∈I

174 | 7 Frames in Hilbert spaces Then the following properties hold: (i) {wi fij }i∈I,j∈Ji is a frame for X; (ii) {wi eij }i∈I,j∈Ji is a frame for X; and (iii) {Wi }i∈I is a frame of subspaces with respect to {wi }i∈I for X. Proof. Fix any f ∈ X and consider πWi (⋅) the orthogonal projection onto Wi . Then, by using Remark 3.3.1, we have 󵄩 󵄩2 ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ≤ ‖f ‖2 ∑ wi2 . i∈I i∈I ⏟⏟⏟⏟⏟⏟⏟⏟⏟

(7.2.2)

0, † ‖πW ‖ i

so 󵄩 󵄩2 󵄩 † 󵄩󵄩−2 ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ≥ ‖f ‖2 ∑ wi2 󵄩󵄩󵄩πW 󵄩 , i󵄩 i∈I i∈I ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

f ∈ ⋃ N(πWi )⊥ . i∈I

(7.2.3)

>0

Thus, both equations (7.2.2) and (7.2.3) imply that {Wi }i∈I is a frame of subspaces with respect to {wi }i∈I for X. This completes the proof of (iii). The proof of (i) and (ii) follow immediately from Theorem 7.2.1. 7.2.3 Synthesis operator Let X be a separable Hilbert space and let I, J, and Ji be countable (or finite) index sets. For each Bessel fusion sequence Ww = {(Wi , wi )}i∈I of X, we define the representation space associated with Ww as follows: l2 (X, I) = {{fk }k∈I such that fk ∈ X and ∑ ‖fk ‖2 < ∞} k∈I

with inner product given by ⟨{fk }k∈I , {gk }k∈I ⟩ = ∑⟨fi , gi ⟩. i∈I

The set l2 (X, ℕ) is replaced by l2 (X). Let E = {ej }j∈J be an orthonormal basis for X. Define eij = {δik ej }k∈I for all i ∈ I, j ∈ J, where δik is the Kronecker delta. Then ℰE = {eij }i∈I,j∈J is an orthonormal basis for l2 (X, I). The sequence ℰE is called the associated orthonormal basis to E in l2 (X, I).

7.2 Frame of subspace or fusion frame

| 175

Definition 7.2.3. Let Ww be a Bessel fusion sequence for X. The synthesis operator TWw : l2 (X, I) 󳨀→ X is defined by TWw ({fi }i∈I ) = ∑ wi πWi (fi ) for all {fi }i∈I ∈ l2 (X, I). i∈I

∗ The adjoint operator TW : X 󳨀→ l2 (X, I) given by w ∗ TW (f ) = {wi πWi (f )}i∈I w

is called the analysis operator.

7.2.4 A characterization of Bessel fusion sequences We can also characterize Bessel fusion sequences in terms of their synthesis operators as in frame theory. Theorem 7.2.2 (M. S. Asgari [34, Theorem 2.3]). A sequence Ww is a Bessel fusion sequence with Bessel fusion bound 𝔹 for X if and only if the synthesis operator TWw is a well-defined bounded operator from l2 (X, I) into X and ‖TWw ‖ ≤ √𝔹. Proof. This claim follows immediately from the fact that, for each J ⊂ I with |J| < ∞ and each {fi }i∈I ∈ l2 (X, I), we have 󵄩󵄩 󵄩󵄩2 󵄨󵄨 󵄨󵄨2 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩∑ wi πWi (fi )󵄩󵄩󵄩 = sup 󵄨󵄨󵄨⟨g, ∑ wi πWi (fi )⟩󵄨󵄨󵄨 󵄩󵄩 󵄩󵄩 ‖g‖=1󵄨󵄨 󵄨󵄨 i∈J i∈J 󵄨󵄨 󵄨󵄨2 󵄨 󵄨 = sup 󵄨󵄨󵄨∑ wi ⟨πWi (g), fi ⟩󵄨󵄨󵄨 󵄨 󵄨󵄨 ‖g‖=1󵄨 i∈J

2

󵄩 󵄩 ≤ sup (∑ wi 󵄩󵄩󵄩πWi (g)󵄩󵄩󵄩‖fi ‖) ‖g‖=1 i∈J

󵄩 󵄩2 ≤ sup ∑ wi2 󵄩󵄩󵄩πWi (g)󵄩󵄩󵄩 ∑ ‖fi ‖2 ‖g‖=1 i∈J

2

i∈J

≤ 𝔹 ∑ ‖fi ‖ . i∈J

The opposite implication is obvious. If Ww = {(Wi , wi )}i∈ℕ is a Bessel fusion sequence in X, then the operator ∗ TW T : l2 (X) 󳨀→ l2 (X) w Ww

176 | 7 Frames in Hilbert spaces given by ∗ TW T ({fi }i∈ℕ ) = { ∑ wk wi πWk πWi (fi )} w Ww

k∈ℕ

i∈ℕ

is a bounded operator. If E = {ej }j∈ℕ is an orthonormal basis for X, then the Gram matrix associated with Ww with respect to ℰE is defined by {wi wm ⟨πWi (ej ), πWm (en )⟩}i,j,m,n∈ℕ . Theorem 7.2.3 (M. S. Asgari [34, Theorem 2.4]). The following conditions are equivalent: (i) Ww is a Bessel fusion sequence with Bessel fusion bound 𝔹; and (ii) the Gram matrix associated with Ww with respect to ℰE defines a bounded operator on l2 (X), with norm at most 𝔹. Proof. The implication (i) 󳨐⇒ (ii) follows from Theorem 7.2.2. To prove (ii) 󳨐⇒ (i), 2 ∗ suppose that {fi }∞ i=1 ∈ l (X). Then, for every n, m ∈ ℕ , m > n, we have 󵄩󵄩 n 󵄩󵄩4 󵄩󵄩 m 󵄩󵄩4 m 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ∑ wk πW (fk ) − ∑ wk πW (fk )󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ wk πW (fk )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k k k 󵄩󵄩k=1 󵄩󵄩 󵄩󵄩k=n+1 󵄩󵄩 k=1 󵄨󵄨 󵄨󵄨2 m m 󵄨󵄨 󵄨󵄨 = 󵄨󵄨󵄨⟨ ∑ wk πWk (fk ), ∑ wi πWi (fi )⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 i=n+1 󵄨 k=n+1 󵄨 2 󵄨󵄨 m 󵄨 m 󵄨󵄨 󵄨󵄨 󵄨 = 󵄨󵄨󵄨 ∑ ⟨fk , ∑ wk wi πWk πWi (fi )⟩󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 i=n+1 󵄨k=n+1 󵄨 󵄩 󵄩󵄩2 m 󵄩 m 󵄩 m 󵄩󵄩 󵄩 ≤ ( ∑ ‖fk ‖2 )( ∑ 󵄩󵄩󵄩 ∑ wk wi πWk πWi (fi )󵄩󵄩󵄩 ) 󵄩 󵄩󵄩 k=n+1 k=n+1󵄩 󵄩i=n+1 󵄩 2

m

2

2

≤ 𝔹 ( ∑ ‖fk ‖ ) . k=n+1

By Theorem 7.2.2, Ww is a Bessel fusion sequence with Bessel fusion bound 𝔹. Let Ww be a fusion frame for X with fusion frame bounds 𝔸 and 𝔹. The fusion frame operator SWw for Ww is defined by SWw : X 󳨀→ X,

∗ SWw (f ) = TWw TW (f ) = ∑ wi2 πWi (f ), w i∈I

which is a positive, self-adjoint, invertible operator on X with 𝔸I ≤ SWw ≤ 𝔹I.

7.2 Frame of subspace or fusion frame

| 177

This provides, for all f ∈ X, the reconstruction formula as follows: −1 −1 π (f ) = SW S (f ) ∑ wi2 SW w Wi w Ww i∈I

=f −1 = SWw SW (f ) w

−1 = ∑ wi2 πWi SW (f ). w i∈I

(7.2.4)

7.2.5 A characterization of fusion frames We now give a characterization of fusion frames in terms of the associated synthesis and analysis operators. Theorem 7.2.4 (M. S. Asgari [34, Theorem 2.6]). Let W = {Wi }i∈I be a family of closed subspaces in X and let V = {wi }i∈I be a family of weights. Then the following conditions are equivalent: (i) Ww is a fusion frame for X; (ii) the synthesis operator TWw is a bounded linear operator from l2 (X, I) onto X; and ∗ (iii) the analysis operator TW is injective with closed range. w Proof. This claim can be proved in a way analogous to frame theory. Corollary 7.2.1. The optimal fusion frame bounds for Ww are 󵄩 † 󵄩󵄩−2 2 𝔸 = 󵄩󵄩󵄩TW 󵄩 = γ(TWw ) w󵄩 and 𝔹 = ‖SWw ‖ = ‖TWw ‖2 , where γ(⋅) is the reduced minimum modulus. Proof. By using Theorem 7.2.4, we have 󵄩 󵄩2 𝔸 = inf ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ‖f ‖=1

i∈I

󵄩 ∗ 󵄩2 = inf 󵄩󵄩󵄩TW (f )󵄩󵄩󵄩 w ‖f ‖=1 2

∗ = γ(TW ) w

= γ(TWw )2 󵄩 † 󵄩󵄩−2 = 󵄩󵄩󵄩TW 󵄩 w󵄩

178 | 7 Frames in Hilbert spaces and 󵄩2 󵄩 𝔹 = sup ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ‖f ‖=1 i∈I

󵄨 󵄨 = sup 󵄨󵄨󵄨⟨SWw (f ), f ⟩󵄨󵄨󵄨 ‖f ‖=1

= ‖SWw ‖

= ‖TWw ‖2 .

This achieves the proof. The definition shows that, if Ww is a fusion frame for X, then W = {Wi }i∈I is complete in X, that is, span{Wi }i∈I = X. We say that Ww is a fusion frame sequence if it is a fusion frame for span{Wi }i∈I . Theorem 7.2.4 leads to a statement about fusion frame sequences. Corollary 7.2.2. A sequence Ww is a fusion frame sequence if and only if TWw : l2 (X, I) 󳨀→ X,

TWw ({fi }i∈I ) = ∑ wi πWi (fi ) i∈I

is a well-defined bounded operator with a closed range. Proof. This follows immediately from Theorem 7.2.4. 7.2.6 Oblique projections and biorthogonality Theorem 7.2.5 (M. S. Asgari [34, Theorem 2.9]). Let Ww = {(Wi , wi )}i∈I , Zλ = {(Zi , λi )}i∈I be two fusion frame sequences for X and let W = span{Wi }i∈I and Z = span{Zi }i∈I . The following conditions are equivalent: (i) X = W ⊕ Z ⊥ ; (ii) X = Z ⊕ W ⊥ ; ∗ (iii) the operator TZ∗λ TWw is a bounded, invertible operator from R(TW ) onto R(TZ∗λ ); and w (iv) cos(θ(W, Z)) > 0 and cos(θ(Z, W)) > 0. Proof. (i) ⇐⇒ (ii) ⇐⇒ (iv) This follows from Corollary 4.12.3. (i) 󳨐⇒ (iii) By Corollary 7.2.2, TZ∗λ TWw is a bounded operator. Therefore, it suf∗ fices to show that TZ∗λ TWw is bijective. Let {fi }i∈I ∈ R(TW ) and let h = TWw ({fi }i∈I ). If w ∗ ⊥ TZλ TWw ({fi }i∈I ) = 0, then h ∈ Z , so h = 0 and it follows that {fi }i∈I ∈ N(TWw ). Since ∗ ⊥ N(TWw ) = R(TW ) , w ⊥

∗ ∗ {fi }i∈I ∈ R(TW ) ∩ R(TW ) = {0}, w w

7.2 Frame of subspace or fusion frame

| 179

which implies that TZ∗λ TWw is injective. Now, let {fi }i∈I ∈ R(TZ∗λ ). Then there is some f ∈ W such that TZ∗λ (f ) = {fi }i∈I and, by (7.2.4), we have {fi }i∈I = TZ∗λ (f ) −1 = TZ∗λ (∑ wi2 πWi SW (f )) w i∈I

−1 = TZ∗λ TWw ({wi πWi SW (f )}i∈I ). w

It follows that TZ∗λ TWw is surjective. (iii) 󳨐⇒ (i) Suppose that h ∈ X. Then h = g1 + g2 , where g1 ∈ Z and g2 ∈ Z ⊥ . Put TZ∗λ (g1 ) = {fi }i∈I and −1

u = TWw (TZ∗λ TWw ) ({fi }i∈I ). Then u ∈ W and we have TZ∗λ (g1 − u) = TZ∗λ (g1 ) − TZ∗λ (u) = {fi }i∈I − {fi }i∈I = 0, which implies that g1 − u ∈ Z ⊥ . Thus, h = u + (g1 − u + g2 ) ∈ W + Z ⊥ . Let f ∈ W ∩ Z ⊥ . Since ∗ −1 TW S−1 (f ) = {wi πWi SW (f )}i∈I , w Ww w

we have −1 ∗ {wi πWi SW (f )}i∈I ∈ R(TW ). w w

Thus, by (7.2.4), we compute −1 −1 TZ∗λ TWw ({wi πWi SW (f )}i∈I ) = TZ∗λ (∑ wi2 πWi SW (f )) = TZ∗λ (f ) = 0. w w i∈I

This shows that f = 0, so X = W ⊕ Z ⊥ . Lemma 7.2.1 (P. G. Cassaza and G. Kutyniok [54, Lemma 3.4]). Let {Wi }i∈I be a family of subspaces in X and let {wi }i∈I be a family of weights. If {Wi }i∈I is a frame of subspaces with respect to {wi }i∈I for X, then it is complete.

180 | 7 Frames in Hilbert spaces Proof. If {Wi }i∈I is not complete, then there exists some f ∈ X, f ≠ 0, with f ⊥ span{Wi }i∈I . It follows that 󵄩2 󵄩 ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 = 0. i∈I

Consequently, {Wi }i∈I is not a frame of subspaces. In the next theorem we give a characterization of fusion frames which keeps the information about the fusion frame bounds. Theorem 7.2.6 (M. S. Asgari [34, Theorem 2.10]). A sequence Ww is a fusion frame for X with bounds 𝔸, 𝔹 if and only if the following conditions are satisfied: (i) W = {Wi }∈I is complete in X; and (ii) the synthesis operator TWw is well defined on l2 (X, I) and, for every {fi }i∈I ∈ N(TWw )⊥ , 󵄩 󵄩2 𝔸 ∑ ‖fi ‖2 ≤ 󵄩󵄩󵄩TWw ({fi }i∈I )󵄩󵄩󵄩 ≤ 𝔹 ∑ ‖fi ‖2 . i∈I

i∈I

(7.2.5)

Proof. Let Ww be a fusion frame with bounds 𝔸 and 𝔹. Then, by Lemma 7.2.1, W is complete in X. Theorem 7.2.2 shows that the right-handed inequality (7.2.5) holds and Theorem 7.2.4 implies that ∗ N(TWw )⊥ = R(TW ), w

that is, N(TWw )⊥ = {{wi πWi (f )}i∈I : f ∈ X}. For every f ∈ X, we also have 2

󵄩 󵄩2 󵄨 󵄨2 (∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ) = 󵄨󵄨󵄨⟨SWw (f ), f ⟩󵄨󵄨󵄨 i∈I

󵄩 󵄩2 ≤ 󵄩󵄩󵄩SWw (f )󵄩󵄩󵄩 ‖f ‖2 1󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩SWw (f )󵄩󵄩󵄩 ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 . 𝔸 i∈I

Hence, 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 𝔸 ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩SWw (f )󵄩󵄩󵄩 = 󵄩󵄩󵄩TWw ({wi πWi (f )}i∈I )󵄩󵄩󵄩 . i∈I

To prove the converse implication, assume that the conditions (i) and (ii) are satisfied. Then, by the right-handed inequality (7.2.5) and Theorem 7.2.2, Ww satisfies the upper

7.2 Frame of subspace or fusion frame

| 181

fusion frame condition with bound 𝔹. For the lower fusion frame condition, we prove that TWw is onto. First we show that R(TWw ) is closed. Suppose that {gn }∞ n=1 is a sequence ⊥ in R(TWw ). Then we can find a sequence {fn }∞ in N(T ) such that Ww n=1 gn = TWw (fn ) ∞ for all n ∈ ℕ. Now, if {gn }∞ n=1 converges to some g ∈ X, then (7.2.5) implies that {fn }n=1 ∞ 2 is a Cauchy sequence. Therefore, {fn }n=1 converges to some f ∈ l (X, I) which, by the continuity of TWw , gives

TWw (f ) = g. Thus, R(TWw ) is closed. By Theorem 7.2.4, Ww is a fusion frame for R(TWw ) and, hence, from R(TWw ) = span{Wi }i∈I we deduce that X = R(TWw ). † † By (2.18.2), we know that the operators TW TWw and TWw TW are the orthogonal prow

w

jections onto N(TWw )⊥ and R(TWw ), respectively. Thus, for each {fi }i∈I ∈ l2 (X, I), we have 󵄩 † 󵄩2 󵄩 󵄩2 󵄩 󵄩2 † 𝔸 󵄩󵄩󵄩TW T ({fi }i∈I )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩TWw TW T ({fi }i∈I )󵄩󵄩󵄩 = 󵄩󵄩󵄩TWw ({fi }i∈I )󵄩󵄩󵄩 . w Ww w Ww We also have † N(TW ) = R(TWw )⊥ = {0}, w

so 1 󵄩󵄩 † 󵄩󵄩2 󵄩󵄩TWw 󵄩󵄩 ≤ . 𝔸 From Lemma 2.18.1, we obtain 󵄩󵄩 ∗ † 󵄩󵄩 󵄩󵄩 † ∗ 󵄩󵄩 󵄩󵄩 † 󵄩󵄩 󵄩󵄩(TWw ) 󵄩󵄩 = 󵄩󵄩(TWw ) 󵄩󵄩 = 󵄩󵄩TWw 󵄩󵄩. Hence, 1 󵄩󵄩 ∗ † 󵄩󵄩2 󵄩󵄩(TWw ) 󵄩󵄩 ≤ . 𝔸 ∗ † ∗ Again, by using (2.18.2), the operator (TW ) TWw is the orthogonal projection onto w ⊥

∗ N(TW ) = R(TWw ) = X. w

182 | 7 Frames in Hilbert spaces Hence, for all f ∈ X, we compute 󵄩 ∗ † ∗ 󵄩2 ‖f ‖2 = 󵄩󵄩󵄩(TW ) TWw (f )󵄩󵄩󵄩 w 󵄩2 󵄩 ∗ † 󵄩󵄩2 󵄩󵄩 ∗ ≤ 󵄩󵄩󵄩(TW ) 󵄩󵄩 󵄩󵄩TWw (f )󵄩󵄩󵄩 w 1󵄩 ∗ 󵄩2 ≤ 󵄩󵄩󵄩TW (f )󵄩󵄩󵄩 w 𝔸 1 󵄩2 󵄩 = ∑ wi2 󵄩󵄩󵄩πWi (f )󵄩󵄩󵄩 . 𝔸 i∈I This shows that the lower fusion frame condition is satisfied.

7.2.7 A characterization in terms of orthonormal fusion bases In the following theorems, we give two more characterizations of fusion frames in terms of orthonormal fusion bases. First, we show that each fusion frame is the image of the orthonormal fusion basis under the synthesis operator (which is a bounded surjective operator). Theorem 7.2.7 (M. S. Asgari [34, Theorem 2.11]). Let Ww be a fusion frame for X. Then there is an orthonormal fusion basis N = {Ni }i∈I for l2 (X, I) such that TWw (Ni ) = Wi for every i ∈ I. Proof. Let E = {ej }j∈J be an orthonormal basis for X. Then ℱi = {πWi (ej )}j∈J

is a Parseval frame for Wi and, hence, Wi = span{πWi (ej )}j∈J . Let ℰE = {eij }i∈I,j∈J be the associated orthonormal basis to E = {ej }j∈J of l2 (X, I) and let Ni = span{eij }j∈J . Then N = {Ni }i∈I is an orthonormal fusion basis for l2 (X, I). Now, if f ∈ Ni , then we write f = ∑⟨f , eij ⟩eij . j∈J

Thus, TWw (f ) = ∑⟨f , eij ⟩TWw (eij ) j∈J

= ∑ wi ⟨f , eij ⟩πWi (ej ). j∈J

7.2 Frame of subspace or fusion frame

| 183

This shows that TWw (f ) ∈ Wi . Finally, if g ∈ Wi , then we have g = ∑⟨g, πWi (ej )⟩πWi (ej ) j∈J

=∑ j∈J

1 ⟨g, ej ⟩TWw (eij ) wi

= TWw (∑ j∈J

1 ⟨g, ej ⟩eij ). wi

Thus, g ∈ TWw (Ni ). Altogether, we have TWw (Ni ) = Wi . Theorem 7.2.8 (M. S. Asgari [34, Theorem 2.12]). Let Ww be a fusion frame for X and let Ei = {eij }j∈Ji be an orthonormal basis for Wi for all i ∈ I. Then there exists an orthonormal fusion basis N = {Ni }i∈I for X and a bounded, surjective operator U : X 󳨀→ X such that U(Ni ) = Wi . Proof. According to Theorem 7.2.1, Ew = {wi eij }i∈I,j∈Ji is a frame for X. Let U = {uij }i∈I,j∈Ji be an arbitrary orthonormal basis for X. By Theorem 7.1.4, there is a bounded, surjective operator U : X 󳨀→ X such that U(uij ) = wi eij for all i ∈ I, j ∈ Ji . Write Ni = span{uij }j∈Ji . Then N = {Ni }i∈I is an orthonormal fusion basis for X and U(Ni ) = Wi . 7.2.8 Duals of fusion frames Theorem 7.2.9 (P. Gavruta [91, Theorem 2.4]). Let {(Wi , wi )}i∈I be a fusion frame with frame bounds 𝔸, 𝔹. If T ∈ ℒ(X) is an invertible operator, then {(TWi , wi )}i∈I is a fusion frame with frame bounds 𝔸 ‖T ∗ ‖2 ‖T ∗−1 ‖2

󵄩 󵄩2 󵄩 󵄩2 and 𝔹󵄩󵄩󵄩T ∗ 󵄩󵄩󵄩 󵄩󵄩󵄩T ∗−1 󵄩󵄩󵄩 .

Proof. From Lemma 3.3.1, we have 󵄩󵄩 󵄩 ∗󵄩 ∗ 󵄩 󵄩󵄩πWi T f 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩T 󵄩󵄩󵄩‖πTWi f ‖. Hence, 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 𝔸 󵄩󵄩󵄩T ∗ f 󵄩󵄩󵄩 ≤ ∑ wi2 󵄩󵄩󵄩πWi T ∗ f 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩T ∗ 󵄩󵄩󵄩 ∑ wi2 ‖πTWi f ‖2 i∈I

i∈I

and, since T ∗ is invertible, we obtain ∑ wi2 ‖πTWi f ‖2 ≥ i∈I

𝔸 ‖f ‖2 . ‖T ∗ ‖2 ‖T ∗−1 ‖2

184 | 7 Frames in Hilbert spaces On the other hand, from Lemma 3.3.1 we obtain, with T −1 instead of T, πTWi = πTWi T ∗−1 πWi T ∗ . Hence, 󵄩 󵄩 󵄩󵄩 ‖πTWi f ‖ ≤ 󵄩󵄩󵄩T ∗−1 󵄩󵄩󵄩󵄩󵄩󵄩πWi T ∗ f 󵄩󵄩󵄩. Therefore, 󵄩 󵄩2 󵄩2 󵄩 ∑ wi2 ‖πTWi f ‖2 ≤ 󵄩󵄩󵄩T ∗−1 󵄩󵄩󵄩 ∑ wi2 󵄩󵄩󵄩πWi T ∗ f 󵄩󵄩󵄩 i∈I

i∈I

󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩T ∗−1 󵄩󵄩󵄩 𝔹󵄩󵄩󵄩T ∗ f 󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 ≤ 𝔹󵄩󵄩󵄩T ∗−1 󵄩󵄩󵄩 󵄩󵄩󵄩T ∗ 󵄩󵄩󵄩 ‖f ‖2 .

This completes the proof. Corollary 7.2.3. The dual fusion frame of fusion frame W = {(Wi , wi )}i∈I with 𝔸, 𝔹 frame bounds is a fusion frame with the frame bounds 𝔸 −1 ‖2 ‖SW ‖2 ‖SW

and

󵄩 −1 󵄩󵄩2 𝔹‖SW ‖2 󵄩󵄩󵄩SW 󵄩󵄩 .

−1 Proof. In Theorem 7.2.9, we take T = SW .

Corollary 7.2.4. Let {(Wi , wi )}i∈I be a fusion frame with frame bounds 𝔸, 𝔹, with U being a unitary operator on X. Then UW := {(UWi , wi )}i∈I is a fusion frame with frame bounds 𝔸, 𝔹 and frame operator USW U ∗ . Proof. For the first part, we apply Theorem 7.2.9 with T = U. For the second part, we apply Proposition 3.3.4 for f ∈ X as follows: SUW f = ∑ wi2 πUWi (f ) i∈I

= ∑ wi2 UπWi U ∗ (f ) i∈I

= USW U ∗ f , which completes the proof. We now give a form of the reconstruction formula with the help of the dual fusion frame. In view of Lemma 3.3.1, we have −1 −1 πWi SW = πWi SW πS−1 Wi . W

7.2 Frame of subspace or fusion frame

| 185

Hence, −1 −1 SW πWi = πS−1 Wi SW πWi . W

Consequently, the reconstruction formula has the following form: −1 f = ∑ wi2 πS−1 Wi SW πWi (f ), W

i∈I

f ∈ X.

(7.2.6)

This leads us to introduce the following definition. Definition 7.2.4. Let W = {(Wi , wi )}i∈I be a fusion frame and let SW be the frame operator. We consider also V = {(Vi , vi )}i∈I a Bessel fusion sequence. We say that V is an alternate dual of W if we have −1 f = ∑ wi vi πVi SW πWi (f ),

(7.2.7)

i∈I

for all f ∈ X. By the relation (7.2.6), we know the dual fusion frame of W is an alternate dual frame. We have also the following result. Proposition 7.2.2 (P. Gavruta [91, Proposition 2.8]). The alternate dual of a fusion frame is a fusion frame. Proof. In view of (7.2.7), we get −1 ‖f ‖2 = ∑ wi vi ⟨SW πWi (f ), πVi (f )⟩ i∈I

󵄩 −1 󵄩󵄩 󵄩 ≤ ∑ wi vi 󵄩󵄩󵄩SW πWi (f )󵄩󵄩󵄩󵄩󵄩󵄩πVi (f )󵄩󵄩󵄩 i∈I

≤

󵄩 −1 󵄩2 (∑ wi2 󵄩󵄩󵄩SW πWi (f )󵄩󵄩󵄩 ) i∈I

1 2

1

󵄩 󵄩2 2 (∑ vi2 󵄩󵄩󵄩πVi (f )󵄩󵄩󵄩 ) i∈I 1

󵄩 −1 󵄩󵄩√ 󵄩2 2 2󵄩 ≤ 󵄩󵄩󵄩SW 󵄩󵄩 𝔹‖f ‖(∑ vi 󵄩󵄩󵄩πVi (f )󵄩󵄩󵄩 ) , i∈I

where 𝔹 is the upper bound of the frame W. This completes the proof.

7.2.9 Fusion frames and operators In this section, we study the relationship between operators and fusion frames for a Hilbert space X. We first consider the behavior of fusion frames under a bounded linear operator with closed range.

186 | 7 Frames in Hilbert spaces Theorem 7.2.10. Let Ww be a fusion frame for X with fusion frame bounds 𝔸 and 𝔹 and let U : X 󳨀→ Y be a bounded operator with closed range such that U ∗ UWi ⊂ Wi for all i ∈ I. Then UWw = {(UWi , wi )}i∈I is a fusion frame sequence with fusion frame bounds 𝔸‖U † ‖−2 ‖U‖−2 and 𝔹‖U † ‖2 ‖U‖2 , respectively. Proof. Since U ∗ UWi ⊂ Wi , πUW U = UπWi . i

In view of (2.18.2), the operator UU † is the orthogonal projection onto R(U). Therefore, for every g ∈ R(U), we get 󵄩 󵄩2 󵄩2 󵄩 ∑ wi2 󵄩󵄩󵄩πUW (g)󵄩󵄩󵄩 = ∑ wi2 󵄩󵄩󵄩πUW UU † (g)󵄩󵄩󵄩 i i i∈I

i∈I

󵄩 󵄩2 = ∑ wi2 󵄩󵄩󵄩UπWi U † (g)󵄩󵄩󵄩 i∈I

󵄩 󵄩2 ≤ ‖U‖2 ∑ wi2 󵄩󵄩󵄩πWi U † (g)󵄩󵄩󵄩 i∈I

󵄩 󵄩2 ≤ 𝔹‖U‖2 󵄩󵄩󵄩U † 󵄩󵄩󵄩 ‖g‖2 . For the lower fusion frame condition, suppose that g ∈ R(U). Then equation (2.18.2) implies ∗

∗

g = (UU † ) (g) = (U † ) U ∗ (g). Moreover, in view of Lemma 3.3.1, we have πWi U ∗ = πWi U ∗ πUW . i

Hence, we get ∗ 󵄩 󵄩2 ‖g‖2 = 󵄩󵄩󵄩(U † ) U ∗ (g)󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩U † 󵄩󵄩󵄩 󵄩󵄩󵄩U ∗ (g)󵄩󵄩󵄩 ‖U † ‖2 󵄩 󵄩2 ≤ ∑ w2 󵄩󵄩π U ∗ (g)󵄩󵄩󵄩 𝔸 i∈I i 󵄩 Wi

= ≤ which completes the proof.

‖U † ‖2 󵄩 󵄩2 ∑ w2 󵄩󵄩π U ∗ πUW (g)󵄩󵄩󵄩 i 𝔸 i∈I i 󵄩 Wi ‖U‖2 ‖U † ‖2 󵄩 󵄩2 ∑ wi2 󵄩󵄩󵄩πUW (g)󵄩󵄩󵄩 , i 𝔸 i∈I

7.2 Frame of subspace or fusion frame

| 187

Corollary 7.2.5. Let Ww be a fusion frame for X with fusion frame bounds 𝔸 and 𝔹 and let P denote the orthogonal projection onto a closed subspace V such that PWi ⊂ Wi for all i ∈ I. Then PWw = {(PWi , wi )}i∈I is a fusion frame for V with fusion frame bounds 𝔸, 𝔹. Proof. This follows immediately from Theorem 7.2.10. Under the same assumptions as Theorem 7.2.10, if SWw and SUWw are fusion frame operators associated with Ww and UWw , respectively, then, for every f ∈ X, we have USWw (f ) = ∑ wi2 UπWi (f ) = ∑ wi2 πUW U(f ) = SUWw U(f ). i∈I

i

i∈I

(7.2.8)

Proposition 7.2.3 (P. G. Cassaza and G. Kutyniok [54, Proposition 3.22]). Let {Wi }i∈I be a family of subspaces in X and let {wi }i∈I be a family of weights. Then the following conditions are equivalent: (i) {Wi }i∈I is a Parseval frame of subspaces with respect to {wi }i∈I for X; and (ii) SWw = I. Corollary 7.2.6. With the same assumptions as in Theorem 7.2.10, if Ww is a Parseval fusion frame for X, then UWw is a Parseval fusion frame sequence in Y. Proof. This follows immediately from both (7.2.8) and Proposition 7.2.3. In the following result we restrict ourselves to finite fusion frames. Corollary 7.2.7. Let Ww be a fusion frame for X with fusion frame bounds 𝔸, 𝔹. If U ∈ ℒ(X, Y) is a bounded, surjective operator, then, for every g ∈ Y, we have 󵄩 󵄩−2 󵄩 󵄩2 𝔸 󵄩󵄩󵄩U † 󵄩󵄩󵄩 ‖U‖−2 ‖g‖2 ≤ ∑ wi2 󵄩󵄩󵄩πUW (g)󵄩󵄩󵄩 . i i∈I

Hence, if |I| < ∞, then UWw = {(UWi , wi )}i∈I is a fusion frame for Y. Proof. This follows from the arguments in Theorem 7.2.10. Corollary 7.2.8. Let Ww be a fusion frame for X with fusion frame bounds 𝔸, 𝔹. If U ∈ ℒ(Y, X) is a bounded, surjective operator, then U ∗ Ww = {(U ∗ Wi , wi )}i∈I is a fusion frame sequence for Y with fusion frame bounds 𝔸‖U † ‖−2 ‖U‖−2 and 𝔹‖U † ‖2 ‖U‖2 , respectively. Proof. This follows immediately from Theorem 7.2.9.

188 | 7 Frames in Hilbert spaces 7.2.10 Non-orthogonal fusion frame Now, we give a formal definition of non-orthogonal fusion frames, which are a modified type of fusion frames with a sequence of non-orthogonal projection operators. Definition 7.2.5. Let {Wi }i∈I be a sequence of closed subspaces in X and let {wi }i∈I be a family of weights, i. e., wi > 0 for all i ∈ I. For each i ∈ I, let PWi be a (non-orthogonal or orthogonal) projection onto Wi . Then we say that {PWi , wi }i∈I is a non-orthogonal fusion frame for X, if there exist constants 0 < 𝒜 ≤ ℬ < ∞ such that 2󵄩

2

󵄩2

2

𝒜‖f ‖ ≤ ∑ wi 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 ≤ ℬ‖f ‖ ,

for all f ∈ X.

i∈I

The numbers 𝒜, ℬ are called the non-orthogonal fusion frame bounds. Theorem 7.2.11. Let {PWi , wi }i∈I be a non-orthogonal fusion frame for X with frame bounds 𝒜 and ℬ, let {Zi }i∈I be a family of closed subspaces in X, and let {vi }i∈I be a family of weights such that 0 < wi ≤ vi ≤ √2wi . Suppose that there exists an 0 < R < 𝒜 such that 󵄩 󵄩2 ∑ vi2 󵄩󵄩󵄩PWi (f ) − PZi (f )󵄩󵄩󵄩 ≤ R‖f ‖2 ,

for all f ∈ X,

i∈I

where PWi (respectively PZi ) denotes the (orthogonal or non-orthogonal) projection onto Wi (respectively Zi ). Then {PZi , vi }i∈I is a non-orthogonal fusion frame with frame R 2 ) and ℬ(√2 + √ ℬR )2 . bounds 𝒜(1 − √ 𝒜

Proof. Fix f ∈ X. By using Minkowski’s inequality, we deduce that 1

󵄩 󵄩2 2 (∑ vi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 ) i∈I

≤

󵄩 (∑ vi2 󵄩󵄩󵄩PWi (f ) i∈I

1

1

󵄩 󵄩2 2 󵄩2 2 − PZi (f )󵄩󵄩󵄩 ) + (∑ vi2 󵄩󵄩󵄩PZi (f )󵄩󵄩󵄩 ) . i∈I

Thus, we get 1

1

1

󵄩 󵄩 󵄩 󵄩2 2 󵄩2 2 󵄩2 2 (∑ vi2 󵄩󵄩󵄩PZi (f )󵄩󵄩󵄩 ) ≥ (∑ vi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 ) − (∑ vi2 󵄩󵄩󵄩PWi (f ) − PZi (f )󵄩󵄩󵄩 ) i∈I

i∈I

≥

󵄩 󵄩2 (∑ wi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 ) i∈I

1 2

i∈I

−

󵄩 (∑ vi2 󵄩󵄩󵄩PWi (f ) i∈I

≥ (√𝒜 − √R)‖f ‖. Consequently, 2

R 󵄩 󵄩2 ∑ vi2 󵄩󵄩󵄩PZi (f )󵄩󵄩󵄩 ≥ 𝒜(1 − √ ) ‖f ‖2 . i∈I

𝒜

1

󵄩2 2 − PZi (f )󵄩󵄩󵄩 )

7.2 Frame of subspace or fusion frame

| 189

Similarly, we have 2

R 󵄩2 󵄩 ∑ vi2 󵄩󵄩󵄩PZi (f )󵄩󵄩󵄩 ≤ ℬ(√2 + √ ) ‖f ‖2 . ℬ

i∈I

Indeed, in view of Minkowski’s inequality, we have 1

1

1

󵄩2 2 󵄩 󵄩2 2 󵄩2 2 󵄩 󵄩 (∑ vi2 󵄩󵄩󵄩PZi (f )󵄩󵄩󵄩 ) ≤ (∑ vi2 󵄩󵄩󵄩PWi (f ) − PZi (f )󵄩󵄩󵄩 ) + (∑ vi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 ) . i∈I

i∈I

i∈I

(7.2.9)

Further, we have 󵄩2 󵄩2 󵄩2 󵄩 󵄩 󵄩 ∑ vi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 = ∑(vi2 − wi2 )󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 + ∑ wi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 . i∈I

i∈I

i∈I

Since wi ≤ vi ≤ √2wi , we get 󵄩 󵄩2 ∑ vi2 󵄩󵄩󵄩PWi (f )󵄩󵄩󵄩 ≤ 2ℬ‖f ‖2 , i∈I

so, from equation (7.2.9), we obtain 1

󵄩 󵄩2 2 (∑ vi2 󵄩󵄩󵄩PZi (f )󵄩󵄩󵄩 ) ≤ (√R + √2ℬ)‖f ‖. i∈I

Consequently, {PZi , vi }i∈I is a non-orthogonal fusion frame with frame bounds 𝒜(1 − R 2 √𝒜 ) and ℬ(√2 + √ ℬR )2 .

8 Summability of series 8.1 Series in principal vectors We consider a linear compact operator K acting in a Hilbert space X. It is well known that the spectrum of a compact operator is discrete. Let λ1 , λ2 , . . . be the characteristic numbers (i. e., the reciprocals of the non-zero eigenvalues of K) of the operator K and let e1 , e2 , . . . , es , . . .

(8.1.1)

be the corresponding eigenvectors and associated vectors (called principal vectors). The question of the completeness of system (8.1.1), in the range of the operator K, has already been studied in Chapter 6. It should be observed, however, that, since the system of principal vectors is not orthogonal, it does not follow from the completeness of this system, i. e., the possibility of approximating any vector f of the form f = KΦ by linear combinations of its elements, that f can be expanded in a Fourier series in the elements of this system. Moreover, the formal Fourier series f ∼ ∑ cs es ,

(8.1.2)

generally speaking, diverges. In this section, we deduce under certain conditions on the operator K that the Fourier series (8.1.2) is summable to the vector f by Abel’s method.

8.2 Jordan chain Let μ1 be a non-zero eigenvalue of a compact operator K acting in a Hilbert space X. By the Riesz theorem, the space X can be decomposed into the direct sum of two subspaces which are invariant with respect to the operator K. We have X = R1 + R1 ,

(8.2.1)

where R1 is a finite-dimensional (root) subspace, consisting of all vectors f which are mapped into zero by some power of the operator K − μ1 . The subspace R1 has the property that the operator K − μ1 , when considered on it, has a bounded inverse. Let m1 be the dimension of R1 and let K (1) be the operator generated by K in the invariant subspace R1 . One can choose a basis R1 consisting of Jordan chains of eigenvectors and associated vectors of the operator K (1) (a Jordan basis). Every such chain https://doi.org/10.1515/9783110493863-008

192 | 8 Summability of series e(1) , e(2) , . . . , e(p) is obviously a Jordan chain for the operator K and is transformed by K by the following formulas: Ke(1) = μ1 e(1) ,

Ke(2) = μ1 e(2) + e(1) ,

Let 0 ≠ λ ∈ ℂ such that

1 λ

...,

Ke(p) = μ1 e(p) + e(p−1) .

(8.2.2)

∈ ρ(K). We have

K(I − λK)−1 =

1 1 1 [(I − λK)−1 − I] = 2 [( − K) λ λ λ

−1

− λ].

(8.2.3)

8.3 Summation of series by Abel’s method Let T be an operator with a dense domain and a discrete spectrum in a Hilbert space X. In the following, {Hn := ⋃ N((T − λn )m )}, m≥1

n = 1, 2, . . . ,

are used to denote the system of all root subspaces of the operator T which are associated with the eigenvalues numbered in increasing order and repeated in accordance with their multiplicities. The sequence {Φn }, n = 1, 2, . . . , denotes the system of root vectors of the operator T, obtained by consecutive numbering of the basis of the subspaces {Hn }, for n = 1, 2, . . . , composed from the Jordan sequences. The Fourier coefficients with respect to the system {Φn }, n = 1, 2, . . . , are defined for any vector Φ ∈ X by cn =

⟨Φ, Φ∗n ⟩ , ⟨Φn , Φ∗n ⟩

where the system {Φ∗n }, n = 1, 2, . . . , is the system of root vectors of the operator T ∗ . Even if the system {Φn }, n = 1, 2, . . . , is complete, it is well known that it is not possible to say anything about the convergence of the Fourier series ∞

∑ cn Φn

n=1

to the vector Φ. Let α be a strictly positive real number. We introduce the following polynomials with respect to the real parameter t: etξ dm −tξ −α α 1 Pm ( , t) = e , ξ m! dξ m −α

and we compare the series ∞

∑ cn Φn

n=1

(m = 0, 1, . . .),

(8.3.1)

8.4 Coefficients of a series in principal vectors |

193

with the series ∞

∑ cn (t)Φn

(8.3.2)

n=1

whose coefficients are defined in the following way. If Φn is an eigenvector of the operator T associated with the eigenvalue λn without a Jordan chain, we set α

cn (t) = e−tλn cn ,

(8.3.3)

where cn =

⟨Φ, Φ∗n ⟩ . ⟨Φn , Φ∗n ⟩

If {Φn , Φ1n , . . . , Φsn },

s ≥ 0,

is a Jordan chain corresponding to the eigenvalue λn , then we set α

s−j

α cn+j (t) = e−tλn ∑ Pm (λn , t)cn+j+m m=0

(0 ≤ j ≤ s).

(8.3.4)

Definition 8.3.1. We say that the series ∞

∑ cn Φn

n=1

is summable to Φ by Abel’s method of order α if there exists a sequence of integers m0 , m1 , . . . , mn , . . . , with m0 = 1 such that, for any t > 0, the series ∞ mn −1

u(t) = ∑ ∑ cj (t)Φj n=1 j=mn−1

(8.3.5)

converges and lim u(t) = Φ.

t→0+

8.4 Coefficients of a series in principal vectors Let K be a compact operator acting in a Hilbert space X. We denote by G the region in the complex λ-plane obtained by removing from the sector −

π π − ε < arg λ < +ε 2ρ 2ρ

(8.4.1)

194 | 8 Summability of series

Bν+1 Bν

0

π 2ρ

π 2ρ

+ε γν

γ

Aν Aν+1 Figure 8.1

a neighborhood of the origin so small that the function (I − λK)−1 f is regular in this neighborhood (ε in (8.4.1) is a sufficient small positive number). Further, we denote the (infinite) contour bounding G (cf. Figure 8.1) by γ and consider the integral u(t) =

α 1 ∫ e−λ t K(I − λK)−1 fdλ. 2πi

(8.4.2)

γ

By virtue of the estimate (2.16.3), if, in equation (8.4.1) defining the position of the contour γ, we choose ε according to the conditions π + ε < π, 2ρ

π π +ε < , 2ρ 2α

(8.4.3)

8.4 Coefficients of a series in principal vectors |

195

then the integral (8.4.2) converges for all t > 0. Let λi be the pole of the resolvent (I − λK)−1 . We consider the integral Ri =

α 1 ∫ e−λ t K(I − λK)−1 fdλ, 2πi

(8.4.4)

γi

where the contour γi does not contain any poles of the resolvent other than λi and is traversed in the clockwise direction. Let Ri be the root subspace corresponding to λi and let es1 , es1 +1 , . . . , es2

(8.4.5)

be a Jordan basis in Ri . Decomposing the Hilbert space X into a direct sum Ri + Ri (cf. (8.2.1)), we represent f as f = f1 + f2 , where f1 ∈ Ri and f2 ∈ Ri . Since the function α

e−λ t K(I − λK)−1 f2 is regular inside the contour γi , the integral (8.4.4) reduces to the following: Ri =

α 1 ∫ e−λ t K(I − λK)−1 f1 dλ. 2πi

γi

Further, setting for convenience λ = ξ −1 in the last integral and using equation (8.2.3), we obtain Ri = −

−α 1 ∫ e−ξ t (ξ − K)−1 f1 dξ . 2πi

(8.4.6)

γ̃i

Supposing that condition (8.4.3) is fulfilled, we prove the following lemma. Lemma 8.4.1 (V. B. Lidskii [153, Lemma 5]). If f = KΦ, then α

lim u(t) = lim+ ∫ e−λ t K(I − λK)−1 fdλ = f .

t→0+

t→0

γ

Proof. We note that (I − λK)−1 K 2 = λ−2 [(I − λK)−1 − (I + λK)] and, therefore, inserting KΦ in place of f in the integrand of (8.4.2), we obtain lim+ u(t) =

t→0

1 (I − λK)−1 Φ dλ. ∫ 2πi λ2 γ

196 | 8 Summability of series The second integral on the right equals zero, since the integrand is regular in the intersection of the region G with the disc |λ| < R and tends to zero sufficiently fast in the sector under consideration as R → ∞. As for the first integral, by virtue of the estimate (2.16.3) it converges uniformly in t for t ≥ 0. As a consequence, u(t) =

−1 α (I − λK) α (I + λK)Φ 1 Φ 1 dλ − dλ. ∫ e−λ t ∫ e−λ t 2 2πi 2πi λ λ2 γ

γ

This integral obviously equals the residue of the integrand at λ = 0. Combining this residue, we easily find that lim u(t) = KΦ = f .

t→0+

We now proceed to find the residues of the integrand of (8.4.2) at the poles of the resolvent. Lemma 8.4.2 (V. B. Lidskii [153, Lemma 6]). The residue of the function α

e−λ t K(I − λK)−1 f at the pole λi of the resolvent (I − λK)−1 equals s2

− ∑ cs (t)es , s=s1

(8.4.7)

where es1 , es1 +1 , . . . , es2 is a Jordan basis for the operator K in the root subspace Ri and cs (t) are coefficients computed by equations (8.3.3) and (8.3.4). Proof. We consider the integral (8.4.4). The Hilbert space X can be decomposed into a direct sum Ri + Ri (cf. (8.2.1)), where Ri is the root subspace corresponding to λi . Let f ∈ X. We may write f = f1 + f2 , where f1 ∈ Ri and f2 ∈ Ri . We expand f1 in the basis (8.4.5) s2

f1 = ∑ cs es s=s1

and note that the coefficients cs coincide with the corresponding coefficients of the series (8.3.2). Let eq , eq+1 , . . . , eq+k

8.4 Coefficients of a series in principal vectors |

197

be a Jordan chain of the basis (8.4.5). We separate the part of the integral (8.4.6) corresponding to this chain. Using equation (8.2.2), it is easy to show that τ

1 e −1 )τ−j+1 q+j (ξ − λ j=0 i

(ξ − K)−1 eq+τ = ∑

(0 ≤ τ ≤ k).

Therefore, using a well-known expression for products of analytic functions, we obtain τ k k −α −α dξ 1 1 e ∫ e−ξ t (ξ − K)−1 ∑ cq+τ eq+τ dξ = ∑ cq+τ ∑ ∫ e−ξ t −1 )τ−j+1 q+j 2πi 2πi (ξ − λ τ=0 τ=0 j=0 i γ̃i

γ̃i

k

τ

1 dτ−j −ξ −α t (e )|ξ =λ−1 eq+j . i (τ − j)! dξ τ−j j=0

= ∑ cq+τ ∑ τ=0

Changing the order of summation and setting τ − j = m, we transform the given expression to the following form: k

k−j

1 dm −ξ −α t (e )|ξ =λ−1 cq+j+m ]eq+j . i m! dξ m m=0

∑[ ∑

j=0

This formula leads to the same expressions for the coefficients cq+j (t) as equations (8.3.1) and (8.3.4). Taking into consideration that the integral (8.4.4) is taken in a clockwise direction, we arrive at equation (8.4.7). Lemma 8.4.3 (V. B. Lidskii [153, Lemma 7]). Let K be a compact operator in a Hilbert space X. Suppose that K belongs to the Carleman class Cp for some p > 0 and suppose that the values of the quadratic form ⟨KΦ, Φ⟩ lie in a sector Sρ of the complex z-plane defined by 󵄨 󵄨 π Sρ := {z ∈ ℂ such that 󵄨󵄨󵄨arg(z)󵄨󵄨󵄨 ≤ }, 2ρ with ρ > max(p, 21 ). Then u(t) =

N

∞ ν+1 α 1 ∫ e−λ t K(I − λK)−1 fdλ = ∑ ( ∑ cs (t)es ), 2πi ν=1 s=N +1 γ

(8.4.8)

ν

where the integral is taken along the contour γ defined by equations (8.4.3) and (8.4.1). Here, the series on the right converges absolutely, i. e., 󵄩󵄩 ∞󵄩 󵄩󵄩 Nν+1 󵄩󵄩 󵄩 ∑ 󵄩󵄩󵄩 ∑ cs (t)es 󵄩󵄩󵄩 < ∞. 󵄩 󵄩󵄩 ν=1󵄩 󵄩s=Nν +1 󵄩

(8.4.9)

198 | 8 Summability of series Proof. We represent the resolvent according to equation (3.6.2). We have fλ =

DK (λ) . Δ K (λ)

By Theorem 3.6.2, the estimates (3.6.4) are valid for the functions Δ K (λ) and DK (λ). By well-known theorems of the theory of functions, the function Δ K (λ) can be estimated from below. Let R be some positive number and 1 > δ > 0. Then, in the ring R > |λ| > (1 − δ)R, we find a circle ̃ |λ| = R,

(8.4.10)

on which we have the following inequality: 󵄨󵄨 󵄨 − log{max |Δ K (2eReiψ )|}(2+log 12e ) δ 󵄨󵄨Δ K (λ)󵄨󵄨󵄨 > e

(0 ≤ ψ ≤ 2π).

Together with estimates (3.6.3) for Δ K (λ) and DK (λ), we now obtain, on the circle (8.4.10), the following estimate for the norm of the resolvent: 󵄩󵄩 ) −1 󵄩 ε(λ)|λ|p (2+log 12e δ , 󵄩󵄩(I − λK) 󵄩󵄩󵄩 ≤ e

(8.4.11)

where ε(λ) → 0 under the condition that |λ| → ∞ in the system of circles of the form of (8.4.10). Fixing R and δ, we consider the sequence of numbers Rν = R(1 − δ)−ν+1

(ν = 0, 1, 2, . . .),

which tends to infinity. By the theorem cited above, in each ring Rν < |λ| < Rν+1 ̃ ν , on which inequality (8.4.11) will hold. If we denote by γν the one finds a circle |λ| = R boundary of the intersection of the ring ̃ ν ≤ |λ| ≤ R ̃ ν+1 R with the region G, then γν consists of two arcs of the circles ̃ν |λ| = R and ̃ ν+1 |λ| = R

8.4 Coefficients of a series in principal vectors |

199

and two segments of the rays which bound G (see Figure 8.1). If we denote by Nν the number of poles of the resolvent (I − λK)−1 (taking multiplicity into consideration) situated in the part of G which is cut off by the arc Aν Bν , then, by Lemma 8.4.2, we will have N

ν+1 α 1 ∫ e−λ t K(I − λK)−1 fdλ = ∑ cs (t)es . 2πi s=N +1

γν

ν

Here, the integral is taken along the contour Aν Bν Bν+1 Aν+1 in a clockwise direction. Let us estimate this integral. To do so, we split it into four parts and estimate separately each of the integrals thus obtained. For the integral along the arc Aν Bν , we have 󵄩󵄩 1 󵄩󵄩 α 󵄩 󵄩 ‖Iν ‖ := 󵄩󵄩󵄩 ∫ e−λ t K(I − λK)−1 fdλ󵄩󵄩󵄩 󵄩󵄩 2πi 󵄩󵄩 Aν Bν

≤ |λ|e

−t Re λα ε(λ)|λ|p (2+log

e

12e ) δ

‖f ‖.

Since, by condition (8.4.3), we have −

π π < arg λ < 2α 2α

on this arc, we can find w0 > 0 such that |arg λ| ≤

π − w0 . 2α

As a consequence, we obtain Re λα ≥ |λ|α cos[(

π − w0 )α] = |λ|α sin(αw0 ). 2α

Further, taking into account that ̃ ν < Rν+1 = Rν (1 − δ)−1 , Rν < R we have α

p

‖Iν ‖ ≤ Rν (1 − δ)−1 e−tRν sin(αw0 ) eε(Rν )Rν (2+log ̃

12e )(1−δ)−p δ

‖f ‖.

(8.4.12)

A similar estimate is obviously valid for the integral Iν+1 taken along the arc Aν+1 Bν+1 . For the integrals Jν and Jν󸀠 along the segment Aν Aν+1 and Bν Bν+1 we have, by (2.16.3), |Aν+1 |

‖Jν ‖ ≤

α Rν 1 󵄨 α 󵄨 ‖f ‖ |dλ| ≤ e−tRν sin(w0 α) [(1 − δ)−2 − 1]‖f ‖ ∫ 󵄨󵄨󵄨e−λ t 󵄨󵄨󵄨 2π sin ϕ 2π sin ϕ

|Aν |

(8.4.13)

200 | 8 Summability of series and the same is true for Jν󸀠 . It is easy to see that, in view of inequalities (8.4.12) and (8.4.13), the series ∞

󵄩 󵄩 ∑ (‖Iν ‖ + ‖Iν+1 ‖ + ‖Jν ‖ + 󵄩󵄩󵄩Jν󸀠 󵄩󵄩󵄩)

ν=1

̃ ν ) → 0). Thereconverges for t > 0 (we recall that, in equation (8.4.12), α ≥ p and ε(R fore, the series (8.4.9) also converges. It is quite obvious that relation (8.4.8) also follows from our considerations, so the lemma is completely proved. Theorem 8.4.1 (V. B. Lidskii [153, Theorem 3]). Let K be a compact operator in a Hilbert space X. Suppose that K belongs to the Carleman class Cp for some p > 0 and suppose that the values of the quadratic form ⟨Kφ, φ⟩ lie in a sector Sρ of the complex z-plane defined by 󵄨 󵄨 π Sρ := {z ∈ ℂ such that 󵄨󵄨󵄨arg(z)󵄨󵄨󵄨 ≤ }, 2ρ with ρ > max(p, 21 ). Then, for every f in the range of the operator K, the corresponding Fourier series with respect to the system of root vectors of K is summable by Abel’s method of order α to the vector f , where p ≤ α < ρ. Proof. Let G be the region in the complex λ-plane obtained by removing from the sector (8.4.1) a neighborhood of the origin so small that the function (I − λK)−1 f is regular in this neighborhood (ε in (8.4.1) is a sufficient small positive number). Further, we denote the (infinite) contour bounding G (cf. Figure 8.1) by γ and consider the integral (8.4.2). The direction of integration is chosen so that G remains to the right of the contour. The rest of the proof of the theorem is connected with the calculation of the integral (8.4.2) by the theory of residues. We break it up into four parts. First, by Lemma 2.16.1, we show that the resolvent (I − λK)−1 is bounded on the contour γ and, therefore, the integral (8.4.2) exists for any t > 0. Using Lemma 8.4.1, we see the integral tends to f when t → 0+ . By Lemma 8.4.2, the expressions for the residues of the integrand at the poles of the resolvent yield polynomials of the form of (8.3.1). Finally, by Lemma 8.4.3, there exists a sequence of contours which stretches to infinity, on which the estimate 󵄩󵄩 −1 󵄩 (o|λ|α ) 󵄩󵄩(I − λK) 󵄩󵄩󵄩 = e holds and which allows us to represent the integral (8.4.2) as a converging sum of residues, i. e., to obtain a formula of the form (8.3.5). In view of Lemma 8.4.1, this fact leads us to the proof of the theorem.

9 ν-convergence operators 9.1 ν-convergence and spectral properties 9.1.1 Semi-Fredholm sequence For T ∈ ℒ(X) and k∈ℕ∪{−∞, ∞}, let ρks−F (T) denote the set of λ∈ℂ for which λ−T ∈Φ± (X) and i(λ − T) = k. Put ρ+s−F (T) = ⋃ ρks−F (T) 1≤k≤∞

and ρ−s−F (T) =

⋃

ρks−F (T).

−∞≤k≤−1

Theorem 9.1.1 (S. Sánchez-Perales and S. V. Djordjevic̀ [177, Theorem 3.4]). Let T ∈ ν Φb (X) and let {Tn }n be a sequence in ℒ(X) such that Tn → T. Then, for any integer −∞ ≤ k ≤ ∞ and any λ ∈ ρks−F (T), there exists n0 ∈ ℕ such that λ ∈ ρks−F (Tn ), for all n ≥ n0 . ν

Proof. Since Tn → T, it is easy to see that (Tn − T)Tn → 0 implies (λ − Tn )T → (λ − T)T, for any λ ∈ ℂ. Suppose that λ ∈ ρks−F (T), −∞ ≤ k ≤ ∞. Then (λ − T)T is a semi-Fredholm operator. Case 1. Let k be an integer. Then (λ − T)T ∈ Φb (X). By Theorem 2.3.4, there exists n0 ∈ ℕ such that (λ − Tn )T ∈ Φb (X) https://doi.org/10.1515/9783110493863-009

202 | 9 ν-convergence operators and i((λ − Tn )T) = i((λ − T)T) for all n ≥ n0 . Moreover, from Theorem 2.3.3 (i) and Theorem 2.3.3 (ii), we know λ − Tn ∈ Φb (X). Thus, by Theorem 2.3.2, it follows that i(λ − Tn ) + i(T) = i(λ − T) + i(T) = k + i(T) and, hence, i(λ − Tn ) = k for all n ≥ n0 . Case 2. Let k = ∞. Then α(λ − T) = ∞, β(λ − T) = ∞,

and λ − T ∈ Φb− (X). Since β(T) < ∞, by Theorem 2.3.3 (ii), it follows that α((λ − T)T) = ∞. Clearly, in view of Theorem 2.3.2, we have (λ − T)T ∈ Φb− (X) and, consequently, by Theorem 2.3.5, there is a positive integer n0 such that, for each n ≥ n0 , (λ − Tn )T ∈ Φb− (X) and α((λ − Tn )T) = ∞. Again, from Theorem 2.3.3 (ii), β(λ − Tn ) < ∞. If α(λ − Tn ) < ∞, then (λ − Tn )T ∈ Φb (X), which is a contradiction. Hence, λ ∈ ρ∞ s−F (Tn ). Case 3. Let k = −∞. Then λ − T ∈ Φb+ (X), α(λ − T) < ∞, and β(λ − T) = ∞. Moreover, (λ − T)T ∈ Φb+ (X)

9.1 ν-convergence and spectral properties | 203

and, by Theorem 2.3.3 (i), β((λ − T)T) = ∞. Again, by Theorem 2.3.5, there is a positive integer n0 such that, for each n ≥ n0 , (λ − Tn )T ∈ Φb+ (X) and i((λ − Tn )T) = i((λ − T)T) = −∞. Additionally, (λ − Tn )T ∈ Φb+ (X) and T ∈ Φb (X) implies λ − Tn ∈ Φb+ (X) with i(λ − Tn ) = −∞. Hence, λ ∈ ρ−∞ s−F (Tn ). Theorem 9.1.2 (S. Sánchez-Perales and S. V. Djordjevic̀ [177, Theorem 3.5]). Let T ∈ ν Φb (X). If {Tn }n is a sequence in ℒ(X) such that Tn → T, then ⋃

k∈ℤ∗ ⋃{±∞}

ρks−F (T) ⊂ lim inf σ(Tn ).

Proof. Let λ ∈ ρks−F (T) for some k ∈ ℤ∗ ∪ {±∞}. Suppose that λ ∉ lim inf σ(Tn ). Then there exists an increasing sequence of natural numbers n1 < n2 < ⋅ ⋅ ⋅ such that λ ∉ σ(Tni ) for all i ∈ ℕ∗ . This implies that λ ∈ ρ0s−F (Tni ) ν

for every i ∈ ℕ∗ . On the other hand, since Tni → T, it follows, by Theorem 9.1.1, that there exists i0 ∈ ℕ such that λ ∈ ρks−F (Tni ) for all i ≥ i0 . Therefore, λ ∈ ρ0s−F (Tni ) ∩ ρks−F (Tni ) for all i ≥ i0 , which is a contradiction. Consequently, ρks−F (T) ⊂ lim inf σ(Tn ) and, by Theorem 2.17.2, ρks−F (T) ⊂ lim inf σ(Tn ).

204 | 9 ν-convergence operators ν

Theorem 9.1.3. Let T ∈ Φ(X). If {Tn }n is a sequence in ℒ(X) such that Tn → T, then ρ+s−F (T) ⊂ lim inf σap (Tn ). Remark 9.1.1. For T ∈ Φ(X), we have σ(T)\σap (T) ⊂ lim inf σ(Tn ) for all sequences {Tn }n in ℒ(X) such that ν

Tn → T. This follows from Theorem 9.1.2, since the set ⋃

ρks−F (T)

−∞≤k≤−1

contains σ(T)\σap (T). 9.1.2 Convergence of the spectrum Let T be an operator satisfying the following condition: 1 󵄩󵄩 −1 󵄩 , 󵄩󵄩(λ − T) 󵄩󵄩󵄩 ≤ dist(λ, σ(T))

(9.1.1)

where λ∈ρ(A). Equation (9.1.1) holds for operators T which are normal (T ∗ T −TT ∗ = 0), subnormal (T has a normal extension), or hypernormal (T ∗ T − TT ∗ ≥ 0). Theorem 9.1.4 (S. Sánchez-Perales and S. V. Djordjevic̀ [177]). Let T, Tn , n ∈ ℕ, be ν

bounded operators that satisfy condition (9.1.1). If Tn → T and 0 is an accumulation point of σ(T), then σ(Tn ) → σ(T). Proof. Let λ ∈ σ(T)\{0} and suppose that λ ∉ lim inf σ(Tn ). Then there exist ε > 0 and an increasing sequence of natural numbers n1 < n2 < n3 < ⋅ ⋅ ⋅ such that, for every k ∈ ℕ, 𝔹(λ, ε) ∩ σ(Tnk ) = 0. Take k ∈ ℕ∗ . Then ε < dist(λ, σ(Tnk )) = min{|λ − s| s ∈ σ(Tnk )}

= min{|μ| μ ∈ σ(λ − Tnk )} = dist(0, σ(λ − Tnk )).

9.1 ν-convergence and spectral properties | 205

Therefore, 0 ∉ σ(λ − Tnk ), so

1 dist(0, σ(λ − Tnk )) 1 < . ε

󵄩󵄩 −1 󵄩 󵄩󵄩(λ − Tnk ) 󵄩󵄩󵄩 ≤

On the other hand, 2

[(T − Tnk )(λ − Tnk )−1 ] = (T − Tnk )(λ − Tnk )−1 (T − Tnk )(λ − Tnk )−1 1 = (T − Tnk ) [Tnk (λ − Tnk )−1 + I](T − Tnk )(λ − Tnk )−1 λ 1 = [(T − Tnk )Tnk (λ − Tnk )−1 (T − Tnk ) λ +(T − Tnk )T − (T − Tnk )Tnk ](λ − Tnk )−1 . Consequently, 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 −1 2 󵄩 [󵄩(T − Tnk )Tnk 󵄩󵄩󵄩󵄩󵄩󵄩(λ − Tnk )−1 󵄩󵄩󵄩‖T − Tnk ‖ 󵄩󵄩[(T − Tnk )(λ − Tnk ) ] 󵄩󵄩󵄩 ≤ |λ| 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 +󵄩󵄩󵄩(T − Tnk )T 󵄩󵄩󵄩 + 󵄩󵄩󵄩(T − Tnk )Tnk 󵄩󵄩󵄩]󵄩󵄩󵄩(λ − Tnk )−1 󵄩󵄩󵄩 1 1 󵄩󵄩 󵄩 [󵄩(T − Tnk )Tnk 󵄩󵄩󵄩‖T − Tnk ‖ ≤ |λ| 󵄩 ε 󵄩 󵄩 󵄩 󵄩 1 +󵄩󵄩󵄩(T − Tnk )T 󵄩󵄩󵄩 + 󵄩󵄩󵄩(T − Tnk )Tnk 󵄩󵄩󵄩] . ε

(9.1.2)

ν

Since Tn → T, it follows that 󵄩󵄩 󵄩 󵄩󵄩(T − Tnk )T 󵄩󵄩󵄩 → 0, 󵄩󵄩 󵄩 󵄩󵄩(T − Tnk )Tnk 󵄩󵄩󵄩 → 0,

and {‖Tnk ‖}k , {‖T − Tnk ‖}k are bounded. Therefore, the right-hand side of (9.1.2) tends to 0. Thus, there exists k ∗ ∈ ℕ such that and, hence,

󵄩󵄩 −1 2 󵄩 󵄩󵄩[(T − Tnk∗ )(λ − Tnk∗ ) ] 󵄩󵄩󵄩 < 1 ρ((Tnk∗ − T)(λ − Tnk∗ )−1 ) < 1,

which implies that λ ∈ ρ(T) and this is a contradiction. This shows that σ(T)\{0} ⊂ lim inf σ(Tn ). Now, since 0 is an accumulation point of σ(T), σ(T) = σ(T)\{0} ⊂ lim inf σ(Tn ) ⊂ lim inf σ(Tn ).

206 | 9 ν-convergence operators 9.1.3 ν-convergence Now, let us study some basic properties of the ν-convergence. Lemma 9.1.1 (M. Ahues, A. Largillier and B. V. Limaye [11, Lemma 2.2]). If Un →U, then ν

ν

Un → U. Conversely, if 0 ∈ ρ(U) and Un → U, then Un →U. Proof. Let Un →U. Since

‖Un ‖ ≤ ‖Un − U‖ + ‖U‖, 󵄩󵄩 󵄩 󵄩󵄩(Un − U)U 󵄩󵄩󵄩 ≤ ‖Un − U‖‖U‖, and 󵄩󵄩 󵄩 󵄩󵄩(Un − U)Un 󵄩󵄩󵄩 ≤ ‖Un − U‖‖Un ‖, it follows that ν

Un → U. ν

Conversely, let 0 ∈ ρ(U) and Un → U. Then U is invertible and 󵄩 󵄩 ‖Un − U‖ = 󵄩󵄩󵄩(Un − U)UU −1 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩(Un − U)U 󵄩󵄩󵄩󵄩󵄩󵄩U −1 󵄩󵄩󵄩, so that Un →U. Lemma 9.1.2. If ‖(Tn − T)T‖ → 0 and ‖(Tn − U)U‖ → 0, then, for every x ∈ R(T) ∩ R(U), Tx = Ux. Proof. Let x ∈ R(T) ∩ R(U). Then there exist y, z ∈ X such that x = Ty = Uz and 󵄩 󵄩 0 ≤ ‖Ux − Tx‖ = 󵄩󵄩󵄩U 2 z − T 2 y󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩U 2 z − Tn Uz + Tn Ty − T 2 y󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(Tn − U)Uz 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Tn − T)Ty󵄩󵄩󵄩 → 0 which completes the proof.

as n → ∞,

9.1 ν-convergence and spectral properties | 207 ν

ν

Lemma 9.1.3. If Tn → T, Tn → U, Tn T = TTn , and Tn U = UTn , then T 2 = U 2. Proof. Let x ∈ X. Then 󵄩 󵄩 󵄩 󵄩 0 ≤ 󵄩󵄩󵄩T 2 x − U 2 x󵄩󵄩󵄩 = 󵄩󵄩󵄩T 2 x − Tn Tx + Tn Tx − Tn2 x + Tn2 x − Tn Ux + Tn Ux − U 2 x 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(Tn − T)Tx󵄩󵄩󵄩 + 󵄩󵄩󵄩(Tn − T)Tn x 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Tn − U)Tn x 󵄩󵄩󵄩 󵄩 󵄩 +󵄩󵄩󵄩(Tn − U)Ux󵄩󵄩󵄩 → 0 as n → ∞, which completes the proof. Remark 9.1.2. Let Tn , n ∈ ℕ, be operators defined on ℂ ⊗ ℂ as 0 0

T0 = (

0 ) 0

and 0 0

Tn = (

1 n) ,

n ∈ ℕ∗ .

0

Then ‖Tn ‖ ≤

1 n

for all n ≥ 1. Thus, Tn → T0 and, hence, ν

Tn → T0 . On the other hand, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Tn − T1 )T1 󵄩󵄩󵄩 = 0 = 󵄩󵄩󵄩(Tn − T1 )Tn 󵄩󵄩󵄩. Therefore, ν

Tn → T1 . Observe that T1 ≠ T0 . These operators clearly satisfy Tn T1 = T1 Tn and Tn T0 = T0 Tn .

208 | 9 ν-convergence operators ν

Lemma 9.1.4 (M. Ahues, A. Largillier and B. V. Limaye [11, Lemma 2.2]). Let Un → U ν

and Vn →V. Then Un + Vn → U + V if and only if (Un − U)V→0. In particular, ν

ν

(i) if Un → U and Vn → 0, then Un + Vn → U; and ν

ν

(ii) if Un → 0, Vn → V, and Un V → 0, then Un + Vn → V. Proof. Since ‖Un + Vn ‖ ≤ ‖Un ‖ + ‖Vn ‖, we see that the sequence (‖Un + Vn ‖)n is bounded. Assume that (Un − U)V→0. As 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Un + Vn − U − V)(U + V)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(Un − U)U 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Un − U)V 󵄩󵄩󵄩 + ‖Vn − V‖‖U + V‖, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Un + Vn − U − V)(Un + Vn )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(Un − U)Un 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Un − U)Vn 󵄩󵄩󵄩 + ‖Vn − V‖(‖Un ‖ + ‖Vn ‖), where 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Un − U)Vn 󵄩󵄩󵄩 ≤ ‖Un − U‖‖Vn − V‖ + 󵄩󵄩󵄩(Un − U)V 󵄩󵄩󵄩. We see that ν

Un + Vn → U + V. ν

Conversely, assume that Un + Vn → U + V. Since (Un − U)V = (Un + Vn − U − V)(U + V) − (Un − U)U − (Vn − V)(U + V), we obtain (Un − U)V→0. The particular cases (i) and (ii) follow easily. ν

Lemma 9.1.5 (M. Ahues, A. Largillier and B. V. Limaye [11, Exercise 2.12]). If Un → U ν

and Un → V, then

σ(U) = σ(V).

9.2 Jeribi, Wolf, and Weyl essential spectra ν

ν

Note that ν-convergence is a pseudo-convergence, i. e., if Un → U and Un → V, then not necessarily U = V. In fact, it suffices to consider the following examples. Let X be a Banach space, C ∈ ℒ(X), An , Bn ∈ ℒ(X), and 𝒰n be a sequence of operators defined on X ⊗ X by An 0

𝒰n = (

0 ) Bn

9.2 Jeribi, Wolf, and Weyl essential spectra

| 209

and let 0 0

C ). 0

𝒱=(

If An → 0 and Bn → 0, then ‖𝒰n ‖ = max{‖An ‖, ‖Bn ‖} → 0, so 0 0

𝒰n → 𝒰 = (

0 ). 0

This implies that ν

𝒰n → 𝒰 .

In other words, using the fact that 2

𝒱 = 0,

the use of [11, Exercise 2.4] makes us conclude that ν

𝒰n → 𝒱 .

However, 𝒰 ≠ 𝒱 .

Theorem 9.2.1. If T, U ∈ ℒ(X) are operators with closed range and {Tn }n is a sequence ν

ν

in ℒ(X) such that Tn → T and Tn → U, then

σp (T)\{0} = σp (U)\{0}. Proof. Let λ ∈ σp (T)\{0}. Then Tx = λx for some x ≠ 0. We prove that x ∈ R(T) ∩ R(U). From (Tn − T)T → 0, we have Tn x → λx,

210 | 9 ν-convergence operators so TTn x → λTx. Therefore, 󵄩 󵄩 󵄩 󵄩 ‖UTn x − λTx‖ ≤ 󵄩󵄩󵄩UTn x − Tn2 x󵄩󵄩󵄩 + 󵄩󵄩󵄩Tn2 x − TTn x 󵄩󵄩󵄩 + ‖TTn x − λTx‖ 󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩(Tn − U)Tn x󵄩󵄩󵄩 + 󵄩󵄩󵄩(Tn − T)Tn x 󵄩󵄩󵄩 + ‖TTn x − λTx‖ → 0

as n → ∞.

Thus, λ2 x = λTx ∈ R(U) = R(U) and, hence, x ∈ R(T) ∩ R(U). On the other hand, by Lemma 9.1.2, (λ − U)x = (λ − T)x = 0, which implies that λ ∈ σp (U). Lemma 9.2.1. If T, U ∈ ℒ(X) satisfy Tn = Un for some positive integer n, then 0 ∈ σap (T) if and only if 0 ∈ σap (U). Moreover, if T and U have a closed range and T|R(T n−1 )∩R(U n−1 ) = U|R(T n−1 )∩R(U n−1 ) , then σap (T) = σap (U). Proof. Let 0 ∈ σap (T). Then, by the spectral mapping theorem, we have 0 ∈ σap (T n ) = σap (U n ) and, again, 0 ∈ σap (U). Now, suppose that T|R(T n−1 )∩R(U n−1 ) = U|R(T n−1 )∩R(U n−1 ) .

9.2 Jeribi, Wolf, and Weyl essential spectra | 211

We set W = R(T n−1 ) ∩ R(U n−1 ). Clearly, W is a closed subspace of X and, since T n = U n , we have T(W) ⊂ W,

U(W) ⊂ W,

T n (X) ⊂ W, and U n (X) ⊂ W. Therefore, by Theorem 2.22.1, for λ ≠ 0, it follows that λ − T is bounded below if and only if λ − T|W is bounded below if and only if λ − U|W is bounded below if and only if λ − U is bounded below. Thus, σap (T)\{0} = σap (U)\{0}. We start our investigation with the following lemma, which constitutes a preparation for the proof of the main results of this section. Lemma 9.2.2. Let U, V ∈ ℒ(X). If there exists n0 ∈ ℕ∗ such that U n0 = V n0 , E = R(U n0 −1 ) ∩ R(V n0 −1 ) is closed, and U|E = V|E , then: (i) σi (U) = σi (V) for i = f , w; (ii) if X satisfies the Dunford–Pettis property (respectively X = Lp (Ω, dμ), p ∈ [1, ∞)) and 𝒲∗ (X) = 𝒲 (X) (respectively 𝒲∗ (Lp (Ω, dμ)) = 𝒮 (Lp (Ω, dμ))), then σj (U) = σj (V). Proof. (i) Let λ ∈ ̸ σw (U). We shall divide the proof into two cases. Case 1. If λ ≠ 0, we have λ − U ∈ Φb (X) and i(λ − U) = 0. Since U n0 = V n0 , it follows that U(E) ⊆ E, V(E) ⊆ E,

n0

U (X) ⊆ E, and

V n0 (X) ⊆ E. By using Corollary 2.22.2, we know λ − U is a Fredholm operator if and only if λ − U|E is a Fredholm operator and i(λ − U) = i(λ − U|E ) = 0.

212 | 9 ν-convergence operators Thus, λ − V|E is a Fredholm operator and i(λ − V|E ) = 0, so λ − V is a Fredholm operator and i(λ − V) = i(λ − V|E ) = 0. Consequently, λ ∈ ̸ σw (V). Case 2. If λ = 0, then U is a Fredholm operator and i(U) = 0. This implies that U n0 is a Fredholm operator and i(U n0 ) = n0 i(U) = 0. Hence, V n0 is a Fredholm operator and i(V n0 ) = 0, which implies that V is a Fredholm operator and i(V) =

i(V n0 ) = 0, n0

so 0 ∈ ̸ σw (V). Therefore, σw (V) ⊆ σw (U). The opposite inclusion is analogous. By following the same reasoning, we obtain the following equality: σf (U) = σf (V). (ii) The result follows from the same reasoning as (i) and Theorems 2.20.1 and 2.20.2. Theorem 9.2.2. Let U, V ∈ ℒ(X) be two operators such that R(U) ∩ R(V) is closed and ν let (Un )n be a sequence of bounded operators commuting with both U and V. If Un → U ν

and Un → V, then

σi (U) = σi (V)

for i = f , w.

Moreover, if X satisfies the Dunford–Pettis property (respectively X = Lp (Ω, dμ), p ∈ [1, ∞)) and 𝒲∗ (X) = 𝒲 (X) (respectively 𝒲∗ (Lp (Ω, dμ)) = 𝒮 (Lp (Ω, dμ))), then σj (U) = σj (V).

9.2 Jeribi, Wolf, and Weyl essential spectra | 213

Proof. Let E = R(U) ∩ R(V). We first prove that U|E = V|E . Let x ∈ R(U) ∩ R(V). Then there exist y, z ∈ X such that Uz = Vy = x. Hence, 󵄩 󵄩 ‖Ux − Vx‖ = 󵄩󵄩󵄩(U − Un )Uz + (Un − V)Vy󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(U − Un )Uz 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Un − V)Vy󵄩󵄩󵄩. ν

ν

Besides, since Un → U and Un → V, we have 󵄩 󵄩 󵄩 󵄩 0 ≤ ‖Ux − Vx‖ ≤ (󵄩󵄩󵄩(U − Un )Uz 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Un − V)Vy󵄩󵄩󵄩) → 0

when n → ∞.

Therefore, Ux = Vx for all x ∈ R(U) ∩ R(V), so U|E = V|E . Second, we claim that U 2 = V 2. Let x ∈ X. Since Un U = UUn and Un V = VUn , we have 󵄩 󵄩 0 ≤ 󵄩󵄩󵄩U 2 x − V 2 x󵄩󵄩󵄩 ≤ Θn → 0

when n → ∞,

where 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 Θn = 󵄩󵄩󵄩(U − Un )Ux󵄩󵄩󵄩 + 󵄩󵄩󵄩(U − Un )Un x󵄩󵄩󵄩 + 󵄩󵄩󵄩(Un − V)Un x 󵄩󵄩󵄩 + 󵄩󵄩󵄩(Un − V)Vx 󵄩󵄩󵄩, so U 2x = V 2x for all x∈X, which concludes the proof of the claim. Now, the use of Lemma 9.2.2 allows us to conclude that σi (U) = σi (V) for i = f , w. The rest of the proof follows immediately from Theorems 2.20.1 and 2.20.2.

214 | 9 ν-convergence operators Remark 9.2.1. (i) Let U and V satisfy the hypotheses of Theorem 9.2.2. Then U ∈ ℛ(X) if and only if V ∈ ℛ(X). (ii) Theorem 9.2.2 paves the way to study the relationship between the Jeribi, Wolf, and Weyl essential spectra of Un and U.

9.3 Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators The goal of this section is to discuss the Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators that are ν-convergent in a Banach space X. We should ν note that, if U ∈ ℒ(X), {Un } ⊆ ℒ(X), 0 ∈ ΦbU , and Un → U, then not necessarily σf (Un ) ⊆ σf (U). In fact, it suffices to consider the following examples. Let R : l2 (ℕ) 󳨀→ l2 (ℕ) be the right shift operator defined by R(x1 , x2 , x3 , . . .) = (0, x1 , x2 , x3 , . . .). For each n ∈ ℕ∗ , let Un = (1 − n1 )R and U = R. Then U, Un ∈ ℒ(X), 0 ∈ ΦbU , and ν

Un → U, but σf (Un ) = {λ ∈ ℂ such that |λ| =

n−1 } n

and σf (U) = {λ ∈ ℂ such that |λ| = 1}. Therefore, σf (Un ) ⊈ σf (U) for all n ∈ ℕ∗ . However, we have the following result.

ν

Theorem 9.3.1. Let U ∈ ℒ(X) and (Un )n be a sequence in ℒ(X) such that 0∈ΦbU . If Un → U and 𝒪 is an open set of ℂ with 0 ∈ 𝒪, then there exists n0 ∈ ℕ such that, for all n ≥ n0 , σi (Un ) ⊆ σi (U) + 𝒪,

for i = f , w.

Moreover, if X satisfies the Dunford–Pettis property (respectively X = Lp (Ω, dμ), p ∈ [1, ∞)) and 𝒲∗ (X) = 𝒲 (X) (respectively 𝒲∗ (Lp (Ω, dμ)) = 𝒮 (Lp (Ω, dμ))), then σj (Un ) ⊆ σj (U) + 𝒪.

9.3 Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators | 215

Proof. For i = w, assume that the assertion fails. Then, by passing to a subsequence, it may be assumed that, for each n ∈ ℕ, there exists λn ∈ σw (Un ) such that λn ∈ ̸ σw (U) + 𝒪. Since (λn )n is bounded, we may assume that lim λ n→+∞ n

= λ.

This implies that λ ∈ ̸ σw (U) + 𝒪. Using the fact that 0 ∈ 𝒪, we have λ ∈ ̸ σw (U) and, therefore, λ − U ∈ Φb (X) and i(λ − U) = 0. Let Vn = (λn − λ)U + (U − Un )U.

(9.3.1)

It follows from equation (9.3.1) that the operator (λn − Un )U can be expressed in the form (λn − Un )U = (λ − U)U + Vn . Using the fact that Vn →0 and (λ − U)U ∈ Φb (X), there exists n0 ∈ ℕ such that, for all n ≥ n0 , we have (λn − Un )U ∈ Φb (X) and i[(λn − Un )U] = i[(λ − U)U]. Moreover, β(U) < ∞ and then, by using Theorem 2.3.3 (ii), we conclude that λn − Un ∈ Φb (X)

216 | 9 ν-convergence operators with i(λn − Un ) = i(λ − U). Hence, λn − Un ∈ Φb (X) and i(λn − Un ) = i(λ − U) = 0, so λn ∈ ̸ σw (Un ), which is a contradiction. This enables us to conclude that σw (Un ) ⊆ σw (U) + 𝒪, ∀n ≥ n0 . The proof of this inclusion σf (Un ) ⊆ σf (U) + 𝒪 follows from the same reasoning as above. The rest of the proof follows immediately from Theorems 2.20.1 and 2.20.2. As a direct consequence of Theorem 9.3.1, we have the following. Corollary 9.3.1. Let U ∈ ℒ(X) and let (Un )n be a sequence of linear operators in ℒ(X) such ν

that 0 ∈ ΦbU and Un → U. If 𝒪 ⊆ ℂ is an open set with 0 ∈ 𝒪, V ∈ ℒ(X), and VU ∈ ℱ b (X) (or UV ∈ ℱ b (X)), then there exists n0 ∈ ℕ such that, for all n ≥ n0 , σi (Un + V) ⊆ σi (U + V) + 𝒪 for i = f , w. Proof. By applying Theorem 9.3.1, there exists n0 ∈ ℕ such that σi (Un ) ⊆ σi (U) + 𝒪, for i = f , w. Now, we prove that σi (Un + V) = σi (Un ) for i = f , w. Since U is a Fredholm linear operator, there exists U0 ∈ ℒ(X) such that UU0 = I − K,

9.3 Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators | 217

where K ∈ 𝒦(X). Using the fact that VUU0 = V(I − K) = V − VK ∈ ℱ b (X) and VK ∈ ℱ b (X), we infer that V ∈ ℱ b (X). Hence, σi (Un + V) = σi (Un ), for i = f , w. For UV ∈ ℱ b (X), the proof can be checked in the same way. The aim of the following theorem is to extend Theorem 9.3.1 to a sequence of closed linear operators. Theorem 9.3.2. Let V be a closed linear operator, let (Vn )n be a sequence of closed linear operators on X, and let 𝒪 ⊆ ℂ be an open set with 0∈ 𝒪. If, for some λ0 ∈ρ(Vn )∩ρ(V), we have ν

(λ0 − Vn )−1 → (λ0 − V)−1 , then there exists n0 ∈ ℕ such that, for all n ≥ n0 , σi (Vn ) ⊆ σi (V) + 𝒪,

for i = f , w.

Moreover, if X satisfies the Dunford–Pettis property (respectively X = Lp (Ω, dμ), p ∈ [1, ∞)) and 𝒲∗ (X) = 𝒲 (X) (respectively 𝒲∗ (Lp (Ω, dμ)) = 𝒮 (Lp (Ω, dμ))), then σj (Vn ) ⊆ σj (V) + 𝒪. Proof. For i = w, take γn ∈ σw (Vn ) such that γn ∈ ̸ σw (V) + 𝒪. Since σ is ν-upper semi-continuous at (V − λ0 )−1 (see Definition 2.17.5), there exists k > 0 such that k −1 ≤ |γn − λ0 |−1 , so (γn )n is bounded. Therefore, it can be assumed that γn → γ. Then γ ∈ ̸ σw (V) + 𝒪 and, hence, γ ∈ ̸ σw (V).

218 | 9 ν-convergence operators This implies that γ − λ0 ∈ ̸ σw (V − λ0 ), so (γ − λ0 )−1 ∈ ̸ σw ((V − λ0 )−1 ). We set λn = (γn − λ0 )−1 , λ = (γ − λ0 )−1 ,

Un = (Vn − λ0 )−1 , and U = (V − λ0 )−1 . Then, arguing as in the proof of Theorem 9.3.1, there exists n0 ∈ ℕ such that, for all n ≥ n0 , λn ∈ ̸ σw (Un ), i. e., (γn − λ0 )−1 ∈ ̸ σw ((Vn − λ0 )−1 ), which implies γn − λ0 ∈ ̸ σw (Vn − λ0 ), so γn ∈ ̸ σw (Vn ), which is a contradiction. The proof of the inclusion σf (Un ) ⊆ σf (U) + 𝒪 follows from the same reasoning as above. The rest of the proof follows immediately from Theorems 2.20.1 and 2.20.2. Example 9.3.1. Let (Vn )n be a sequence of linear operators in l2 (ℕ), defined by Vn : 𝒟(Vn ) ⊂ l2 (ℕ) 󳨀→ l2 (ℕ), { { { { { jej , j ≠ n, { { {Vn ej = { −nej , j = n, { { { +∞ { { { {𝒟(Vn ) = {(xj )j≥1 such that ∑ j2 |xj |2 < ∞}. { j=1 {

9.3 Jeribi, Wolf, and Weyl essential spectra of a sequence of linear operators | 219

A simple calculation gives 󵄨󵄨 1 2n 1 󵄨󵄨󵄨󵄨 󵄨 󵄩󵄩 −1 −1 󵄩 − → 0, 󵄨= 󵄩󵄩(i − Vn ) − (i − V0 ) 󵄩󵄩󵄩 = 󵄨󵄨󵄨 󵄨󵄨 i + n i − n 󵄨󵄨󵄨 1 + n2 where i = √−1. Clearly, ν

(i − Vn )−1 → (i − V0 )−1 . According to Theorem 9.3.2, there exists n0 ∈ ℕ such that σi (Vn ) ⊆ σi (V0 ) + 𝒪,

for i = f , w

and for all n ≥ n0 . Remark 9.3.1. Consider the right shift R : l2 (ℕ) 󳨀→ l2 (ℕ) defined by R(x1 , x2 , x3 , . . .) = (0, x1 , x2 , x3 , . . .). For each n ∈ ℕ∗ , let Un = (1 −

1 )R, n

let U = R, and let 𝒪 ⊆ ℂ be an open set with 0 ∈ 𝒪. Clearly, 2 ∈ ρ(Un ) ∩ ρ(U). Then ν

(Un − 2)−1 → (U − 2)−1 . By virtue of Theorem 9.3.1, there exists n0 ∈ ℕ such that, for all n ≥ n0 , σf ((Un − 2)−1 ) ⊆ σf ((U − 2)−1 ) + 𝒪. However, σf ((Un − 2)−1 ) ⊈ σf ((U − 2)−1 ) for all n ∈ ℕ∗ . Indeed, σf (Un − 2) = {λ − 2 such that λ ∈ σf (Un )}

󵄨 󵄨 n−1 = {λ ∈ ℂ such that 󵄨󵄨󵄨λ − (−2)󵄨󵄨󵄨 = }. n

Also, 󵄨 󵄨 σf (U − 2) = {λ ∈ ℂ such that 󵄨󵄨󵄨λ − (−2)󵄨󵄨󵄨 = 1}.

220 | 9 ν-convergence operators Now, since g(z) = z −1 is analytic on a neighborhood of σf ((Un − 2)−1 ) and σf ((Un − 2)−1 ), we have σf ((Un − 2)−1 ) = {β−1 such that β ∈ σf (Un − 2)} and σf ((U − 2)−1 ) = {β−1 such that β ∈ σf (U − 2)}. Suppose that there exists n0 ∈ ℕ such that, for all n ≥ n0 , σf ((Un − 2)−1 ) ⊆ σf ((U − 2)−1 ). Take λn ∈ σf ((Un − 2)−1 ). Then there exists βn ∈ σf (Un − 2) such that λn = βn−1 . Also, there exists β ∈ σf (U − 2) such that λn = β−1 . Therefore, βn−1 = β−1 and βn = β. This implies that σf (Un − 2) ∩ σf (U − 2) ≠ 0, which is a contradiction.

9.4 ν-continuity of Wolf and Weyl essential spectra Theorem 9.4.1 (A. Ammar [16]). Let U ∈ ℒ(X) such that 0 ∈ ΦbU . Then σi (⋅) is ν-upper semi-continuous at U for i = f , w. Furthermore, σw (⋅) is ν-continuous at U in each one of the following cases: (i) σf (⋅) is ν-continuous at U; and (ii) there exists n0 ∈ ℕ such that, for all n ≥ n0 , we have σf (U) ⊂ σw (Un ).

9.4 ν-continuity of Wolf and Weyl essential spectra | 221

Proof. Let λ ∈ lim sup(σi (Un )) for i = f , w. Then we may suppose that there exists a sequence (λn )n such that λn ∈ σi (Un ) for all n ∈ ℕ and λn → λ. By Theorem 9.3.1, for each p ∈ ℕ∗ , there exists np ∈ ℕ such that 1 σi (Un ) ⊆ σi (U) + 𝔹(0, ) p for i = f , w and for all n ≥ np . This implies that, for each p ≥ 1, there exist βp ∈ σi (U) and γp ∈ 𝔹(0, p1 ) such that λnp = βp + γp . Therefore, lim βp = lim λnp − lim γp = λ. Since σi (U) is a closed set, we have λ ∈ σi (U) for i = f , w. We conclude that lim sup(σi (Un )) ⊆ σi (U)

for i = f , w.

(i) Now, we prove that σw (U) ⊆ lim inf(σw (Un )). Suppose that λ ∈ ̸ lim inf(σw (Un )). Then there is a neighborhood 𝒱 of λ which does not intersect with infinitely many σw (Un ). Since σf (Un ) ⊂ σw (Un ), 𝒱 does not intersect with infinitely many σf (Un ). Hence,

λ ∈ ̸ lim inf(σf (Un )).

222 | 9 ν-convergence operators Now, since σf (⋅) is ν-continuous at U, we have lim inf(σf (Un )) = σf (U). This implies that λ ∈ ̸ σf (U). This shows that λ − U ∈ Φb (X).

(9.4.1)

Now, we prove that i(λ − U) = 0. Since λ ∈ ̸ lim inf(σw (Un )), there exists an increasing sequence of natural numbers n1 < n2 < ⋅ ⋅ ⋅ such that λ ∈ ̸ σw (Unj ) for all j ∈ ℕ∗ . Then λ − Unj ∈ Φb (X) and i(λ − Unj ) = 0. By Theorem 9.1.1, there exists j0 ∈ ℕ∗ such that i(λ − Unj ) = i(λ − U) for all j ≥ j0 . Therefore, i(λ − U) = 0.

(9.4.2)

It follows from equations (9.4.1) and (9.4.2) and Proposition 2.20.1 that λ ∈ ̸ σw (U), so we obtain σw (U) ⊂ lim inf(σw (Un )). (ii) Assume that the assertion does not hold. Then there exists λ ∈ σw (U) such that λ ∈ ̸ lim inf(σw (Un )). We discuss two cases. Case 1. If λ ∈ σw (U)\σf (U), then, by using a similar reasoning as for (i), we have i(λ − U) = 0, which is a contradiction. Case 2. If λ ∈ σf (U), then λ ∈ ̸ lim inf(σw (Un )).

9.4 ν-continuity of Wolf and Weyl essential spectra |

223

Arguing as above, there exist an increasing sequence of natural numbers n1 < n2 < ⋅ ⋅ ⋅ and ε > 0 such that, for all j ∈ ℕ∗ , we have 𝔹(λ, ε) ∩ σw (Unj ) = 0. Then 𝔹(λ, ε) ∩ σf (U) = 0, which is a contradiction. Remark 9.4.1. As described by K. K. Oberai [165], σw (⋅) is upper semi-continuous. Moreover, if 0 ∈ ΦbU , then, by using Theorem 9.4.1, we know σw (⋅) is ν-upper semicontinuous at U. As an immediate consequence of Theorem 9.4.1, we have the following corollary. ν

Corollary 9.4.1. Let U ∈ ℒ(X) such that 0 ∈ ΦbU and Un → U. If ΦbUn is connected, then lim σf (Un ) = σf (U) if and only if lim σw (Un ) = σw (U). Proof. First note that, by using Theorem 9.4.1, we deduce that lim σf (Un ) = σf (U), which implies that lim σw (Un ) = σw (U). Conversely, by using Theorem 9.4.1, σf (⋅) is ν-upper semi-continuous at U. It is sufficient to show that σf (⋅) is ν-lower semi-continuous at U. We suppose that λ ∈ ̸ lim inf(σf (Un )) and there is a neighborhood 𝒱 of λ which does not intersect with infinitely many σf (Un ). Note that ΦbUn is connected. Then, by using Theorem 2.20.3, one gets σf (Un ) = σw (Un ). Hence, 𝒱 does not intersect with infinitely many σw (Un ). Hence, λ ∈ ̸ lim inf(σw (Un )). Now, by using the fact that σw (⋅) is ν-continuous at U, we have lim inf(σw (Un )) = σw (U).

224 | 9 ν-convergence operators This implies that λ ∈ ̸ σw (U). Hence, λ ∈ ̸ σf (U), so σf (U) ⊂ lim inf(σf (Un )).

9.5 2 × 2 operator matrix In this section, we focus on the relationship between the ν-continuity of 2 × 2 block operator matrices and the ν-continuity of its components. Theorem 9.5.1 (A. Ammar [16]). Let X be a Banach space. In the product space X ⊗ X, we consider a sequence of operators formally defined by An Dn

Cn ), Bn

ℳn = (

A 0

0 ), B

ℳ=(

and A 0

C ), B

ℳC = (

where A, An , B, Bn , C, Cn , Dn ∈ ℒ(X). ν

ν

(i) If An → A, Bn → B, Cn → 0, and Dn → 0, then ν

ℳn → ℳ.

Moreover, if 0 ∈ ΦbA ∩ ΦbB and 𝒪 is an open set of ℂ with 0 ∈ 𝒪, then there exists n0 ∈ ℕ such that, for every n ≥ n0 , σi (ℳn ) ⊆ [σi (A) ∪ σi (B)] + 𝒪, ν

for i = f , w.

ν

(ii) If An → A, Bn → B, Cn → C, Dn → 0, and An C → AC, then ν

ℳn → ℳC .

Moreover, if 0 ∈ ΦbA ∩ ΦbB and 𝒪 is an open set of ℂ with 0 ∈ 𝒪, then there exists n0 ∈ ℕ such that, for every n ≥ n0 , σi (ℳn ) ⊆ [σi (A) ∪ σi (B)] + 𝒪,

for i = f , w.

9.5 2 × 2 operator matrix | 225

Proof. (i) Let An 0

0 ), Bn

𝒰n = (

0 0

Cn ), 0

𝒱n = (

and 0 Dn

𝒲n = (

0 ). 0

ν

First we prove that 𝒰n → ℳ. Indeed, ‖𝒰n ‖ = max{‖An ‖, ‖Bn ‖}, so (‖Un ‖)n is bounded, 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩(𝒰n − ℳ)ℳ󵄩󵄩󵄩 = max{󵄩󵄩󵄩(An − A)A󵄩󵄩󵄩, 󵄩󵄩󵄩(Bn − B)B󵄩󵄩󵄩} → 0, and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩(𝒰n − ℳ)𝒰n 󵄩󵄩 = max{󵄩󵄩(An − A)An 󵄩󵄩, 󵄩󵄩(Bn − B)Bn 󵄩󵄩} → 0. Now, we represent ℳn as ℳn = 𝒰n + 𝒱n + 𝒲n . ν

From the fact that 𝒰n → ℳ, 𝒱n → 0, 𝒲n → 0 and Lemma 9.1.4 it follows immediately that ν

𝒰n + 𝒱n + 𝒲n → 0 + 0 + ℳ.

Then ν

ℳn → ℳ.

Now, if we suppose that 0 ∈ ΦbA ∩ ΦbB , then 0 ∈ Φbℳ . According to Theorem 9.3.1, there exists n0 ∈ ℕ such that, for every n ≥ n0 , σi (ℳn ) ⊆ [σi (A) ∪ σi (B)] + 𝒪,

for i = f , w.

(ii) By using the assertion (i), we have (

An Dn

Cn − C A ) = [( n Bn Dn

Cn 0 )+( Bn 0

ν −C )] → ℳ. 0

Since A [( n Dn

Cn 0 )+( Bn 0

−C 0 )] ( 0 0

C 0 )=( 0 0

An C ) Dn C

226 | 9 ν-convergence operators and A [( 0

0 0 )( B 0

C 0 )] = ( 0 0

AC ), 0

we have A [(( n Dn

Cn 0 )+( Bn 0

−C 0 )) ( 0 0

C A )] → [( 0 0

0 0 )( B 0

C )] . 0

0 0 )+( B 0

C )] , 0

Now, by using Lemma 9.1.4, we have A [( n Dn

Cn 0 )+( Bn 0

−C 0 )+( 0 0

ν C A )] → [( 0 0

so ν

ℳn → ℳC .

Now, if we suppose that 0 ∈ ΦbA ∩ ΦbB , then 0 ∈ Φbℳ . By applying Theorem 9.3.1, there exists n0 ∈ ℕ such that, for every n ≥ n0 , σi (ℳn ) ⊆ [σi (A) ∪ σi (B)] + 𝒪,

for i = f , w.

ν

Remark 9.5.1. (i) If ℳn → ℳC , An C → AC, Dn → 0, and Cn → C, then ν

An → A and ν

Bn → B. ν

In fact, let ℳn → ℳC , 0 −Dn

(

−Cn 0 )→( 0 0

−C ), 0

and 0 0

(ℳn − ℳC ) (

−C 0 )=( 0 0

−(An − A)C ) → 0, −Dn C

so, in view of Lemma 9.1.4, we have An 0

(

0 ν A )→( Bn 0

0 ). B

9.5 2 × 2 operator matrix | 227

Hence, ν

An → A and ν

Bn → B. (ii) Let Un 0

ℳn = (

0 ) Un

and U 0

0 ), U

ℳ=(

ν

where Un and U are defined in Remark 9.3.1. Then ℳn → ℳ, 0 ∈ ΦbU , and σf (ℳn ) = σf (Un ) ⊈ σf (U) = σf (ℳ). Corollary 9.5.1. Let X be a Banach space. In the product space X ⊗ X, we consider a sequence of operators formally defined by An 0

Cn ) Bn

ℳn = (

and A 0

ℳ=(

0 ), B

where An and A are two closed operators acting on X and have the domain 𝒟(An ) and 𝒟(A), respectively, and Bn and B act on X and have the domain 𝒟(Bn ) and 𝒟(B), respectively. Let Cn be a bounded operator in ℒ(X). If λ ∈ ρ(A) ∩ ρ(B) ∩ ρ(An ) ∩ ρ(Bn ) and if ν

(λ − An )−1 → (λ − A)−1 , ν

(λ − Bn )−1 → (λ − B)−1 , and Cn → 0, then, for each open set 𝒪 ⊆ ℂ with 0 ∈ 𝒪, there exists n0 ∈ ℕ such that, for every n ≥ n0 , σi (ℳn ) ⊆ [σi (A) ∪ σi (B)] + 𝒪,

for i = f , w.

228 | 9 ν-convergence operators Proof. It is easy to verify that (λ − An )−1 0

−(λ − An )−1 Cn (λ − Bn )−1 ). (λ − Bn )−1

(λ − ℳn )−1 = ( Since

(λ − An )−1 Cn (λ − Bn )−1 → 0, by Theorem 9.5.1, we have ν

(λ − ℳn )−1 → (λ − ℳ)−1 . Now, the result follows from Theorem 9.3.2. Theorem 9.5.2 (A. Ammar [16]). Let X be a Banach space. In the product space X ⊗ X, we consider a sequence of operators formally defined by An 0

Cn ) Bn

A 0

C ), B

ℳn = (

and ℳC = (

where A, An , B, Bn , C, and Cn ∈ ℒ(X), An C → AC, and Cn → C. If 0 ∈ ΦbA ∩ ΦbB and σi (A) ∩ σi (B) = 0, for i = f , w, then σi (⋅) is ν-continuous at A and B if and only if σi (⋅) is ν-continuous at ℳC . Proof. Since σi (A) ∩ σi (B) = 0, for i = f , w, from the ν-upper semi-continuity of σi (⋅) at A and B, for i = f , w (see Theorem 9.4.1) and for every sequence (An )n in ℒ(X) and every sequence (Bn )n in ℒ(X) ν

ν

such that An → A and Bn → B, there exists n0 ∈ ℕ such that, for all n ≥ n0 , we have σi (ℳn ) = σi (An ) ∪ σi (Bn ), for i = f , w.

(9.5.1)

󳨐⇒ Suppose that σi (⋅) is ν-continuous at A and B. We may show that σi (⋅) is ν

ν-continuous at ℳC for i = f , w. Let ℳn → ℳC . First, by using Theorem 9.4.1, σi (⋅) is ν-upper semi-continuous at ℳC , for i = f , w. It is sufficient to show that σi (⋅) is ν

lower semi-ν-continuous at ℳC for i = f , w. We suppose ℳn → ℳC . Then, by using ν

ν

Remark 9.5.1, we have An → A and Bn → B. Since σi (⋅) is ν-lower semi-continuous at A and B, we have σi (A) ⊂ lim inf σi (An )

9.5 2 × 2 operator matrix | 229

and σi (B) ⊂ lim inf σi (Bn ) for i = f , w. Then [σi (A) ∪ σi (B)] ⊂ [(lim inf σi (An )) ∪ (lim inf σi (Bn ))],

for i = f , w.

Hence, (σi (A) ∪ σi (B)) ⊂ lim inf(σi (An ) ∪ σi (Bn ))

for i = f , w.

This implies that σi (ℳC ) ⊂ lim inf(σi (ℳn )) for i = f , w. Hence, σi (⋅) is ν-lower semi-continuous at ℳC for i = f , w. ⇐󳨐 Suppose that σi (⋅) is ν-continuous at ℳC . We show that σi (⋅) is ν-continuous ν

ν

at A and B, for i = f , w. Let An → A and Bn → B. By using Theorem 9.4.1, σi (⋅) is ν-upper semi-continuous at A and B for i = f , w, so lim sup σi (An ) ⊂ σi (A) for i = f , w lim sup σi (Bn ) ⊂ σi (B)

and

for i = f , w.

(9.5.2)

Now, we prove that σi (⋅) is ν-lower semi-continuous at A for i = f , w. Let λ∈σi (A) for i = f , w. Since σi (A) ⊂ σi (ℳC ), we have λ ∈ σi (ℳC ) for i = f , w. By using Theorem 9.5.1, we have ν

ℳn → ℳC .

It follows from the ν-lower semi-continuity of σi (⋅) at ℳC that λ ∈ lim inf(σi (ℳn )) for i = f , w. Then there exists a sequence (λn )n such that λn ∈ σi (ℳn ) and λn → λ for i = f , w. On the other hand, by equation (9.5.1), there exists N ∈ ℕ such that λn ∈ σi (ℳn ) = σi (An ) ∪ σi (Bn ), for all n ≥ N and i = f , w. We discuss two cases. Case 1. If λn ∈ σi (An ), for all n ≥ N and i = f , w, then λ ∈ lim inf σi (An ) and, hence, σi (⋅) is ν-continuous at A for i = f , w.

230 | 9 ν-convergence operators Case 2. If there exists a subsequence (λnj )j of λn such that λnj ∈ σi (Bnj ), then λ ∈ lim sup σi (Bn )

for i = f , w.

Thus, by using equation (9.5.2), we have λ ∈ σi (B)

for i = f , w.

This shows that λ ∈ σi (B) ∩ σi (A) for i = f , w. This is a contradiction. By the same reasoning, we can prove that σi (⋅) is ν-lower semicontinuous at B for i = f , w.

10 Γ-hypercyclic set of linear operators 10.1 Γ-hypercyclic set of a bounded linear operator 10.1.1 Properties of Γhyp (T ) Let α ∈ ℂ and r > 0. Consider 𝕊ℂ (α, r) = {λ ∈ ℂ such that |λ − α| = r}. We begin by giving some properties of the set Γhyp (T) given in (2.21.1). Theorem 10.1.1. Let X be a separable Banach space and let T ∈ ℒ(X). Then Γhyp (T) ⊂ {λ ∈ ℂ such that σ(T − λ) ∩ 𝕊ℂ (0, 1) ≠ 0}. Proof. Let λ ∈ Γhyp (T) and let T − λ be hypercyclic. Then, from [157], we have σ(T − λ) ∩ 𝕊ℂ (0, 1) ≠ 0. Hence, Γhyp (T) ⊂ {λ ∈ ℂ such that σ(T − λ) ∩ 𝕊ℂ (0, 1) ≠ 0}. by

In the following theorem, we will denote, for any subset M of ℂ, the set |M| defined |M| := {|λ| such that λ ∈ M}.

We continue by giving the following well-known technical result. Theorem 10.1.2. Let X be a separable Banach space and let T ∈ ℒ(X). If λ ∈ Γhyp (T) and |σ(T − λ)| is countable, then σ(T − λ) ⊂ 𝕊ℂ (0, 1). Proof. Suppose that |σ(T − λ)| is countable and write 󵄨󵄨 󵄨 󵄨󵄨σ(T − λ)󵄨󵄨󵄨 = {(αn ), n ∈ ℤ with αi < αi+1 }. Let αi = |μi − λ| ∈ |σ(T − λ)| be an isolated point and Σi := {μ ∈ σ(T) such that |μ − λ| = αi }. If Σi is a connected component in σ(T − λ) and T − λ is hypercyclic, then Σi ∩ 𝕊ℂ (0, 1) ≠ 0. Thus, αi = 1, so σ(T − λ) ⊂ 𝕊ℂ (0, 1). https://doi.org/10.1515/9783110493863-010

232 | 10 Γ-hypercyclic set of linear operators Remark 10.1.1. Clearly, every compact operator is essentially quasi-nilpotent. Moreover, it was shown in [10, Corollary 7.50] that the spectrum of an essentially quasinilpotent operator is at most countable and if it is countable, say, {λ1 , λ2 , . . .}, then λn → 0. It follows from Theorem 10.1.2 that, if T is an essentially quasi-nilpotent operator with infinite spectrum, then Γhyp (T) ⊂ {α}. Corollary 10.1.1. Let X be a separable Banach space and let T ∈ ℒ(X). If σ(T) is countable, then Γhyp (T) ⊂ ρ(T). Proof. Let λ ∈ Γhyp (T). Since σ(T) is countable, σ(T − λ) is countable, so |σ(T − λ)|. Since T − λ is a hypercyclic operator, by Theorem 10.1.2, σ(T − λ) ⊂ 𝕊ℂ (0, 1). We deduce that 0 does not belong to σ(T − λ), so λ ∈ ρ(T). Corollary 10.1.2. Let X be a separable Banach space and let T ∈ ℒ(X). We have: (i) if σ(T) contains more than two points, then Γhyp (T) is a singleton; and (ii) if σ(T) = {α}, then Γhyp (T) = 𝕊ℂ (α, 1). Proof. (i) Let λ1 , λ2 ∈ Γhyp (T). Then, from Theorem 10.1.2, we have σ(T) ⊂ 𝕊ℂ (λ1 , 1) ∩ 𝕊ℂ (λ2 , 1). If λ1 ≠ λ2 , then card(𝕊ℂ (λ1 , 1) ∩ 𝕊ℂ (λ2 , 1)) ≤ 2. Hence, card(σ(T)) ≤ 2. The assertion (ii) is clear. 10.1.2 Essentially quasi-nilpotent operators Proposition 10.1.1. If the operator T is essentially quasi-nilpotent and Γhyp (T) ≠ 0, then T is quasi-nilpotent. Proof. Suppose that T is essentially quasi-nilpotent and λ0 ∈Γhyp (T). It is easy to check, by Proposition 2.21.2, that hypercyclic operators must have a dense range. Then, for any λ ∈ ℂ∗ , T − λ has a dense range and is a Fredholm operator, so T − λ is surjective. Since i(T − λ) = 0, T − λ is injective and thus invertible, λ ∉ σ(T). Hence, σ(T) = {0}, so T is quasi-nilpotent.

10.1 Γ-hypercyclic set of a bounded linear operator | 233

Proposition 10.1.2 (V. Matache [156]). Any compact operator commuting with an arbitrary hypercyclic operator is quasi-nilpotent. According to [104], for any fixed, unimodular, complex λ, there exists some compact operator K such that K − λ is a hypercyclic operator. Consequently, we have 𝕊ℂ (0, 1) ⊂

⋃ Γhyp (K).

K∈𝒦(X)

In fact, we have the following theorem. Theorem 10.1.3. Let X be a separable Banach space. Then 𝕊ℂ (0, 1) =

⋃ Γhyp (K).

K∈𝒦(X)

Proof. Let λ ∈ ⋃K∈𝒦(X) Γhyp (K). Then there is some compact operator such that K − λ is hypercyclic. By using Proposition 10.1.1, K is quasi-nilpotent. From the spectral mapping theorem, σ(K − λ) = {−λ}. This, together with Theorem 10.1.2, makes us conclude that λ ∈ 𝕊ℂ . Remark 10.1.2. If T is a polynomially compact operator, i. e., there exist P ∈ ℂ[ℤ]\{0} such that P(T) is compact, then Γhyp (T) ⊂

⋃

λ0 ∈P −1 {0}

𝕊ℂ (λ0 , 1).

Indeed, let λ ∈ Γhyp (T). Then T − λ is hypercyclic and it follows from Proposition 10.1.1 that σ(P(T)) = {0}. By the spectral mapping theorem, we deduce that σ(T) ⊂ P −1 {0} and σ(T − λ) ⊂ P −1 {0} − λ. Since σ(T − λ) ∩ 𝕊ℂ (0, 1) ≠ 0, we have (P −1 {0} − λ) ∩ 𝕊ℂ (0, 1) ≠ 0. It follows that there exist λ0 ∈ 𝕊ℂ (λ0 , 1) such that P(λ0 ) = 0.

234 | 10 Γ-hypercyclic set of linear operators Theorem 10.1.4. Let X be a separable Banach space and let T ∈ ℒ(X). Then Γhyp (T) ⊂ 𝔹(0, ‖T‖ + 1). Proof. Let λ ∈ Γhyp (T). Suppose that |λ| ≥ ‖T‖ + 1. Then 󵄩󵄩 T 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 < 1. 󵄩󵄩 λ 󵄩󵄩 Thus, I − Tλ is invertible with bounded inverse, so T − λ is invertible with bounded inverse. The operator T − λ is hypercyclic and so is (T − λ)−1 . Since |λ| > ‖T‖, we have 1 󵄩󵄩 −1 󵄩 . 󵄩󵄩(T − λ) 󵄩󵄩󵄩 ≤ |λ| − ‖T‖ Thus, (T − λ)−1 is a contraction, which is a contradiction. We refine the increase in Theorem 10.1.2 using the spectral radius of the operator. Theorem 10.1.5. Let T ∈ ℒ(X) be a bounded operator acting in a Banach space X. Then Γhyp (T) ⊂ 𝔹(0, r(T) + 1), where r(T) denotes the spectral radius of the operator T. Proof. Let λ ∈ Γhyp (T). Then σ(T − λ) ∩ 𝕊ℂ (0, 1) ≠ 0, so there exists μ such that |μ| = 1 and λ + μ ∈ σ(T). Hence, λ ∈ σ(T − μ). Thus, |λ| ≤ r(T − μI)

≤ max{|λ − μ|, λ ∈ σ(T)},

so |λ| ≤ r(T) + 1.

10.1 Γ-hypercyclic set of a bounded linear operator | 235

10.1.3 Operators with small Γhyp (T ) We start with the following trivial results. Proposition 10.1.3. Let T ∈ ℒ(X) be a bounded operator acting in a Banach space X such that σp (T ∗ ) ≠ 0, where as usual T ∗ denotes the adjoint operator of T. Then Γhyp (T) = 0. Proof. It suffices to observe that σp (T − μ)∗ = σp (Tμ )∗ − μ and to use Proposition 2.21.2. Theorem 10.1.6. Let T be a bounded linear operator acting in a separable Hilbert space X. If T is hyponormal, then Γhyp (T) = 0. Proof. If λ ∈ Γhyp (T), then T − λ is hypercyclic and hyponormal. Hence, from [50], the operator T − λ is not hypercyclic and thus a contradiction. Definition 10.1.1. We say that T has Bishop’s property (β), if, for each open subset U of ℂ and every sequence of analytic functions fn : U 󳨀→ X for which (T − λ)fn (λ) → 0 as n → ∞, locally uniformly on U, it follows that fn (λ) → 0 as n → ∞, again locally uniformly on U. Theorem 10.1.7. Let T be a bounded linear operator acting in a separable Banach space X satisfying the property (β). If Γhyp (T) ≠ 0, then σ(T) is included in a circle. Proof. Let λ ∈ Γhyp (T). Then T − λ has the property (β) and is hypercyclic. Then, from [158], we have σ(T − λ) ⊂ 𝕊ℂ (0, 1). It follows that σ(T) ⊂ 𝕊ℂ (λ, 1).

236 | 10 Γ-hypercyclic set of linear operators 10.1.4 Operators with big Γhyp (T ) G. Godefroy and J. H. Shapiro in [93] gave a sufficient condition for hypercyclicity that is a consequence of Kitai’s criterion [141]. In the next proposition, we will denote by H an infinite-dimensional separable complex Hilbert space. The results show that bounded operators on X with a sufficiently large supply of eigenvectors are hypercyclic. Proposition 10.1.4. For any bounded operators T on X, let H+ (T) be the vector space spanned by the kernels N(T − λ) with |λ| > 1 and let H− (T) be the vector space spanned by the kernels N(T − λ) with |λ| < 1. If H+ (T) and H− (T) are dense subspaces of X, then T is hypercyclic. This result is used by G. Godefroy and J. H. Shapiro in their paper [93], but it is not stated explicitly there. S. Grivaux in [99] sketched a proof of Proposition 10.1.4. Twice the backward shift on l2 is an example of a hypercyclic operator (see [174]). The backward shift is not hypercyclic because all of its orbits are bounded. In what follows, we denote by Sr∗ the adjoint of the forward shift. From the previous criterion, it is easy to show the following. Theorem 10.1.8. 𝔹(0, 1 + α) Γhyp (αSr∗ ) = { C(1 − α, 1 + α)

if α > 1, if α < 1,

where C(1 − α, 1 + α) is the crown centered at 0 and having radii 1 − α and 1 + α. Remark 10.1.3. It is clear that, for any complex λ, Γhyp (T − λ) = Γhyp (T) − λ and it was shown in [33] that T is hypercyclic if and only if T n is hypercyclic for all n ∈ ℕ. However, in general, the set Γhyp (T n ) is different from (Γhyp (T))n . For example, if we consider T = αSr∗ with α < 1, then, by Theorem 10.1.8, 2

Γhyp (T 2 ) ≠ (Γhyp (T)) . p Let (en )∞ n=1 be the canonical basis of one of the spaces l , 1 ≤ p < +∞, or c0 and let w = (wn )n≥1 be a bounded sequence of positive numbers. The backward shift Bw with weights wn (see [176]) is defined by

Bw e1 = 0

10.1 Γ-hypercyclic set of a bounded linear operator | 237

and Bw en = wn en−1

for n ≥ 2.

Theorem 10.1.9. Γhyp (Bw ) = 𝕊ℂ (0, 1). Proof. For any λ∈ℂ∗ , I +λBw is hypercyclic. For |λ| = 1 and any rotation of a hypercyclic operator being hypercyclic [151], we know that λ + Bw is hypercyclic. It follows that 𝕊ℂ (0, 1) ⊂ Γhyp (Bw ). Since the operator Bw is quasi-nilpotent, we have σ(λ + Bw ) = {λ}. Hence, by using Theorem 10.1.1, |λ| = 1, so Γhyp (Bw ) = 𝕊ℂ (0, 1). Corollary 10.1.3. For any complex λ such that |λ| = 1, we have Γhyp (Bw + λ) = 𝕊ℂ (λ, 1). Proof. The operator Bw + λI − μI is hypercyclic if and only if |μ − λ| = 1. Then Γhyp (Bw + λ) = 𝕊ℂ (λ, 1). 10.1.5 Aluthge transforms It was shown in [39] that, if A and B are two bounded linear operators and λ ≠ 0, the operators λ − AB and λ − BA have many basic properties in common. Proposition 10.1.5. Let X be a separable Banach space and let A and B be two bounded linear operators with dense range. Then BA is hypercyclic if and only if AB is hypercyclic. Proof. There is a vector x such that Orb(BA, x) is dense in X. It is obvious that Orb(AB, Ax) = A Orb(BA, x). The continuity of the operator A involves A Orb(BA, x) ⊇ A(Orb(BA, x)) = R(A),

238 | 10 Γ-hypercyclic set of linear operators Orb(AB, Ax) ⊇ R(A). It follows that R(A) ⊆ Orb(AB, Ax), so X = Orb(AB, Ax). We deduce that the operator AB is hypercyclic. From Proposition 10.1.5, we have the following theorem. Theorem 10.1.10. Let X be a separable Banach space and let A and B be two bounded linear operators such that R(A) and R(B) are dense in X. Then Γhyp (AB) = Γhyp (BA). Remark 10.1.4. It follows from Theorem 10.1.10 that, if X is a separable Hilbert space, 1 T = U|T|∈ ℒ(X) is the polar decomposition of the operator T. If the two sets R(|T| 2 ) and 1 ̃ where T̃ is the Aluthge R(U|T| 2 ) are dense, then λ ∈ Γhyp (T) if and only if λ ∈ Γhyp (T), transform of the operator T.

10.1.6 Operator equations ABA = A2 and BAB = B2 Theorem 10.1.11. Let X be a separable Banach space and let A and B be two bounded linear operators satisfying the following equations: {

ABA = A2 , BAB = B2 .

Then Γhyp (A) = Γhyp (B) = Γhyp (AB) = Γhyp (BA) = 0. Proof. If λ ∈ Γhyp (A), then A − λ is hypercyclic. Hence, A has a dense range. It follows that A = B = I. Corollary 10.1.4. Let X be a separable Banach space. The operators P and Q are two idempotent operators and A = PQ and B = QP. Then Γhyp (A) = Γhyp (B) = 0.

10.1 Γ-hypercyclic set of a bounded linear operator | 239

10.1.7 Upper triangular matrices The class of all hypercyclic operators are invariant under similarity. Moreover, if A, B ∈ ℒ(X), B is hypercyclic, and AC = CB for some C ∈ ℒ(X) with dense range, then so is A. Therefore, λ ∈ Γhyp (B)

λ ∈ Γhyp (A).

󳨐⇒

In particular, if A and B are quasi-similar (i. e., YA = YB and AZ = ZB for two quasi-affinities Y and Z) (injective and having a dense range), then Γhyp (B) = Γhyp (A). Let MC = (

A 0

C ) B

A M0 = ( 0

0 ) B

and

such that AC = CB. For any λ ∈ ℂ, we have A−λ 0

MC − λ = (

C I )=( B−λ 0

C A−λ )( I 0

Since (

I 0

C ) I

I ( 0

−C ), I

is invertible with inverse

we derive that Γhyp (MC ) = Γhyp (M0 ).

0 I )( B−λ 0

−C ). I

11 Analytic operators in Béla Szökefalvi-Nagy’s sense Let T0 , T1 , T2 , . . . be some linear operators on a Banach space X. In this chapter, we are interested in the perturbed operator T(ε) := T0 + εT1 + ε2 T2 + ⋅ ⋅ ⋅ + εk Tk + ⋅ ⋅ ⋅ ,

(11.0.1)

where ε ∈ ℂ.

11.1 Invariance of the closure 11.1.1 Hypotheses Let T0 be a linear operator on a separable Banach space X such that: (H1) T0 is closed with domain 𝒟(T0 ) dense in X. Let T1 , T2 , T3 , . . . be some linear operators on X having the same domain 𝒟 and satisfying the following hypothesis: (H2) 𝒟(T0 ) ⊂ 𝒟 and there exist a, b, q > 0 and β ∈ ]0, 1] such that, for all k ≥ 1, ‖Tk φ‖ ≤ qk−1 (a‖φ‖ + b‖T0 φ‖β ‖φ‖1−β ),

for all φ ∈ 𝒟(T0 ).

(11.1.1)

11.1.2 Closeness We have the following theorem. Theorem 11.1.1. Assume that the assumptions (H1) and (H2) hold. Then, for |ε| < q−1 , the series ∑ ε k Tk φ

k≥0

converges for all φ ∈ 𝒟(T0 ). If T(ε)φ denotes its limit, then T(ε) is a linear operator with domain 𝒟(T0 ) and, for |ε| < (q + βb)−1 , the operator T(ε) is closed. Proof. Let |ε|
0, there exists a positive constant H such that 󵄨󵄨 󵄨 󵄨󵄨f (x) − f (y)󵄨󵄨󵄨 ≤ H|x − y|. Next, we are going to state the second main result, namely, the Riesz basis exponential family of the perturbed transformation T(ε), by applying the general results stated in Theorem 12.3.1. Theorem 12.3.2. Suppose that the hypotheses (H1) and (H2) for β = 1 are satisfied. Assume further that the eigenvalues (λn )∞ n=1 of T0 are isolated and with multiplicity one and the family {eif (λn )t }∞ forms a Riesz basis in L2 (0, T), T > 0. Then the exponential 1 family {eif (λn (εn ))t }1

∞

forms a Riesz basis in L2 (0, T) for some T > 0, where (λn (εn ))n corresponding to a sequence of eigenvalues of the perturbed transformation T(εn ) can be developed as an entire series of (εn )∞ n=1 . Proof. First, we observe that 󵄨󵄨 if (λn (εn ))t 󵄨 󵄨 󵄨 − eif (λn )t 󵄨󵄨󵄨 = 󵄨󵄨󵄨eif (λn )t (ei(f (λn (εn ))−f (λn ))t − 1)󵄨󵄨󵄨 󵄨󵄨e 󵄨 󵄨 ≤ t 󵄨󵄨󵄨f (λn (εn )) − f (λn )󵄨󵄨󵄨 󵄨 󵄨 ≤ tH 󵄨󵄨󵄨λn (εn ) − λn 󵄨󵄨󵄨. Second, for |εn | ∈ ]0,

1 A [ ⊂ ]0, [, (q + αn + wn2 rn Mn αn ) βwn2 rn2 Mn αn n√BTtH + (q + αn + wn2 rn Mn αn )A

we complete the proof by the same reasoning as the one of Theorem 12.3.1 and we get the desired result.

12.3 Riesz basis property of families of non-harmonic exponentials | 263

12.3.2 The value of ε is fixed In the sequel, we are concerned with the Riesz basis property of the family of nonharmonic exponentials {eiλn (ε)t }∞ 1 . To this end, we consider the following hypothesis: (H4) for all n ∈ ℕ∗ , there exists a sequence (rn )n≥1 in ℝ∗+ such that {z ∈ ℂ such that |z − λn | ≤ rn } ∩ σ(T0 ) = {λn }, { { { { { sup(q + αn + ω2n rn Mn αn ) < ∞, { { { n≥1 { T { {∞ { 2 { 2 2 2 2t(rn +h) { { dt < ∞. { ∑ (ωn rn Mn αn ) ∫ t e { n=1 0 The first result of this section is formulated as follows. Theorem 12.3.3. Suppose that the hypotheses (H1)–(H4) are satisfied. Assume further that the eigenvalues (λn )∞ n=1 of T0 are separated with multiplicity one. Then there exist a constant D > 0 and a sequence of eigenvalues (λn (ε))∞ n=1 having the form λn (ε) = λn + ελn,1 + ε2 λn,2 + ⋅ ⋅ ⋅ 2 such that, for all |ε| ∈ ]0, D[, the system {eiλn (ε)t }∞ 1 forms a Riesz basis in L (0, T).

Proof. Let n ∈ ℕ∗ , let λn be the nth eigenvalue of T0 , and let 𝒞n = 𝒞 (λn , rn ) be the closed circle with center λn and radius rn , given by rn :=

dist(λn , σ(T0 )\{λn }) . 2

Since the family {eiλn t }n forms a Riesz basis in L2 (0, T), by using Theorem 4.12.2, there exist numbers A, B > 0 such that ∞

󵄨 󵄨2 A‖u‖2 ≤ ∑ 󵄨󵄨󵄨⟨u, eiλn t ⟩󵄨󵄨󵄨 ≤ B‖u‖2 n=1

for all u ∈ L2 (0, T). Moreover, in view of Theorem 4.12.10, the sequence {λn }∞ 1 lies in a strip parallel to the real axis, so there exists a positive constant h such that, for all n ≥ 1, |Im λn | ≤ h, where h := supn |Im λn |. We set |ε| ∈ ]0, D[, where D=

A 2 2 2 T 2 2t(rn +h) dt 2√B ∑∞ n=1 (ωn rn Mn αn ) ∫0 t e

+ A supn≥1 (q + αn +

ω2n rn Mn αn )

.

264 | 12 Bases of the perturbed operator T (ε) Since |ε|
0 and a sequence of eigenvalues (λn (ε))∞ n=1 having the form λn (ε) = λn + ελn,1 + ε2 λn,2 + ⋅ ⋅ ⋅ 2 such that, for all |ε| ∈ ]0, D[, the system {eif (λn (ε))t }∞ 1 forms a Riesz basis in L (0, T).

Proof. Let n ∈ ℕ∗ and let λn be the nth eigenvalue of T0 and |ε| ∈ ]0, D[, where D=

A H √B

T3 3

2 2 2 2 ∑∞ n=1 (ωn rn Mn αn ) + A supn≥1 (q + αn + ωn rn Mn αn )

.

We have 󵄨󵄨 if (λn (ε))t 󵄨 󵄨 󵄨 − eif (λn )t 󵄨󵄨󵄨 = 󵄨󵄨󵄨eif (λn )t (ei(f (λn (ε))−f (λn ))t − 1)󵄨󵄨󵄨. 󵄨󵄨e Consequently, since f is an H-Lipschitz function, we obtain 󵄨󵄨 if (λn (ε))t 󵄨 󵄨 󵄨 − eif (λn )t 󵄨󵄨󵄨 ≤ t 󵄨󵄨󵄨f (λn (ε)) − f (λn )󵄨󵄨󵄨 󵄨󵄨e 󵄨 󵄨 ≤ tH 󵄨󵄨󵄨λn (ε) − λn 󵄨󵄨󵄨. Then, by using the same reasoning as in Theorem 12.3.3, we conclude that the system 2 {eif (λn (ε))t }∞ 1 forms a Riesz basis in L (0, T).

12.3 Riesz basis property of families of non-harmonic exponentials | 267

12.3.3 Values of ε are both fixed and variable Theorem 12.3.5. Suppose that the hypotheses (H1)–(H3) are satisfied. If the eigenvalues (λn )n of T0 are isolated and with multiplicity one, then there exist a constant CN > 0 (N > 1), a sequence of complex numbers (εn )∞ n=1 , and two sequences of eigenvalues ∞ {λn (ε)}∞ and {λ (ε )} having the form n n n=1 n=1 λn (ε) = λn + ελn1 + ε2 λn2 + ⋅ ⋅ ⋅ ,

λn (εn ) = λn + εn λn1 + εn2 λn2 + ⋅ ⋅ ⋅ such that, for |ε| ∈ ]0, CN ], the systems (i) {eiλn (ε)t }N1 ∪ {eiλn (εn )t }∞ N+1 and iλn (ε)t N iλn t ∞ (ii) {e }1 ∪ {e }N+1 form Riesz bases in L2 (0, T). Proof. (i) Let n ∈ ℕ∗ , N > 1, let λn be the nth eigenvalue of T0 , and let 𝒞n = 𝒞 (λn , rn ) be the closed circle with center λn and radius rn , given by rn :=

dist(λn , σ(T0 )\{λn }) . 2

2 Since the family {eiλn t }∞ n=1 forms a Riesz basis in L (0, T), due to Theorem 4.12.2, there exist numbers A, B > 0 such that ∞

󵄨 󵄨2 A‖u‖2 ≤ ∑ 󵄨󵄨󵄨⟨u, eiλn t ⟩󵄨󵄨󵄨 ≤ B‖u‖2 n=1

for all u ∈ L2 (0, T). Furthermore, by using Theorem 4.12.10, the sequence {λn }∞ 1 lies in a strip parallel to the real axis. Thus, there exists a positive constant h such that, for all n ≥ 1, |Im λn | ≤ h, where h := supn |Im λn |. We set CN = min

n∈[1,N]

ηω2n rn2 Mn αn n√TBtetr1,n

A , + A(q + αn + ω2n rn Mn αn )

where ∞

4 2 k k=1

η2 = ∑

and r1,n = rn + h. Let n ∈ [1, N]. If |ε| ∈ ]0, CN ], we have |ε|
1), a sequence of complex numbers (εn )∞ n=1 , and two sequences of ∞ eigenvalues {λn (ε)}∞ and {λ (ε )} having the form n n n=1 n=1 λn (ε) = λn + ελn1 + ε2 λn2 + ⋅ ⋅ ⋅ ,

λn (εn ) = λn + εn λn1 + εn2 λn2 + ⋅ ⋅ ⋅ such that, for |ε| ∈ ]0, CN ], the systems (i) {eif (λn (ε))t }N1 ∪ {eif (λn (εn ))t }∞ N+1 and (ii) {eif (λn (ε))t }N1 ∪ {eif (λn )t }∞ N+1 form Riesz bases in L2 (0, T). Proof. (i) Let n ∈ ℕ∗ , N > 1, and let λn be the nth eigenvalue of T0 . We have 󵄨󵄨 if (λn (ε))t 󵄨 󵄨 󵄨 − eif (λn )t 󵄨󵄨󵄨 = 󵄨󵄨󵄨eif (λn )t (ei(f (λn (ε))−f (λn ))t − 1)󵄨󵄨󵄨. 󵄨󵄨e On the other hand, since f is an H-Lipschitz function, we obtain 󵄨󵄨 if (λn (ε))t 󵄨 󵄨 󵄨 − eif (λn )t 󵄨󵄨󵄨 ≤ t 󵄨󵄨󵄨f (λn (ε)) − f (λn )󵄨󵄨󵄨 󵄨󵄨e 󵄨 󵄨 ≤ tH 󵄨󵄨󵄨λn (ε) − λn 󵄨󵄨󵄨. To prove the first item, we set CN = min

n∈[1,N]

A , ηω2n rn2 Mn αn n√TBtH + A(q + αn + ω2n rn Mn αn )

where ∞

1 . 2 k k=1

η2 = ∑

12.4 Unconditional basis with parentheses | 271

We let |ε| ∈ ]0, CN ], so, applying the same reasoning as in Theorem 12.3.5 with |εn | ∈ ]0,

ηω2n rn2 Mn αn n√TBtH

A [ + A(q + αn + ω2n rn Mn αn )

for all n ≥ 1, we get the desired result. (ii) To prove (ii), it suffices to choose the same constant CN and to apply Theorem 4.12.3.

12.4 Unconditional basis with parentheses We assume that: (H5) T0 has a compact resolvent. The first result of this section is given by the following theorem. Theorem 12.4.1. Suppose that the hypotheses (H1), (H2), and (H5) are satisfied. Assume that there exist a sequence of circles 𝒞 (O, rk ), k = 1, 2, . . . , with radius rk going to infinity and a constant c > 0 such that 󵄩󵄩 −1 󵄩 −β 󵄩󵄩(T0 − λ) 󵄩󵄩󵄩 ≤ c|λ| ,

for |λ| = rk .

(12.4.1)

Then, for |ε| small enough, the spectrum of the operator T(ε) is discrete and, for any φ ∈ 𝒟(T02 ), there exists a subsequence of partial sums of the series ∑ Pn φ n

converging to φ in the sense of X, where Pn denotes the spectral projection of T(ε). Proof. Let λ ∈ ρ(T0 ). We have T(ε) − λ = T0 − λ + εT1 + ε2 T2 + ⋅ ⋅ ⋅

= (I + εT1 (T0 − λ)−1 + ε2 T2 (T0 − λ)−1 + ⋅ ⋅ ⋅)(T0 − λ).

Using equation (11.1.1), we get 󵄩󵄩 k 󵄩 −1 󵄩 k k−1 󵄩 −1 󵄩 −1 󵄩β 󵄩 −1 󵄩1−β 󵄩󵄩ε Tk (T0 − λ) φ󵄩󵄩󵄩 ≤ |ε| q (a󵄩󵄩󵄩(T0 − λ) φ󵄩󵄩󵄩 + b󵄩󵄩󵄩T0 (T0 − λ) φ󵄩󵄩󵄩 󵄩󵄩󵄩(T0 − λ) φ󵄩󵄩󵄩 ), for all k ≥ 1 and φ ∈ X. As T0 (T0 − λ)−1 = I + λ(T0 − λ)−1 , from equation (12.4.1) we obtain 󵄩󵄩 −1 󵄩 1−β 󵄩󵄩T0 (T0 − λ) 󵄩󵄩󵄩 ≤ 1 + c|λ| ,

272 | 12 Bases of the perturbed operator T (ε) for β ∈ ]0, 1[ and |λ| = rk → ∞. Hence, for |λ| = rk sufficiently large, we have 󵄩󵄩 −1 󵄩 ̃ 1−β , 󵄩󵄩T0 (T0 − λ) 󵄩󵄩󵄩 ≤ c|λ| where c̃ = 1 + c, so 󵄩󵄩 k k k−1 −β −1 󵄩 ̃ + b)‖φ‖, 󵄩󵄩ε Tk (T0 − λ) φ󵄩󵄩󵄩 ≤ |ε| q c(a|λ| for all φ ∈ X and k ≥ 1. Then, using the Neumann identity, we get (T(ε) − λ)

−1

= (T0 − λ)−1 (I + K),

−β ̃ for |ε| < (q + c(a|λ| + b))−1 , where

K = ∑ (−1)k Sk k≥1

and S = ∑ εk Tk (T0 − λ)−1 . k≥1

Taking into account hypothesis (H5), for −1

−β ̃ |ε| < (q + c(a|λ| + b)) ,

we deduce that T(ε) has a compact resolvent and −1 󵄩 󵄩󵄩 󵄩 −1 󵄩 ̃ −β . 󵄩󵄩(T(ε) − λ) 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(T0 − λ) 󵄩󵄩󵄩 ≤ c|λ|

Consequently, the result follows immediately from Theorems 11.1.1 and 3.3.1, which completes the proof of the theorem. The rest of this section is formulated as follows. Theorem 12.4.2. Suppose that the hypotheses (H1), (H2), and (H5) are satisfied. If T0 is self-adjoint and lim sup r→∞

n(𝔹(0, r), T0 ) < ∞, r 1−β

then, for |ε| small enough, the system of generalized eigenvectors of the operator T(ε) forms an unconditional basis with parentheses in X. Proof. Let λ ∈ ρ(T0 ). We set S := (T(ε) − λ) − (T0 − λ). Using equation (11.1.1), we obtain ‖Sφ‖ ≤ ∑ |ε|k qk−1 (a‖φ‖ + b‖T0 φ‖β ‖φ‖1−β ) k≥1

12.4 Unconditional basis with parentheses | 273

β 󵄩 󵄩 ≤ ∑ |ε|k qk−1 (a‖φ‖ + b(󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 + |λ|‖φ‖) ‖φ‖1−β ), k≥1

for all φ ∈ 𝒟(T0 ). Since β 󵄩β 󵄩 󵄩 󵄩 (󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 + |λ|‖φ‖) ≤ 󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 + |λ|β ‖φ‖β ,

we get ‖Sφ‖ ≤

|ε| |ε|b 󵄩󵄩 󵄩β 1−β (a + b|λ|β )‖φ‖ + 󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 ‖φ‖ , 1 − |ε|q 1 − |ε|q

(12.4.2)

for every φ ∈ 𝒟(T0 ) and |ε| < q1 . Moreover, λ ∈ ρ(T0 ), so T0 − λ is invertible and (T0 − λ)−1 is bounded. Then 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩(T0 − λ) x 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(T0 − λ) 󵄩󵄩󵄩‖x‖, for all x ∈ X. For x = (T0 − λ)φ, where φ ∈ 𝒟(T0 ), we obtain 󵄩 󵄩󵄩 󵄩 ‖φ‖ ≤ 󵄩󵄩󵄩(T0 − λ)−1 󵄩󵄩󵄩󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩, for all φ ∈ 𝒟(T0 ), so 󵄩 󵄩β 󵄩 󵄩β ‖φ‖β ≤ 󵄩󵄩󵄩(T0 − λ)−1 󵄩󵄩󵄩 󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 and 󵄩 󵄩β 󵄩 󵄩β ‖φ‖ ≤ 󵄩󵄩󵄩(T0 − λ)−1 󵄩󵄩󵄩 󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 ‖φ‖1−β .

(12.4.3)

Hence, substituting equation (12.4.3) into equation (12.4.2), we get ‖Sφ‖ ≤

|ε| 󵄩 󵄩β 󵄩 󵄩β ((a + b|λ|β )󵄩󵄩󵄩(T0 − λ)−1 󵄩󵄩󵄩 + b)󵄩󵄩󵄩(T0 − λ)φ󵄩󵄩󵄩 ‖φ‖1−β , 1 − |ε|q

for all φ∈ 𝒟(T0 ) and |ε| < q1 . Thus, the operator S is β-subordinate to T0 −λ. On the other hand, since T0 is self-adjoint, T0 − λ is normal. Consequently, using Theorem 4.14.1, we conclude that the system of generalized eigenvectors of T(ε) − λ forms an unconditional basis with parentheses in X. This completes the proof of the theorem.

13 Frame of the perturbed operator T (ε) 13.1 Frames of eigenvectors of the perturbed operator T (ε) 13.1.1 Frames of analytic eigenvectors in the sense of Kato In this section, we provide some conditions that ensure the existence of frames of eigenvectors of the perturbed operator T(ε), when the eigenvalues of T0 are with multiplicity one, with finite (or infinite) multiplicity. In order to prove the existence of frames of eigenvectors related to the perturbed operator T(ε), we consider the following hypothesis: (H6) the system of eigenvectors {φn }∞ n=1 , associated with the eigenvalues (λn )n of T0 , forms a frame for X. The main results of this part are formulated as follows. Theorem 13.1.1. Assume that the hypotheses (H1), (H2), and (H6) are satisfied. If the eigenvalues (λn )n of T0 are isolated and with multiplicity one, then there exist a sequence ∞ of complex numbers (εn )∞ n=1 and a sequence of eigenvectors {φn (εn )}n=1 having the form φn (εn ) = φn + εn φn,1 + εn2 φn,2 + ⋅ ⋅ ⋅ such that the system {φn (εn )}∞ n=1 forms a frame for X. Proof. Let n ∈ ℕ∗ . In the sequel, we designate by λn the nth eigenvalue of the operator T0 and by 𝒞n = 𝒞 (λn , rn ) the circle with center λn and with radius rn =

dn , 2

where dn = d(λn , σ(T0 )\{λn }) is the distance between λn and σ(T0 )\{λn }. Since (T0 − z)−1 is an analytic function of z, ‖(T0 − z)−1 ‖ is a continuous function of z, so we denote 󵄩 󵄩 Mn := max 󵄩󵄩󵄩(T0 − z)−1 󵄩󵄩󵄩, z∈𝒞n

󵄩 󵄩 󵄩 󵄩 Nn := max 󵄩󵄩󵄩T0 (T0 − z)−1 󵄩󵄩󵄩 = max 󵄩󵄩󵄩I + z(T0 − z)−1 󵄩󵄩󵄩, z∈𝒞n

αn := aMn +

bNnβ Mn1−β ,

z∈𝒞n

and ωn = ‖φn ‖, https://doi.org/10.1515/9783110493863-013

276 | 13 Frame of the perturbed operator T (ε) where φn is an eigenvector of T0 associated with the eigenvalue λn . In view of Corollary 11.2.1, for |ε|
0 y→0

p− (x) =

lim p[M(x, y)],

y a, 𝜕y

u(−a) = u(a) = 0,

Sommerfeld condition for p.

https://doi.org/10.1515/9783110493863-014

(14.1.1) (14.1.2) (14.1.3) (14.1.4) (14.1.5) (14.1.6) (14.1.7)

290 | 14 Perturbation method for sound radiation by a vibrating plate in a light fluid Using equations (14.1.1), (14.1.4), (14.1.5), and (14.1.7), the pressure p(M) is given by the following integrals: a

ω2 ρ0 2 p(M) = −i ∫ H0 (k0 √(x − x󸀠 ) + y2 )u(x󸀠 )dx󸀠 , 2

for y > 0,

(14.1.8)

ω2 ρ0 2 ∫ H0 (k0 √(x − x󸀠 ) + y2 )u(x󸀠 )dx󸀠 , 2

for y < 0,

(14.1.9)

−a a

p(M) = +i

−a

where H0 (⋅) is the Hankel function of the first kind and order 0 given by +∞

e−xt −1 i(x− π4 ) H0 (x) = e dt ∫ t π 0 √t(1 + i 2 )

(14.1.10)

and k0 =

ω c0

is the wave number of the fluid. Using (14.1.2), (14.1.3), (14.1.6), (14.1.8), and (14.1.9) leads to the following boundary value problem: a

iρ ρ d2 u f (x) 󵄨 󵄨 (x) + ω2 { 1 u(x) + 0 ∫ H0 (k0 󵄨󵄨󵄨x − x 󸀠 󵄨󵄨󵄨)u(x 󸀠 )dx󸀠 } = , 2 T1 T1 T1 dx −a

u(−a) = u(a) = 0.

∀ −a < x < a, (14.1.11) (14.1.12)

In the sequel we shall need the following operators: T0 : 𝒟(T0 ) ⊂ L2 (]−a, a[) 󳨀→ L2 (]−a, a[), { { { { d2 ψ ψ 󳨀→ T0 ψ(x) = − 2 (x), { { dx { { 1 2 𝒟 (T ) = H (]−a, a[) ∩ H (]−a, a[) { 0 0 and K : L2 (]−a, a[) 󳨀→ L2 (]−a, a[), { { { a i 󵄨 󵄨 { { ψ 󳨀→ Kψ(x) = H0 (k 󵄨󵄨󵄨x − x 󸀠 󵄨󵄨󵄨)ψ(x󸀠 )dx󸀠 . ∫ { 2 −a {

(14.1.13)

From the problem (14.1.11)–(14.1.12), P. J. T. Filippi in [89] has considered the following eigenvalue problem.

14.2 Compactness results | 291

Find the values λ ∈ ℂ for which there is a solution u ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[), u ≠ 0, for the equation T0 u = λ(I + εK)u, ω2 ρ

(14.1.14)

2ρ

where λ = T 1 and ε = ρ 0 . According to the definition given in [160, Chapter 9, Sec1 1 tion 4], λ is the eigenvalue and u is the eigenmode. Note that λ and u both depend on the value of ε, so we denote λ := λ(ε) and u := u(ε).

14.2 Compactness results The purpose of the first part of this section is to give some compactness results about the Hankel operator defined in (14.1.13), which play a crucial role in our subsequent analysis. Theorem 14.2.1. The Hankel operator K is compact on L2 (]−a, a[). Proof. Let us first note that the Hankel function of the first kind and order 0, H0 (⋅), is equivalent, in some neighborhood ]x − ε, x + ε[ ε > 0 of x = x 󸀠 , to the function 󵄨 󵄨 log(k 󵄨󵄨󵄨x − x󸀠 󵄨󵄨󵄨) (cf. [162]), so the operator K, defined in (14.1.13), can be written in the following form: Kψ(x) = K1 ψ(x) + K2 ψ(x) + K3 ψ(x), where K1 , K2 , and K3 denote the operators defined as follows: K1 : L2 (]−a, a[) 󳨀→ L2 (]−a, a[), { { { a i 󵄨 󵄨 { { ψ 󳨀→ K1 ψ(x) = ∫ H0 (k 󵄨󵄨󵄨x − x 󸀠 󵄨󵄨󵄨)χ(−a,x−ε) (x 󸀠 )ψ(x 󸀠 )dx󸀠 , { 2 −a { 2 2 K2 : L (]−a, a[) 󳨀→ L (]−a, a[), { { { a i 󵄨 󵄨 { { H0 (k 󵄨󵄨󵄨x − x 󸀠 󵄨󵄨󵄨)χ(x−ε,x+ε) (x 󸀠 )ψ(x 󸀠 )dx󸀠 , ψ 󳨀→ K ψ(x) = ∫ 2 { 2 −a { and K3 : L2 (]−a, a[) 󳨀→ L2 (]−a, a[), { { { a i 󵄨 󵄨 { { ψ 󳨀→ K3 ψ(x) = ∫ H0 (k 󵄨󵄨󵄨x − x 󸀠 󵄨󵄨󵄨)χ(x+ε,a) (x 󸀠 )ψ(x 󸀠 )dx󸀠 , { 2 −a { where χ(−a,x−ε) (⋅), χ(x−ε,x+ε) (⋅), and χ(x+ε,a) (⋅) denote, respectively, the characteristic functions of the intervals (−a, x − ε), (x − ε, x + ε), and (x + ε, a). In order to complete

292 | 14 Perturbation method for sound radiation by a vibrating plate in a light fluid the proof, it is sufficient to show that Ki , i = 1, 2, 3, are compact on L2 (]−a, a[). We set r = 45 , σ = 1, p = 2, and q = 2. Then, from Lemma 2.5.1 and the equality 2

(k|x − x󸀠 |) 3 log(k|x − x 󸀠 |) 󵄨 󵄨 log(k 󵄨󵄨󵄨x − x󸀠 󵄨󵄨󵄨) = , 2 (k|x − x 󸀠 |) 3 we obtain the compactness of the Hankel operator K from L2 (]−a, a[) into L2 (]−a, a[). We claim that, for all x, x󸀠 ∈ ]−a, a[ such that |x − x 󸀠 | ≥ ε, there is a positive constant c = c(ε) > 0 depending on ε such that 󵄨󵄨 󵄨 󸀠󵄨 󵄨 󵄨󵄨H0 (k 󵄨󵄨󵄨x − x 󵄨󵄨󵄨)󵄨󵄨󵄨 ≤ c.

(14.2.1)

Indeed, the expression of the Hankel function H0 (⋅) is given in [164] by (14.1.10). Let x ∈ ]−a, a[ such that |x − x󸀠 | ≥ ε. Then we have the following: +∞

󸀠

e−k|x−x |t 1 󵄨󵄨 󵄨 󸀠󵄨 󵄨 dt ∫ 󵄨󵄨H0 (k 󵄨󵄨󵄨x − x 󵄨󵄨󵄨)󵄨󵄨󵄨 ≤ π 2 t 0 √t √1 + 4 1

+∞

󸀠

󸀠

1 e−k|x−x |t 1 e−k|x−x |t ≤ ∫ dt + dt ∫ π π 2 2 1 √t √1 + t 0 √t √1 + t 4 4 1

+∞

0

1

󸀠 1 dt 1 ≤ ∫ + ∫ e−k|x−x |t dt π √t π

1 1 (2 + ) π k|x − x 󸀠 | 1 1 ≤ (2 + ). π kε

≤

This concludes the proof of the claim. Using equation (14.2.1), we deduce that a a

󵄨 󵄨 󵄨 󵄨2 ∫ ∫ 󵄨󵄨󵄨H0 (k 󵄨󵄨󵄨x − x󸀠 󵄨󵄨󵄨)χ(−a,x−ε) (x󸀠 )󵄨󵄨󵄨 dx 󸀠 dx ≤ 4a2 c2

−a −a

and a a

󵄨 󵄨 󵄨 󵄨2 ∫ ∫ 󵄨󵄨󵄨H0 (k 󵄨󵄨󵄨x − x󸀠 󵄨󵄨󵄨)χ(x+ε,a) (x 󸀠 )󵄨󵄨󵄨 dx󸀠 dx ≤ 4a2 c2 .

−a −a

Thus, K1 and K3 are Hilbert–Schmidt operators. This achieves the proof.

14.3 Spectral properties of the operator T0

| 293

14.3 Spectral properties of the operator T0 Lemma 14.3.1. We have the following assertions: (i) T0 is a closed operator; (ii) the injection from 𝒟(T0 ) into L2 (]−a, a[) is compact; (iii) the resolvent set of T0 is not empty (in fact, 0 ∈ ρ(T0 )); (iv) T0 is a self-adjoint operator with compact resolvent; (v) the continuous spectrum and the residual spectrum of T0 are empty; and (vi) the spectrum of T0 is constituted only of eigenvalues which are positive and denumerable, of which the multiplicity is one, and which have no finite limit points and satisfy 0 < λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λn → +∞

as n → +∞.

Further, λn = (

2

nπ ). 2a

Proof. (i) It suffices to prove that the graph G(T0 ) of T0 is closed. In fact, let λ ∈ ρ(T0 ) and (ψn , T0 ψn ) ∈ G(T0 ) which converges to (ψ, φ). We must show that ψ ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[) and φ = T0 ψ. The sequence (ψn )n converges to ψ in H 1 (]−a, a[) and ψn ∈H01 (]−a, a[), so ψ∈H01 (]−a, a[). The fact that (ψn )n converges to ψ in H 1 (]−a, a[) implies (ψn )n converges to ψ in L2 (]−a, a[). Hence, we infer that (λψn − T0 ψn )n converges to λψ − φ in L2 (]−a, a[). Since λ ∈ ρ(T0 ), (ψn )n converges to (λ − T0 )−1 (λψ − φ) in L2 (]−a, a[). Since (ψn )n converges to ψ in L2 (]−a, a[), we have ψ = (λ − T0 )−1 (λψ − φ) and (λ − T0 )ψ = λψ − φ, which implies that T0 ψ = φ. On the other hand, φ ∈ L2 (]−a, a[), so we get T0 ψ ∈ L2 (]−a, a[) and, as ψ ∈ H 1 (]−a, a[), ψ ∈ H 2 (]−a, a[). This completes the proof of (i).

294 | 14 Perturbation method for sound radiation by a vibrating plate in a light fluid The proofs of the items (ii), (iv), and (v) are evident. (iii) Let φ ∈ N(T0 ). Since T0 is V-elliptic, φ = 0 since φ is continuous. This implies that N(A) = {0}. Let ψ ∈ L2 (]−a, a[). We look for φ ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[) such that T0 φ = ψ. This equality must hold, in particular, for all φ ∈ H01 (]−a, a[). The fundamental lemma of the calculus of variations implies there exist two constants α and β such that x α

φ(x) = αx + β + ∫ ∫ ψ(s)dsdα, φ(−a) = 0,

−a −a

and φ(a) = 0.

We let a α

η = ∫ ∫ ψ(s)dsdα. −a −a

We find ourselves with the following linear system of two equations with two unknowns: −a a

(

1 α 0 )( ) = ( ). 1 β η

The last system has a unique solution since the matrix’s determinant is non-zero. As |η| ≤ (2a)2 ‖ψ‖ and 󵄨󵄨 x α 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ ∫ ψ(s)dsdα󵄨󵄨󵄨 ≤ (2a)2 ‖ψ‖, 󵄨󵄨 󵄨󵄨 󵄨󵄨−a −a 󵄨󵄨 we have 󵄨󵄨 x α 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨φ(x)󵄨󵄨 ≤ a|α| + |β| + 󵄨󵄨󵄨 ∫ ∫ ψ(s)dsdα󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨−a −a 󵄨 Hence, 󵄨󵄨 󵄨 󵄨󵄨φ(x)󵄨󵄨󵄨 ≤ c‖ψ‖ and 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩(0 − T0 ) 󵄩󵄩󵄩 = sup 󵄩󵄩󵄩(0 − T0 ) ψ󵄩󵄩󵄩 ≤ c. ‖ψ‖≤1

This achieves the proof of (iii).

14.3 Spectral properties of the operator T0

| 295

(vi) Let (λn )∞ n=1 be the eigenvalues of T0 . We consider the following eigenvalue problem: {

T0 φ = λn φ, φ ∈ 𝒟(T0 ).

(14.3.1)

The solution of the problem (14.3.1) is formally given by φ(x) = αei√λn x + βe−i√λn x and it satisfies the following boundary conditions: φ(−a) = φ(a) = 0. Thus, we find ourselves with a linear system of two equations with two unknowns which are vanished. To avoid trivial solutions, it suffices to ensure that the determinant of the following system be zero: ei√λn a det ( e−i√λn a

e−i√λn a ei√λn a

) = 0,

so sin(2a√λn ) = 0, i. e., λn = (

2

nπ ) 2a

with n ≠ 0 because zero is not an eigenvalue of T0 (see (iii)). This completes the proof. Theorem 14.3.1. The resolvent of the operator T0 belongs to the Carleman class Cp for any p > 21 . Proof. This follows from both Lemma 14.3.1 (v) and the fact that T0 is a self-adjoint operator. Proposition 14.3.1. Let (λn )n be the eigenvalues of T0 . Then, for each λ ≠ λn , n ∈ ℕ∗ , we have 1 ⟨φ, φn ⟩φn , λ − λn n=1 ∞

(λ − T0 )−1 φ = ∑

296 | 14 Perturbation method for sound radiation by a vibrating plate in a light fluid where (φn )n is an orthonormal system of eigenfunctions corresponding to (λn )n . Moreover, if Im λ ≠ 0 and if λ belongs to a ray with origin zero and of angle θ with θ ≠ 0 and θ ≠ π, we have 1 c(θ) 󵄩󵄩 −1 󵄩 = , 󵄩󵄩(λ − T0 ) 󵄩󵄩󵄩 ≤ |Im λ| |λ| where c(θ) is a positive constant that depends on θ. Proof. This follows immediately from Lemma 14.3.1 (iv). Proposition 14.3.2. There exists a sequence of circles 𝒞 (O, rn ), n ∈ ℕ∗ , with radius rn going to infinity such that a√10 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T0 ) 󵄩󵄩󵄩 ≤ 1 π|λ| 2 and |λ| = rn . Proof. Let n ∈ ℕ∗ . We have λn+1 − λn ≥

π2 n. 2a2

We set rn =

λ − λn λn+1 + λn = λn + n+1 . 2 2

Using both equation (2.18.1) and Lemma 14.3.1 (iv), we have 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T0 ) 󵄩󵄩󵄩 =

1 , dist(λ, σ(T0 ))

where λ belongs to ρ(T0 ). From equation (14.3.2), we obtain 1 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T0 ) 󵄩󵄩󵄩 = dist(λ, σ(T0 )) 2 , ≤ λn+1 − λn where rn = |λ|. On the other hand, rn =

λn+1 + λn 2

= [(n + 1)2 + n2 ] ≤

5π 2 2 n. 8a2

π2 8a2

(14.3.2)

14.4 On a Riesz basis in L2 (]−a, a[)

| 297

Thus, n−2 ≤

5π 2 −1 r . 8a2 n

Since 2a2 π2n 1 2a2 = 2 (n−2 ) 2 π a√5 −1 21 (rn ) ≤ π √2 a√5 ≤ , 1 π √2|λ| 2

(λn+1 − λn )−1 ≤

we have the desired result.

14.4 On a Riesz basis in L2 (]−a, a[) Theorem 14.4.1. The eigenvectors of T0 form a Riesz basis in L2 (]−a, a[). Proof. Let {φn }∞ n=1 be a system of normalized eigenvectors of the operator T0 . Using Theorem 4.7.1, this system forms a basis in L2 (]−a, a[), so, for all u ∈ L2 (]−a, a[), we write ∞

u = ∑ ⟨u, φn ⟩φn . n=0

Then ∞

∞

n=0 ∞

m=0

‖u‖2 = ⟨ ∑ ⟨u, φn ⟩φn , ∑ ⟨u, φm ⟩φm ⟩ ∞

󵄨 󵄨2 = ∑ ⟨u, φn ⟩⟨u, φm ⟩⟨φn , φm ⟩ = ∑ 󵄨󵄨󵄨⟨u, φn ⟩󵄨󵄨󵄨 . n,m=0

n=0

Now, the result follows from Proposition 4.12.2.

14.5 Closeness operators 1 It is easy to see that, for |ε| < ‖K‖ , the operator (I + εK)−1 is invertible, so the problem (14.1.14) becomes the following.

298 | 14 Perturbation method for sound radiation by a vibrating plate in a light fluid Find the values λ(ε) ∈ ℂ for which there is a solution φ ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[), φ ≠ 0, for the following equation: (I + εK)−1 T0 φ = λ(ε)φ.

(14.5.1)

The problem (14.5.1) is equivalent to the following. Find the values λ(ε) ∈ ℂ for which there is a solution φ ∈ H01 (]−a, a[) ∩ H 2 (]−a, a[), φ ≠ 0, for the following equation: (T0 − εKT0 + ε2 K 2 T0 − ⋅ ⋅ ⋅ + (−1)k εk K k T0 + ⋅ ⋅ ⋅)φ = λ(ε)φ. Theorem 14.5.1. (i) For |ε|
0

Consequently, equations (14.9.1)–(14.9.3) imply that the family {φ̃ n }∞ n=1 forms a frame for L2 (]−a, a[).

14.9 Frames | 303

We denote Tn := (−1)n K n T0 , 𝒟(Tn ) = H01 (]−a, a[) ∩ H 2 (]−a, a[), for all n ≥ 1. Remark 14.9.1. There exist positive constants a, b, and q ≥ 0 such that, for all φ ∈ 𝒟(T0 ) and for all k ≥ 1, we have ‖Tk φ‖ ≤ qk−1 (a‖φ‖ + b‖T0 φ‖). In fact, it suffices to choose a = 0, b = ‖K‖, and q = ‖K‖. Now, we are ready to state the objective of this part. Theorem 14.9.1. For |εn | small enough and |ε| small enough, there exist two sequences ∞ of eigenvectors {φ̃ n (εn )}∞ n=1 and {φ̃ n (ε)}n=1 of T(ε) having the following form: φ̃ n (εn ) = φ̃ n + εn φ̃ n,1 + εn2 φ̃ n,2 + ⋅ ⋅ ⋅ , φ̃ n (ε) = φ̃ n + εφ̃ n,1 + ε2 φ̃ n,2 + ⋅ ⋅ ⋅

N ∞ N ∞ such that the systems {φ̃ n (εn )}∞ n=1 , {φ̃ n (ε)}1 ∪ {φ̃ n (εn )}N+1 , and {φ̃ n (ε)}1 ∪ {φ̃ n }N+1 form 2 frames for L (]−a, a[).

Proof. It suffices to apply Theorems 13.1.1, 14.6.2, and 13.1.2, Lemma 14.3.1, Proposition 14.9.1, and Remark 14.9.1. Theorem 14.9.2. For |ε| small enough, there exists a sequence of eigenvectors {φ̃ n (ε)}∞ n=1 of T(ε) having the form φ̃ n (ε) = φ̃ n + εφ̃ n,1 + ε2 φ̃ n,2 + ⋅ ⋅ ⋅ 2 such that the family {φ̃ n (ε)}∞ n=1 forms a frame for L (]−a, a[).

Proof. Let n ∈ ℕ∗ , let λn be the nth eigenvalue of T0 , and let rn = min{

|λn − π3 λn−1 | |λn+1 − π3 λn | , }. 2 2

Since {z ∈ ℂ, |z − λn | ≤ rn } ∩ σ(T0 ) = {λn }, let 𝒞n = 𝒞 (λn , rn ) be the closed circle with center λn and radius rn and let z ∈ 𝒞n . As T0 is self-adjoint, in view of equation (2.18.1), we have 󵄩 󵄩 ‖Rz ‖ = 󵄩󵄩󵄩(T0 − zI)−1 󵄩󵄩󵄩 = Hence, we get αn := aMn + bNn

1 . dist(z, σ(T0 ))

304 | 14 Perturbation method for sound radiation by a vibrating plate in a light fluid := a max ‖Rz ‖ + b max ‖T0 Rz ‖ z∈𝒞n

z∈𝒞n

a = + b max ‖I + zRz ‖ z∈𝒞n rn a ≤ + b max(1 + |z|‖Rz ‖) z∈𝒞n rn λ a + b sup(2 + n ). rn rn n≥1

≤ If rn =

|λn − π3 λn−1 | , 2

then we obtain

sup αn ≤ sup( n≥1

n≥1

≤ sup n≥1

2λn 2a + b(2 + )) |λn − π3 λn−1 | |λn − π3 λn−1 |

2a 2 + b sup(2 + ) λ |λn − π3 λn−1 | n≥1 |1 − π3 λn−1 | n

2a 2 ≤ + b(2 + ) < ∞. ν |1 − π3 | Similarly, for rn =

|λn+1 − π3 λn | , 2

we have

sup αn ≤ sup( n≥1

n≥1

≤ sup n≥1

2λn 2a + b(2 + )) |λn+1 − π3 λn | |λn+1 − π3 λn |

2 2a + b sup(2 + λ ). π n+1 |λn+1 − 3 λn | n≥1 | λ − π3 | n

Thus, we get sup αn ≤ n≥1

2 2a + b(2 + ) < ∞. ν |1 − π3 |

Consequently, we obtain ̃n 2 rn Mn αn ) ≤ ‖K‖ + sup(q + αn + ω n≥1

4a + 2b(2 + ν

π 3

2 ) < ∞. −1

On the other hand, we have ∞

∞

n=1

n=1

̃n 2 = ∑ ω ̃n 2 ∑ rn2 Mn2 ω 2

∞ 1 = ∑ ( ) < ∞. n=1 n

Hence, the result follows immediately from Theorems 13.1.3 and 14.6.2, Lemma 14.3.1, Proposition 14.9.1, and Remark 14.9.1.

15 Applications to mathematical models It is worth mentioning that each section of this chapter has its own equations, notations, and symbols. In other words, the reader should recall that the same symbol does not have the same meaning or significance from one section to another.

15.1 Heat exchanger equation with a boundary feedback In this section, we are concerned with the following type of counter-flow heat exchanger equation with boundary feedback: 𝜕θ1 𝜕θ { (t, x) = −v1 1 (t, x) + h1 (θ2 (t, x) − θ1 (t, x)) { { 𝜕t 𝜕x { { { { 𝜕θ2 { 𝜕θ2 (t, x) = v2 (t, x) + h2 (θ1 (t, x) − θ2 (t, x)) { 𝜕t 𝜕x { { { { { { {θ1 (t, 0) = −k1 θ2 (t, 0), θ2 (t, l) = −k2 θ1 (t, l) {θ1 (0, x) = θ10 (x), θ2 (0, x) = θ20 (x)

for (t, x) ∈ (0, ∞) × [0, l], for (t, x) ∈ (0, ∞) × [0, l],

(15.1.1)

for t ∈ (0, ∞), for x ∈ [0, l],

where v1 , v2 , h1 , h2 , and l are positive physical parameters and k1 and k2 are feedback gains (in [143] they are assumed to be non-negative). For the sake of generality, we assume that k1 , k2 ∈ ℝ and k1 k2 ≠ 0. Let ℋ := L2 [0, l] × L2 [0, l] with inner product l

⟨f , g⟩ := ∫[f1 (x)g1 (x) + f2 (x)g2 (x)]dx, 0

where f := [f1 , f2 ]T and g := [g1 , g2 ]T ∈ ℋ. First, we write the system (15.1.1) as follows: −v1 𝜕θ1 (t, x) −h 𝜕 θ1 (t, x) ( ) = ( 𝜕θ𝜕x )+( 1 2 h2 𝜕t θ2 (t, x) v2 (t, x) 𝜕x

θ (0, x) θ (x) ( 1 ) = ( 10 ) , θ2 (0, x) θ20 (x)

θ1 (t, 0) = −k1 θ2 (t, 0),

h1 θ (t, x) )( 1 ), −h2 θ2 (t, x)

θ2 (t, l) = −k2 θ1 (t, l).

In ℋ, the operator A is defined by −v1 f1󸀠 −h1 󸀠 )+( v2 f2 h2

Af := (

T

h1 f ) ( 1) −h2 f2

for f ∈ 𝒟(A), 󸀠

󸀠

(15.1.2)

2

𝒟(A) := {f := [f1 , f2 ] ∈ ℋ such that f1 , f2 ∈ L [0, l], f1 (0) = −k1 f2 (0), f2 (l) = −k2 f1 (l)}.

(15.1.3)

https://doi.org/10.1515/9783110493863-015

306 | 15 Applications to mathematical models Then (15.1.1) can be written into an evolutionary equation in ℋ as follows: d { Θ(t) = AΘ(t), dt { {Θ(0) = Θ0

t > 0,

with Θ(t) := [θ1 (t, x), θ2 (t, x)]T and Θ0 = (θ10 (x), θ20 (x))T . Set α1 =

h1 , v1

β1 =

h2 , v2

α2 =

1 , v1

and β2 = v1 . For more details related to the results of this section, we refer the reader 2 to [143, 191]. Proposition 15.1.1. The operator A has compact resolvent on ℋ. 15.1.1 The semi-group generated by the operator A Let T ∈ ℒ(ℋ), define the operator √h1 0

T := (

0 ), √h2

and analyze the properties of a semi-group generated by the operator T −1 AT. The operator T −1 AT is expressed as −v1 f1󸀠 −h1 )+( √h1 h2 v2 f2󸀠

T −1 ATf = (

√h1 h2 f1 )( ) f2 −h2

T

−1

for f ∈ 𝒟(T −1 AT), 󸀠

󸀠

2

𝒟(T AT) = {f := [f1 , f2 ] ∈ ℋ such that f1 , f2 , f1 , f2 ∈ L [0, l],

f1 (0) = −k1 √

h2 h f (0), f2 (l) = −k2 √ 1 f1 (l)}. h1 2 h2

Then noting that, for all f ∈ 𝒟(T −1 AT), h 󵄨 h 󵄨 1 󵄨2 1 󵄨2 ⟨T −1 ATf , f ⟩ = − (v1 − v2 k22 1 )󵄨󵄨󵄨f1 (l)󵄨󵄨󵄨 − (v2 − v1 k12 2 )󵄨󵄨󵄨f2 (0)󵄨󵄨󵄨 2 h2 2 h1 l

󵄨2 󵄨 − ∫ 󵄨󵄨󵄨√h1 f1 (x) − √h2 f2 (x)󵄨󵄨󵄨 dx, 0

it is easily seen that, if the feedback gains k1 and k2 satisfy k12 ≤

α1 , β1

k22 ≤

β1 , α1

then ⟨T −1 ATf , f ⟩ ≤ 0

(15.1.4)

15.1 Heat exchanger equation with a boundary feedback | 307

for f ∈ 𝒟(T −1 AT), where α1 :=

h1 v1

β1 :=

h2 . v2

and

Hereafter, it is assumed that the physical parameters satisfy α1 ≠ β1 . Next, we consider the adjoint system of T −1 AT, (T −1 AT)∗ , concretely written as follows: ∗

v1 f1󸀠 −h1 󸀠) + ( √h −v2 f2 1 h2

(T −1 AT) f = ( −1

√h1 h2 f1 )( ) −h2 f2

T

∗

∗

for f ∈ 𝒟((T −1 AT) ), 󸀠

󸀠

2

𝒟((T AT) ) = {f := [f1 , f2 ] ∈ ℋ such that f1 , f2 , f1 , f2 ∈ L [0, l],

f1 (l) = −k2

v2 h1 v h √ f (l), f2 (0) = −k1 1 √ 2 f1 (0)}. v1 h2 2 v2 h1

Then it is easy to see that, for all f ∈ 𝒟((T −1 AT)∗ ), v h 󵄨 v h 󵄨 1 ∗ 󵄨2 1 󵄨2 ⟨(T −1 AT) f , f ⟩ = − v2 (1 − 2 k22 1 )󵄨󵄨󵄨f2 (l)󵄨󵄨󵄨 − v1 (1 − 1 k12 2 )󵄨󵄨󵄨f1 (0)󵄨󵄨󵄨 2 v1 h2 2 v2 h1 l

󵄨 󵄨2 − ∫ 󵄨󵄨󵄨√h1 f1 (x) − √h2 f2 (x)󵄨󵄨󵄨 dx. 0

Therefore, we see that ∗

⟨(T −1 AT) f , f ⟩ ≤ 0 for f ∈ 𝒟((T −1 AT)∗ ) if the feedback gains k1 and k2 satisfy the condition (15.1.4). Also, it is easily verified that both T −1 AT and (T −1 AT)∗ are densely defined closed linear operators. Hence, under the assumption that the condition (15.1.4) is satisfied, the operator −1 T −1 AT generates a contractive C0 -semi-group etT AT . Hence, we have the following result of N. Kunimatsu and H. Sano, given in [143]. Theorem 15.1.1. The operator A generates a uniformly bounded C0 -semi-group etA . 15.1.2 Eigenvalues of the operator A In this section, we calculate the eigenvalues of the operator A, i. e., finding the values λ ∈ ℂ such that Af = λf

for f ≠ 0.

(15.1.5)

308 | 15 Applications to mathematical models Equation (15.1.5) is equivalent to λf1 + v1 f1 + h1 f1 − h1 f2 = 0, { { { λf − v2 f2󸀠 − h2 f1 + h2 f2 = 0, { { 2 { {f1 (0) + k1 f2 (0) = 0, f2 (l) + k2 f1 (l) = 0. 󸀠

(15.1.6)

Moreover, this is equivalent to the following equation: f2󸀠󸀠 (x) + [(α2 − β2 )λ + α1 − β1 ]f2󸀠 (x) − [α2 β2 λ2 + (α1 β2 + β1 α2 )λ]f2 (x) = 0, { { { 󸀠 {[β1 (k1 + 1) + β2 λ]f2 (0) − f2 (0) = 0, { { 󸀠 {[β1 + k2 (β1 + β2 λ)]f2 (l) − k2 f2 (l) = 0, where α2 = is

1 v1

and β2 =

1 . v2

(15.1.7)

The characteristic equation of the first equation of (15.1.7)

r 2 + [(α2 − β2 )λ + α1 − β1 ]r − [α2 β2 λ2 + (α1 β2 + α2 β1 )λ] = 0.

(15.1.8)

The solutions of equation (15.1.8) are given by 1 r = [β1 − α1 + (β2 − α2 )λ ± √Δ(λ)], 2 where 2

Δ(λ) := [α1 + β1 + (α2 + β2 )λ] − 4α1 β1 and √Δ(λ) is taken as the non-negative real part throughout this section. Hereafter, we set 1 { {r1 := 2 [β1 − α1 + (β2 − α2 )λ + √Δ(λ)], { 1 { √ {r2 := 2 [β1 − α1 + (β2 − α2 )λ − Δ(λ)].

(15.1.9)

Here, we consider two cases with respect to r1 and r2 . (i) The case where r1 ≠ r2 . We set the solution f2 (x) of (15.1.7) as f2 (x) = c1 er1 x + c2 er2 x . From the second and third equations of (15.1.7), we then get [β1 + k2 (β1 + β2 λ − r1 )]er1 l β1 (k1 + 1) + β2 λ − r1

(

c 0 [β1 + k2 (β1 + β2 λ − r2 )]er2 t ) ( 1) = ( ) . c2 0 β1 (k1 + 1) + β2 λ − r2

(15.1.10)

15.1 Heat exchanger equation with a boundary feedback | 309

Here, setting the determinant of the coefficient matrix of (15.1.10) equal to 0, we obtain 2 η sinh z + β1 (1 − k1 k2 )z cosh z = 0, l

(15.1.11)

where η := 2{[β1 + k2 (β1 + β2 λ)][β1 (k1 + 1) + β2 λ] + k2 r1 r2 }

− [β1 − α1 + (β2 − α2 )λ][β1 + 2k2 (β1 + β2 λ) + β1 k1 k2 ],

l z := √Δ(λ). 2

(15.1.12) (15.1.13)

Moreover, using (15.1.11)–(15.1.13), we derive the following equation: 4k1 k2

z2 2z sinh2 z + (k2 α1 + β1 k1 )[− (1 − k1 k2 ) cosh z sinh z] l l2 + [α1 β1 (1 + k1 k2 )2 − (k2 α1 + β1 k1 )2 ] sinh2 z =

z2 (1 − k1 k2 )2 . l2

(15.1.14)

Now, by choosing the feedback gains as k1 = √

α1 β1

k2 = √

β1 α1

and

(this choice satisfies the condition (15.1.4)), equation (15.1.14) becomes the following: z sinh z = 0. In this case, from the fact that r1 ≠ r2 leads to z ≠ 0, we obtain sinh z = 0. Therefore, it follows that z = ikπ, for k = ±1, ±2, ±3, . . .. Moreover, using the definition of Δ(λ) and (15.1.13), we get 2

λ = λ±k

k 2 α + β1 2√α1 β1 − l2 π := − 1 ± α2 + β2 α2 + β2

for k ∈ ℕ.

Defining M1 and M2 (where ℕ = M1 ∪ M2 ) as M1 := {k ∈ ℕ such that

l2 α β ≥ k 2 }, π2 1 1

310 | 15 Applications to mathematical models

M2 := {k ∈ ℕ such that

l2 α β < k 2 }, π2 1 1

we have, for k ∈ M1 , 2

λ±k

k 2 (√α1 + √β1 )2 (√α1 − √β1 )2 α + β1 2√α1 β1 − l2 π := − 1 ± ∈ [− ,− ] α2 + β2 α2 + β2 α2 + β2 α2 + β2

and, for k ∈ M2 , 2

λ±k

k 2 α + β1 2√ l2 π − α1 β1 := − 1 ± i. α2 + β2 α2 + β2

(ii) The case of r1 = r2 = 21 [β1 − α1 + (β2 − α2 )λ], i. e., λ=−

(√α1 ∓ √β1 )2 . α2 + β2

Setting the solution f2 (x) of (15.1.7) as f2 (x) = c1 er1 x + c2 xer1 x , from the second and third equations of (15.1.7), we get (

β1 (k1 + 1) + β2 λ − r1 β1 + k2 (β1 + β2 λ − r1 )

−1 c 0 ) ( 1) = ( ) . β1 l + k2 [(β1 + β2 λ − r1 )l − 1] c2 0

(15.1.15)

β

Noting that k1 = √ αβ1 and k2 = √ α1 , we see that: 1

–

for λ = −

(√α1 −√β1 )2 α2 +β2

1

, this value is not the eigenvalue of the operator A, since

det[the coefficient matrix of (15.1.15)] = 4lβ1 √α1 β1 > 0; and –

for λ = −

(√α1 +√β1 )2 α2 +β2

:= λ0 , the eigenvalue of the operator A, det[the coefficient matrix of (15.1.15)] = 0.

From (i) and (ii), we obtain the result of N. Kunimatsu and H. Sano given in [143]. Proposition 15.1.2. The eigenvalues σ(A) = {λ0 , λ±1 , λ±2 , . . .} of the operator A satisfy sup{Re λ such that λ ∈ σ(A)} ≤ −

(√α1 − √β1 )2 . α2 + β2

15.1 Heat exchanger equation with a boundary feedback | 311

15.1.3 The stability of the semi-group generated by the operator A In Theorem 15.1.1, we have shown that the operator A generates a uniformly bounded C0 -semi-group etA (i. e., ‖etA ‖ ≤ M, t ≥ 0) when the feedback gains k1 and k2 are chosen such that k1 = √

α1 β1

k2 = √

β1 . α1

and

Then it follows from the Hille–Yosida theorem (see Theorem 5.1.1) that M M 󵄩󵄩 −1 󵄩 ≤ 󵄩󵄩(λ − A) 󵄩󵄩󵄩 ≤ Re λ ε for Re λ ≥ ε and for any ε > 0. Define the set E by E := {λ ∈ ℂ such that −

(√α1 − √β1 )2 + ε ≤ Re λ ≤ ε, |Im λ| ≤ h}. α2 + β2

Then, since E ⊂ ρ(A) holds for any h > 0, it is easily seen that 󵄩 󵄩 sup 󵄩󵄩󵄩(λ − A)−1 󵄩󵄩󵄩 < ∞. λ∈E

Set Eh := {λ ∈ ℂ such that −

(√α1 − √β1 )2 + ε ≤ Re λ ≤ ε, |Im λ| ≥ h}. α2 + β2

Hereafter, the following lemma will be needed. Lemma 15.1.1 (N. Kunimatsu and H. Sano [143, Lemma 5.1]). For any λ ∈ Eh , the following inequalities hold: 0 ≤ Re √Δ(λ) ≤ 2(α1 + β1 − √α1 β1 ) + (α2 + β2 )ε, 󵄨 󵄨󵄨 √ 󵄨󵄨Im Δ(λ)󵄨󵄨󵄨 ≥ (α2 + β2 )|Im λ|.

312 | 15 Applications to mathematical models Let λ ∈ Eh . Using Lemma 15.1.1, we get the following estimates: α 󵄨 󵄨󵄨 { { 󵄨󵄨k1 μ + 2α1 󵄨󵄨󵄨 ≥ √ 1 |y|, { { { β1 { { { { { 󵄨 󵄨󵄨√ 2 { 󵄨 Δ(λ)󵄨󵄨󵄨 ≤ (√α1 + √β1 ) + (α2 + β2 )ε + |y|, { { {󵄨 { { l |y|2 { { { {|sinh z| ≥ sinh( 2 [2√α1 β1 + (α2 + β2 )ε]√ |y|2 + 4α β ), { { 1 1 { { { 󵄨󵄨 {󵄨󵄨󵄨 1 1 { {󵄨󵄨exp{ [α − β + (α − β )λ]l}󵄨󵄨󵄨 ≤ exp { [2(α + β − √α β ) + (α + β )ε]l}, 1 2 2 1 1 1 1 2 2 󵄨󵄨󵄨 2 1 2 {󵄨󵄨󵄨

(15.1.16) where y := (α2 + β2 ) Im λ. Here, in order to apply Theorem 5.1.4, it must be shown that there exists an h > 0 such that 󵄩 󵄩 sup 󵄩󵄩󵄩(λ − A)−1 󵄩󵄩󵄩 < ∞. λ∈Eh

(15.1.17)

For each λ ∈ Eh and each g ∈ ℋ, we solve f ∈ ℋ such that (λ − A)f = g.

(15.1.18)

Equation (15.1.18) is equivalent to the following equation: f (x) α d f1 (x) ( ) = Λ( 1 ) + ( 2 f2 (x) 0 dx f2 (x) f1 (0) = −k1 f2 (0),

0 g (x) )( 1 ), −β2 g2 (x)

(15.1.19)

f2 (l) = −k2 f1 (l),

where −α1 − α2 λ −β1

Λ=(

α1 ), β1 + β2 λ

f1 (x) ), f2 (x)

f =( and

g (x) g = ( 1 ). g2 (x) In the above, it is easy to see that the solutions of the characteristic equation det(rI − Λ) = 0 of the matrix Λ become r = r1 , r2 , where r1 and r2 are defined in (15.1.9). Noting that r1 ≠ r2 holds for sufficiently large h > 0, the state-transition matrix of Λ is calculated

15.1 Heat exchanger equation with a boundary feedback | 313

as follows: e

Λx

=

μ (er2 x − er1 x ) + 21 (er1 x 2√Δ(λ) ( β1 (er2 x − er1 x ) √Δ(λ)

+ er2 x ) μ (e 2√Δ(λ)

α1 (er1 x √Δ(λ) r1 x r2 x

− er2 x )

), − e ) + 21 (er1 x + er2 x )

(15.1.20)

where μ := α1 + β1 + (α2 + β2 )λ. Lemma 15.1.2. For sufficiently large h > 0, we have 󵄩 󵄩 sup 󵄩󵄩󵄩eΛx 󵄩󵄩󵄩ℒ(ℂ2 ) ≤ M0 ,

(15.1.21)

x∈[0,l]

where M0 is a positive constant. Proof. From (15.1.20), we have [1

k1 ] eΛl [

k μ + 2α1 −k1 1 ]=2 1 exp{ [β1 − α1 + (β2 − α2 )λ]l} sinh z. 1 2 √Δ(λ)

(15.1.22)

Here, noting that sinh z ≠ 0 holds since λ ∈ Eh ⊂ ρ(A) and that k1 μ + 2α1 ≠ 0 for sufficiently large h > 0, we see that [[1

k1 ] eΛl [

−1

−k1 ]] 1

exists. Here, from (15.1.22) and (15.1.16), we have 󵄨󵄨 󵄨󵄨 󵄨󵄨 [[1 󵄨󵄨

exp{ 21 [2(α1 + β1 − √α1 β1 ) + (α2 + β2 )ε]l} 󵄨󵄨 󵄨󵄨 ≤ γ := 󵄨󵄨 󵄨 2√ αβ1 |y|

−1 󵄨

−k k1 ] eΛl [ 1 ]] 1

1

×

(√α1 + √β1 )2 + (α2 + β2 )ε + |y|

2

|y| sinh( 2l [2√α1 β1 + (α2 + β2 )ε]√ |y|2 +4α ) β

.

(15.1.23)

1 1

Noting that |y| = (α2 + β2 )|Imλ| ≥ (α2 + β2 )h, it is easily seen that |y| → ∞ as h → ∞. Therefore, as h goes to infinity, γ in (15.1.23) becomes as follows: γ 󳨀→

exp{ 21 [2(α1 + β1 − √α1 β1 ) + (α2 + β2 )ε]l} 2√ αβ1 sinh( 2l [2√α1 β1 + (α2 + β2 )ε]) 1

.

314 | 15 Applications to mathematical models Hence, it follows that there exists a positive constant h0 > 0 such that, for all h ≥ h0 , 󵄨󵄨 󵄨󵄨 󵄨󵄨[[1 󵄨󵄨 Also, noting that

−1 󵄨

−k k1 ] e [ 1 ]] 1 Λl

󵄨󵄨 μ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≤ M1 , 󵄨󵄨 √Δ(λ) 󵄨󵄨󵄨

󵄨󵄨 󵄨󵄨 ≤ M0 . 󵄨󵄨 󵄨

(15.1.24)

󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ≤ M1 󵄨󵄨 √Δ(λ) 󵄨󵄨󵄨

for sufficiently large h > 0 and that 󵄨󵄨 r1 x 󵄨󵄨 󵄨󵄨e 󵄨󵄨 ≤ M1 and

󵄨󵄨 r2 x 󵄨󵄨 󵄨󵄨e 󵄨󵄨 ≤ M1

for each x ∈ [0, l], we see that, for sufficiently large h > 0, 󵄩 󵄩 sup 󵄩󵄩󵄩eΛx 󵄩󵄩󵄩ℒ(ℂ2 ) ≤ M0 .

x∈[0,l]

The proof is complete. Noting the above and the relation k1 k2 = √

α1 β1 √ = 1, β1 α1

from (15.1.19), we obtain f (x) [ 1 ] = − [[1 f2 (x)

k1 ] eΛl [

−1 l

−k1 −k ]] ∫ eΛx [ 1 1 1 0

x

α + ∫ eΛ(x−ξ ) [ 2 0 0

−k12 Λ(l−ξ ) α2 ]e [ k1 0

0 g (ξ ) ] [ 1 ] dξ −β2 g2 (ξ )

0 g (ξ ) ] [ 1 ] dξ . −β2 g2 (ξ )

(15.1.25)

Lemma 15.1.3. For sufficiently large h > 0, we have 󵄩 󵄩 sup 󵄩󵄩󵄩(λ − A)−1 󵄩󵄩󵄩 ≤ l(α2 + β2 )M0 [(k1 + 1)2 M02 + 1], λ∈Eh

where M0 is a positive constant. Proof. By using equations (15.1.21), (15.1.24), and (15.1.25), we see that 󵄩󵄩 f (x) 󵄩󵄩 󵄩󵄩 g 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩 󵄩 ]󵄩󵄩󵄩 ≤ √l(α2 + β2 )M0 [(k1 + 1)2 M02 + 1]󵄩󵄩󵄩[ 1 ]󵄩󵄩󵄩 󵄩󵄩[ 󵄩󵄩 f2 (x) 󵄩󵄩ℂ2 󵄩󵄩 g2 󵄩󵄩ℋ

for x ∈ [0, l].

(15.1.26)

15.1 Heat exchanger equation with a boundary feedback | 315

Moreover, in view of (15.1.26), we have 1 2

l 󵄩󵄩 f (x) 󵄩󵄩2 󵄩󵄩 f 󵄩󵄩 󵄩󵄩 g 󵄩󵄩 󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 1 󵄩󵄩 󵄩 ]󵄩󵄩󵄩 dx) ≤ l(α2 + β2 )M0 [(k1 + 1)2 M02 + 1]󵄩󵄩󵄩[ 1 ]󵄩󵄩󵄩 . = ( [ ] ∫ 󵄩󵄩[ 󵄩󵄩 󵄩󵄩 󵄩󵄩 f2 (x) 󵄩󵄩ℂ2 󵄩󵄩 f2 󵄩󵄩ℋ 󵄩󵄩 g2 󵄩󵄩ℋ 0

g

This implies that, for all [ g21 ] ∈ ℋ and all λ ∈ Eh , 󵄩󵄩 󵄩 󵄩󵄩 g 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 −1 g 󵄩 2 2 󵄩󵄩(λ − A) [ 1 ]󵄩󵄩󵄩 ≤ l(α2 + β2 )M0 [(k1 + 1) M0 + 1]󵄩󵄩󵄩[ 1 ]󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩 g2 󵄩󵄩ℋ g2 󵄩󵄩ℋ

(15.1.27)

From (15.1.27), we obtain 󵄩 󵄩 sup 󵄩󵄩󵄩(λ − A)−1 󵄩󵄩󵄩 ≤ l(α2 + β2 )M0 [(k1 + 1)2 M02 + 1]. λ∈Eh

This shows the result and completes the proof. By using both Lemma 15.1.3 (see (15.1.17)) and Theorem 5.1.4, we have the following result. Proposition 15.1.3 (N. Kunimatsu and H. Sano [143, Proposition 5.1]). For any ε > 0, there exists a constant Mε > 0 such that 2

(√α1 − √β1 ) 󵄩󵄩 tA 󵄩󵄩 − ε)t} (t ≥ 0). 󵄩󵄩e 󵄩󵄩ℒ(ℋ) ≤ Mε exp{−( α +β 2

2

Remark 15.1.1. This result shows that static feedback with the gains k1 = √ αβ1 and 1

k2 =

√ αβ1 1

decreases the degree of stability of the system.

15.1.4 Auxiliary operator We introduce an auxiliary operator A0 defined by −v1 f1󸀠 −h1 󸀠 )+( v2 f2 0

A0 f := (

0 f ) ( 1) −h2 f2

(15.1.28)

with domain 𝒟(A0 ) = 𝒟(A).

To prove that the eigenvector system of A0 forms a Riesz basis for ℋ, we need the following lemma.

316 | 15 Applications to mathematical models Lemma 15.1.4. Let γ ∈ (0, 1) and En (x) := e2nπγix . Then En (x) forms a Riesz basis for L2 [0, 1]. Proof. It is a direct consequence from the fact that {e2nπix , n ∈ ℤ} is an orthonormal basis in L2 [0, 1]. Theorem 15.1.2. Let A0 be given in (15.1.28). Then A0 has compact resolvent. Proof. The proof is left to the reader. Theorem 15.1.3 (G. Q. Xu and S. P. Yung [191, Theorem 3.1]). Let (15.1.28). Then the spectrum of A0 is given by log(k k )−(α +β )l

1 2 1 1 + (α2nπ i, { (α2 +β2 )l { 2 +β2 )l λn := { { log(|k1 k2 |)−(α1 +β1 )l 2(n+1)π + (α +β )l i, (α2 +β2 )l { 2 2

A0

k1 k2 > 0,

n ∈ ℤ,

k1 k2 < 0,

n ∈ ℤ.

be

given

in

(15.1.29)

Proof. We calculate the eigenvalues and eigenvectors of A0 . Let λ ∈ ℂ such that A0 f = λf for f ≠ 0, i. e., λf1 (x) = −v1 f1 (x) − h1 f1 (x) { { { λf (x) = v2 f2󸀠 (x) − h2 f2 (x) { { 2 { {f1 (0) = −k1 f2 (0), 󸀠

∀x ∈ (0, l), ∀x ∈ (0, l), f2 (l) = −k2 f1 (l).

Solving the equations, we obtain f1 (x) = −k1 e−(α2 λ+α1 )x ,

{

f2 (x) = e(β2 λ+β1 )x

with λ satisfying e(β2 λ+β1 )l − k2 k1 e−(α2 λ+α1 )l = 0. Thus, log(k k )−(α +β )l

1 2 1 1 + (α2nπ i, { (α2 +β2 )l { 2 +β2 )l λn = { { log(|k1 k2 |)−(α1 +β1 )l 2(n+1)π + (α +β )l i, (α2 +β2 )l { 2 2

k1 k2 > 0,

n ∈ ℤ,

k1 k2 < 0,

n ∈ ℤ,

which completes the proof. Theorem 15.1.4 (G. Q. Xu and S. P. Yung [191, Theorem 3.1]). Each eigenvalue of A0 is simple and the corresponding eigenvector system forms a Riesz basis for ℋ.

15.1 Heat exchanger equation with a boundary feedback | 317

Proof. Obviously, each λn given in (15.1.29) of A0 is simple and the corresponding eigenvector is Fn = [

−k1 e−(α2 λn +α1 )x ], e(β2 λn +β1 )x

n ∈ ℤ.

We now show that {Fn , n ∈ ℤ} forms a Riesz basis for ℋ. Since −k e Fn = [ 1 [

log(k1 k2 )+(α1 β2 −α2 β1 )l x (α2 +β2 )l

2nπα2 x (α2 +β2 )l

] [e 2nπβ ] log(k1 k2 )−(α1 β2 −α2 β1 )l 2 x x (α +β )l (α2 +β2 )l e ] [e 2 2 ] 0

0

and the operator T is defined, for F = [f1 , f2 ]T ∈ ℋ, by TF :=

[−k1 e

log(k1 k2 )+(α1 β2 −α2 β1 )l x (α2 +β2 )l

0

[

0

e

log(k1 k2 )−(α1 β2 −α2 β1 )l x (α2 +β2 )l

] [f1 ] f2 ]

it is an invertible bounded linear operator, so we only need to prove that 2nπα2

xi

e (α2 +β2 )l Gn = [ 2nπβ2 ] , xi (α +β )l [e 2 2 ]

n ∈ ℤ,

forms a Riesz basis for ℋ. According to Lemma 15.1.4, 2nπα2

xi

2nπβ2

xi

{e (α2 +β2 )l , n ∈ ℤ} and {e (α2 +β2 )l , n ∈ ℤ} are Riesz bases for L2 [0, l], so {Gn , n ∈ ℤ} also forms a Riesz basis for ℋ. The proof is completed.

15.1.5 Completeness system We will use different approaches to investigate the properties of A and to show that the generalized eigenvectors of A are complete in ℋ. Let B := A − A0 . It is easy to see that B is a bounded linear operator on ℋ. Corollary 15.1.1 (G. Q. Xu and S. P. Yung [191, Corollary 3.1]). Let A0 be defined by (15.1.28) and k1 k2 ≠ 0. Then A0 generates a C0 -group in ℋ, so A also generates a C0 -group.

318 | 15 Applications to mathematical models Proof. According to Theorem 15.1.4, {Fn , n ∈ ℤ} forms a Riesz basis in ℋ. Let {Fn∗ , n ∈ ℤ} be the biorthogonal system associated with {Fn , n ∈ ℤ}. We define an operator S(t) by S(t)f := ∑ eλn t ⟨f , Fn∗ ⟩Fn

∀f = [f1 , f2 ]T ∈ ℋ.

n∈ℤ

(15.1.30)

Since | log(k1 k2 )| + (α1 + β1 )l 󵄨󵄨 λn t 󵄨󵄨 ]} 󵄨󵄨e 󵄨󵄨 ≤ exp{t[ (α + β )l 2

2

∀n ∈ ℤ,

S(t)f in (15.1.30) is well defined. Clearly, S(t) is a strongly continuous group, so the perturbation theory of semi-groups (e. g., see [22]) says that A = A0 + B is a generator of a C0 -group. Corollary 15.1.2 (G. Q. Xu and S. P. Yung [191, Corollary 3.2]). Let A be defined by (15.1.2) and (15.1.3). Then the following assertions are true: (i) A has compact resolvent; (ii) if k1 k2 ≠ 0, then the family of generalized eigenvectors of A is complete in ℋ. Proof. Let λ ∈ ℂ. For any g := [g1 , g2 ]T ∈ ℋ, we consider the resolvent problem (λ − A)f = g, i. e., λf1 + v1 f1󸀠 + h1 f1 − h1 f2 = g1 ,

λf2 − v2 f2󸀠 − h2 f1 + h2 f2 = g2 with boundary conditions f1 (0) + k1 f2 (0) = 0

and f2 (l) + k2 f1 (l) = 0.

Under the previous notations, we have λα + α1 d f1 ( )+( 2 β1 dx f2

−α1 f α g (x) ) ( 1) = ( 2 1 ). −(λβ2 + β1 ) f2 −β2 g2 (x)

Solving them yields x

f f (0) α g (s) ( 1 ) = exp{−A(λ)x} ( 1 ) + ∫ exp{−A(λ)(x − s)} ( 2 1 ) ds, f2 f2 (0) −β2 g2 (s) 0

15.1 Heat exchanger equation with a boundary feedback | 319

where λα2 + α1 β1

A(λ) = (

−α1 ). −(λβ2 + β1 )

Using the boundary conditions, we obtain l

f2 (0) [k2

1 1] exp{−A(λ)l} [ ] + ∫ [k2 −k1 0

α g (s) 1] exp{−A(λ)(x − s)} [ 2 1 ] ds = 0. −β2 g2 (s)

If 1] exp{−A(λ)l} [

Γ(λ) = [k2

1 ] ≠ 0, −k1

then the solution can be expressed as follows: l

k f 1 [ 1] = − ∫ exp{−A(λ)x} [ 2 −k1 k2 f2 Γ(λ) x

0

α g (s) 1 ] exp{−A(λ)(l − s)} [ 2 1 ] ds −β2 g2 (s) −k1

α2 g1 (s) ] ds −β2 g2 (s)

+ ∫ exp{−A(λ)(x − s)} [ 0

= R(λ, A)g. Thus, we deduce that R(λ, A) is compact on ℋ and 󵄩󵄩 󵄩 2l(α +β )|λ| 󵄩󵄩R(λ, A)󵄩󵄩󵄩 = O(e 2 2 ). By Corollary 15.1.1, R(λ, A) is bounded on arg λ = π as k1 k2 ≠ 0. Therefore, Lemma 5.1.1 yields the completeness of the generalized eigenvectors of A. 15.1.6 Asymptotic behavior of the eigenvalues of the operator A To investigate the eigenvalue problem of A, let λ ∈ σ(A), f = [f1 , f2 ]T ∈ ℋ such that (λ − A)f = 0. By using equation (15.1.6), we know f1 and f2 are the solutions of the equation y󸀠󸀠 + [(α2 λ + α1 ) − (β2 λ + β1 )]y󸀠 + [α1 β1 − (α1 λ + α1 )(β2 λ + β1 )]y = 0. Denote 1 b(λ) := [(α2 λ + α1 ) − (β2 λ + β1 )], 2

320 | 15 Applications to mathematical models 1 2 w(λ) := √[α1 + β1 + (α2 + β2 )λ] − 4α1 β1 , 2 1 v(λ) := [α1 + β1 + (α2 + β2 )λ]. 2

(15.1.31) (15.1.32)

Set fj (x) := ξj e(−b(λ)+w(λ))x + ηj e(−b(λ)−w(λ))x ,

j = 1, 2.

Substituting fj (x) into (15.1.6) leads to some algebraic equations. We have {[v(λ) + w(λ)]ξ1 − α1 ξ2 = 0, { { { { { {(v(λ) − w(λ))η1 − α1 η2 = 0, { { { ξ1 + η1 + k1 (ξ2 + η2 ) = 0, { { { { (−b(λ)+w(λ))l + (k2 η1 + η2 )e(−b(λ)−w(λ))l = 0. {(k2 ξ1 + ξ2 )e Thus, the equations in (15.1.6) will have non-zero solutions if and only if α1 + k1 [v(λ) + w(λ)] Γ(λ) := det [ (α1 k2 + v(λ) + w(λ))ew(λ)l

α1 + k1 [v(λ) − w(λ)] ] = 0, [α1 k2 + v(λ) − w(λ)]e−w(λ)l

i. e., 0 = Γ(λ) = [α1 + k1 (v(λ) + w(λ))][α1 k2 + v(λ) − w(λ)]e−w(λ)l

− [α1 + k1 (v(λ) − w(λ))][α1 k2 + v(λ) + w(λ)]ew(λ)l .

From (15.1.31) and (15.1.32), we have w(λ) − v(λ) = O(λ−1 ), when |λ| is sufficiently large, so Γ(λ) = [α1 k1 k2 (α2 + β2 )λ + O(1)]e−v(λ)l − [α1 (α2 + β2 )λ + O(1)]ev(λ)l . Set G(λ) := k1 k2 e−v(λ)l − ev(λ)l . Then we have G(λ) −

Γ(λ) = O(λ−1 ) α1 (α2 + β2 )λ

as |λ| large enough. With these, we are all set to prove the following result.

(15.1.33)

15.2 The shape memory alloys operator

| 321

Theorem 15.1.5 (G. Q. Xu and S. P. Yung [191, Theorem 3.2]). Let A be defined as before and k1 k2 ≠ 0. Then the following assertions hold: (i) for each λ ∈ σ(A), dim N(λ − A) = 1; and (ii) for λ ∈ σ(A), the chain length m(λ) is 1 when |λ| is large enough and λ = λn + O(λn−1 )

(15.1.34)

with λn as given in (15.1.29). Proof. From the previous discussion, we know that the first assertion is true. Also, we note that each λn , n∈ℤ, defined by (15.1.29) is a simple zero of G(λ). Applying Rouché’s theorem to (15.1.33), we conclude that Γ(λ) has simple zero ξn very close to λn as |λ| is large enough. Set ξn := λn + ηn and substitute ξn into (15.1.33). This yields G(ξn ) = O(ξn−1 ). Expanding G(λ) at λn , we have G(ξn ) = G󸀠 (λn )(ξn − λn ) + O((ξn − λn )2 ) = O(ξn−1 ). The estimate (15.1.34) follows.

15.2 The shape memory alloys operator Let A : 𝒟(A) ⊂ L2 (0, 1) 󳨀→ L2 (0, 1), { { { { 𝜕4 ψ ψ 󳨀→ Aψ(x) = γ 4 (x), { { 𝜕x { { 4 𝒟 (A) = {ψ ∈ H (0, 1) such that ψ(0) = ψ(1) = ψ󸀠󸀠 (0) = ψ󸀠󸀠 (1) = 0}, { where γ is a positive constant arising in Landau–Ginzburg potentials. For more details related to the results of this section, we refer the reader to [112, 182].

15.2.1 Elementary results We have the following theorem. Theorem 15.2.1. The operator A is strictly positive.

322 | 15 Applications to mathematical models Proof. Let u ∈ 𝒟(A). Integration by parts of ⟨Au, u⟩ yields 1

󵄩 󵄩2 ⟨Au, u⟩ = ⟨γu󸀠󸀠󸀠󸀠 , u⟩ = γ ∫ u󸀠󸀠󸀠󸀠 udx = γ 󵄩󵄩󵄩u󸀠󸀠 󵄩󵄩󵄩 ≥ 0. 0

Moreover, ⟨Au, u⟩ = 0 implies ‖u󸀠󸀠 ‖ = 0, which in turn implies u = 0 since u|𝜕[0,1] = 0, so A is strictly positive. Theorem 15.2.2. The operator A is self-adjoint. Proof. Let v ∈ 𝒟(A). Then, for any u ∈ 𝒟(A), we have 1

⟨Au, v⟩ = γ ∫ u󸀠󸀠󸀠󸀠 vdx 0

1

= γ ∫ v󸀠󸀠󸀠󸀠 udx 1

0

= ∫ uγv󸀠󸀠 dx 0

= ⟨u, Av⟩, so v ∈ 𝒟(A∗ ) and A∗ v = Av, i. e., A is symmetric. Furthermore, if u ∈ 𝒟(A∗ ), then there exists v ∈ L2 (0, 1) such that, for all w ∈ 𝒟(A), 1

0 = ⟨Aw, u⟩ − ⟨w, v⟩ = ∫(γw󸀠󸀠󸀠󸀠 u − wv)dx.

(15.2.1)

0

This equality must hold in particular for all w ∈H04 (0, 1). By using the fundamental lemma of the calculus of variations, we know there exist four constants a, b, c, and d such that 3

2

x s1 s2 s3

γu(x) = ax + bx + cx + d − ∫ ∫ ∫ ∫(−v(ξ ))dξds3 ds2 ds1 ,

for x ∈ (0, 1),

0 0 0 0

so u ∈ H 4 (0, 1) and, by differentiating four times, the above expression becomes γu󸀠󸀠󸀠󸀠 = v.

15.2 The shape memory alloys operator

Substituting this expression into (15.2.1), we get 1

0 = ∫(w󸀠󸀠󸀠󸀠 u − wu󸀠󸀠󸀠󸀠 )dx 0

󸀠󸀠󸀠 󸀠󸀠 󸀠 󸀠󸀠 = uw|𝜕[0,1] − u󸀠 w|𝜕[0,1] − wu󸀠󸀠󸀠 |𝜕[0,1] + w u|𝜕[0,1]

󸀠󸀠󸀠 󸀠 = uw|𝜕[0,1] + u󸀠󸀠 w|𝜕[0,1] .

Since this equality must hold for all w ∈ 𝒟(A), we conclude that u|𝜕[0,1] = u󸀠󸀠 |𝜕[0,1] = 0. Hence, ∗

𝒟(A ) = 𝒟(A)

and A is self-adjoint. This completes the proof. The eigenvalues of the operator A are μn = γπ 4 n4 ,

n = 1, 2, . . . ,

with corresponding normalized eigenfunctions in L2 (0, 1) given by hn (x) = √2 sin(nπx). Let B : 𝒟(B) ⊂ L2 (0, 1) 󳨀→ L2 (0, 1), { { { { { { 𝜕2 ψ ψ 󳨀→ Bψ(x) = −β (x), { { 𝜕x 2 { { { { 2 {𝒟(B) = {ψ ∈ H (0, 1) such that ψ(0) = ψ(1) = 0}, where β > 0. Theorem 15.2.3. The operator B is strictly positive. Proof. If u ∈ 𝒟(B), then ⟨Bu, u⟩ = ⟨−βu󸀠󸀠 , u⟩ 1

= −β ∫ u󸀠󸀠 udx 0

= β‖u󸀠 ‖2 ≥ 0.

| 323

324 | 15 Applications to mathematical models Moreover, ⟨Bu, u⟩ = 0 implies ‖u󸀠 ‖ = 0 and, therefore, u = 0 since u|𝜕[0,1] = 0. Thus, B is strictly positive. Theorem 15.2.4. The operator B is self-adjoint. Proof. Let v ∈ 𝒟(B). Then, for any u ∈ 𝒟(B), we have ⟨Bu, v⟩ = ⟨−βu󸀠󸀠 , v⟩ 1

= −β ∫ u󸀠󸀠 vdx. 0

After integrating by parts twice and using u|𝜕[0,1] = v|𝜕[0,1] = 0, one obtains 1

⟨Bu, v⟩ = −β ∫ uv󸀠󸀠 dx 0

= ⟨u, −βv󸀠󸀠 ⟩ = ⟨u, Bv⟩. Therefore, v ∈ 𝒟(B∗ ) and B∗ v = Bv, i. e., B is symmetric. Now, if u ∈ 𝒟(B∗ ), then there exists v ∈ L2 (0, 1) such that, for all w ∈ 𝒟(B), 1

0 = ⟨Bw, u⟩ − ⟨w, v⟩ = − ∫(βγw󸀠󸀠 u + wv)dx.

(15.2.2)

0

This equality must hold for all w ∈ H02 (0, 1). The fundamental lemma of the calculus of variations implies there exist two constants a and b such that x s

βu(x) = ax + b − ∫ ∫ v(ξ )dξds, 0 0

for x ∈ (0, 1).

15.2 The shape memory alloys operator

| 325

Hence, u ∈ H 2 (0, 1) and, by differentiating twice, it follows that βu󸀠󸀠 = −v. Substituting into (15.2.2), we get 1

0 = ∫(w󸀠󸀠 u − wu󸀠󸀠 )dx 0

󸀠 = uw|𝜕[0,1] − wu󸀠|𝜕[0,1] 󸀠 = uw|𝜕[0,1] .

Since this equality must hold for all w ∈ 𝒟(B), we conclude that u|𝜕[0,1] = 0. Hence, ∗

𝒟(B ) = 𝒟(B)

and B is self-adjoint. Theorem 15.2.5. The operators A and B satisfy the following equality: B=

β 21 A . √γ

Proof. Using Theorems 15.2.1 and 15.2.2, we know A is positive and self-adjoint, so it 1 possesses a unique positive self-adjoint square root A 2 . Moreover, any positive fractional δ-power Aδ of A is well defined, positive, and self-adjoint. It is easy to see that 2

𝒟(B ) = 𝒟(A)

and B2 u = for all u ∈ 𝒟(A). Hence, B=

β2 Au γ

β 21 A √γ

and this completes the proof of the theorem. Here, we shall be concerned with the operator 0 C := ( −A

I ) −B

326 | 15 Applications to mathematical models with domain 𝒟(C) = 𝒟(A) × 𝒟(B) acting in the Hilbert space [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1), where the inner product on [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1) is defined by 1

1

u φ ⟨( ) , ( )⟩ = ∫ BuBφdx + ∫ vψdx v ψ 0

0

1

1

= γ ∫ u󸀠󸀠 φ󸀠󸀠 dx + ∫ vψdx. 0

0

Note that 𝒟(C) is dense in [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1). This operator appears in the one-dimensional mathematical model for the dynamics of pseudo-elastic materials with Landau–Ginzburg free energy potentials. Theorem 15.2.6. The operator C is the infinitesimal generator of a strongly continuous semi-group of contractions eCt on [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1). Proof. Let η = ( uv ) ∈ 𝒟(C). Then u ∈ 𝒟(A), v ∈ 𝒟(B), and 0 −A

I u u ) ( ) , ( )⟩ −B v v

⟨Cη, η⟩ = ⟨(

v u = ⟨( ) , ( )⟩ −Au − Bv v 1

1

= ⟨A 2 v, A 2 u⟩ + ⟨−Au − Bv, v⟩. 1

Also, since A 2 is self-adjoint, u ∈ 𝒟(A), and B is positive, we have ⟨Cη, η⟩ = ⟨v, Au⟩ − ⟨Au, v⟩ − ⟨Bv, v⟩ = −⟨Bv, v⟩ ≤ 0. Hence, C is dissipative. One can easily verify that the adjoint C ∗ of C is given by 𝒟(C ∗ ) = 𝒟(C) = 𝒟(A) × 𝒟(B) and 0 C ∗ := ( A

−I ). −B

Moreover, if η = ( uv ) ∈ 𝒟(C ∗ ), we have 0 ⟨C ∗ η, η⟩ = ⟨( A

−I u u ) ( ) , ( )⟩ −B v v

15.2 The shape memory alloys operator

| 327

−v u = ⟨( ) , ( )⟩ Au − Bv v 1

1

= ⟨A 2 v, A 2 u⟩ + ⟨Au − Bv, v⟩. 1

Again, since A 2 is self-adjoint, u ∈ 𝒟(A), and B is positive, we have ⟨C ∗ η, η⟩ = −⟨v, Au⟩ + ⟨Au, v⟩ − ⟨Bv, v⟩ = −⟨Bv, v⟩ ≤ 0. Hence, C ∗ is also dissipative. It now follows from the Lumer–Phillips theorem (see Theorem 5.1.2) that C is the infinitesimal generator of a strongly continuous semigroup of contractions eCt on [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1). 15.2.2 Denseness of the generalized eigenvectors Theorem 15.2.7. The resolvent of the operator C belongs to the Carleman class Cp for every p > 21 . Proof. Since C −1 acts from [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1) into 𝒟(A) × 𝒟(B), it is easy to verify that the injection of H 4 (0, 1) into H 2 (0, 1) and the injection of H01 (0, 1) ∩ H 2 (0, 1) into L2 (0, 1) belong to the Carleman class Cp for all p > 21 (see [186]). This completes the proof. Theorem 15.2.8. The system of generalized eigenvectors of the operator C is dense in the space [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1). Proof. From Theorem 15.2.7, we deduce that the resolvent of the operator C is nuclear. Since C generates an analytic semi-group, it follows from Theorem 3.10.1 that the system of generalized eigenvectors of the operator C is dense in [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1). This completes the proof.

15.2.3 Diagonalization of the semi-groups e−tC and e−tC

1 2

Theorem 15.2.9. The series ∞

∑ etC Pk U

k=1

is convergent for any t > 0 and for every U ∈ [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1) and its sum is etC U, where Pk is the Riesz projection corresponding to the isolated eigenvalue λk of C. Proof. Since C generates an analytic semi-group, C −1 belongs to the Carleman class Cp for all p > 21 . In accordance with Lidskii’s theorem (Theorem 5.2.1), we get the desired result.

328 | 15 Applications to mathematical models Let F = ( gf ) ∈ [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1). We consider CU − λU = F, where U = ( uv ). Then v − λu = f ,

{

−γu󸀠󸀠󸀠󸀠 + βv󸀠󸀠 − λv = g.

Hence, the first equation of the above expression becomes v = λu + f . Substituting this expression into the second equation, we get −γu󸀠󸀠󸀠󸀠 + βλu󸀠󸀠 − λ2 u = h, where h = g − βf 󸀠󸀠 + λf . Let u = φ + ψ. Then we have −γφ󸀠󸀠󸀠󸀠 + βλφ󸀠󸀠 − λ2 φ = h, { { { { { φ(0) = 0, { { { 󸀠 φ (0) = 0, { { { { { φ󸀠󸀠 (0) = 0, { { { 󸀠󸀠󸀠 {φ (0) = 0 and −γψ󸀠󸀠󸀠󸀠 + βλψ󸀠󸀠 − λ2 ψ = 0, { { { { { {ψ(0) = 0, { { 󸀠󸀠 ψ (0) = 0, { { { { {ψ(1) = −φ(1), { { { 󸀠󸀠 󸀠󸀠 {ψ (1) = −φ (1). The first equation of the system (15.2.3) has a solution in the form of ψ(x) = c1 er1 x + c2 er2 x + c3 er3 x + c4 er4 x , where r1 = √aλ, β 1 β2 4 + √ 2 − , 2γ 2 γ γ r2 = −r1 , a=

(15.2.3)

15.2 The shape memory alloys operator

| 329

r3 = √bλ, b=

β 1 β2 4 − √ 2 − , 2γ 2 γ γ

and r4 = −r3 . By using the boundary conditions in (15.2.3), we get ψ(x) = c5 sinh(r1 x) + c6 sinh(r3 x) and c5 sinh(r1 ) + c6 sinh(r3 ) = −φ(1), { λ(ac5 sinh(r1 ) + bc6 sinh(r3 )) = −φ󸀠󸀠 (1). The principal determinant of this system is w(λ) = (b − a)λ sinh(√aλ) sinh(√bλ) and 1 2

w(λ) = O(e|λ| ). If a ≠ b, then λ ∈ ρ(C) if and only if w(λ) ≠ 0, i. e., the eigenvalues λk of the operator C are the zeros of the entire function w(λ) and its multiplicities βk are the zero-orders of w(λ). Since these zeros are simple, βk = 1 for any k and since the rank of the last system is at least 1, the dimension of N(C−λk ) is one. Hence, w(λ) and d(λ) have the same zeros with the same multiplicities. From the representation theorem [49], it follows that d(λ) = c7 λs ec8 λ w(λ). Since C −1 ∈ Cp for any p > 21 , by using Theorem 5.2.1, we have d(λ) = O(e|λ|

1 2 +ε

)

for any ε > 0. Since 1 2

w(λ) = O(e|λ| ), we obtain c8 = 0. Consequently, 1 2

d(λ) = O(e|λ| ). Hence, we have the following theorem.

330 | 15 Applications to mathematical models Theorem 15.2.10. There exist M > 0 and L > 0 such that 1

󵄨 󵄨󵄨 L|λ| 2 . 󵄨󵄨d(λ)󵄨󵄨󵄨 ≤ Me Theorem 15.2.11. There exists t0 > 0 such that the series ∞

1 2

∑ e−tC Pk U

k=1

is convergent for any t > t0 and for every U ∈ [H01 (0, 1) ∩ H 2 (0, 1)] × L2 (0, 1) and its sum 1 2

is e−tC U. Proof. The result follows immediately from Theorems 15.2.10 and 5.2.4.

15.3 Expansion of solution for a hyperbolic system We concentrate our attention to give a unified treatment for the following general class of linear hyperbolic systems: u(x, t) h(x, t) d u(x, t) { { ( )+𝒞( )=( ), { { { dt w(x, t) w(x, t) k(x, t) { u(0, t) = α1 w(0, t), { { { { u(L, t) = β1 w(L, t), {

(15.3.1)

where d −c(x) dx ) 0

0 d −c(x) dx

𝒞 := (

with a positive function c(x), which was first studied by A. Intissar in [110] in the case where the function c(x) is considered a constant equal to one with a unique restrictive termination. Since the matrix 0 −c(x)

(

−c(x) ) 0

is symmetric with distinct eigenvalues λi = (−1)i+1 c(x),

i = 1, 2,

introducing the change of variables p −1 1 ( )= ( v 2 −1

−1 u )( ), −1 w

15.3 Expansion of solution for a hyperbolic system

f −1 1 ( ( )= g 2 −1 α :=

1 − α1 , 1 + α1

| 331

−1 h )( ), −1 k

and β :=

1 − β1 , 1 + β1

our system becomes c(x) d p(x, t) { { ( )+( { { { dt v(x, t) 0 { { { { { {

0 f (x, t) d p(x, t) ) ( )=( ), −c(x) dx v(x, t) g(x, t) p(0, t) = αv(0, t),

(15.3.2)

p(L, t) = βv(L, t).

To turn system (15.3.1) into a framework of semi-groups, we introduce the underlying state Hilbert space ℋ := L2 [0, L] × L2 [0, L]. System (15.3.2) is then written as an evolutionary equation in ℋ. We have d W(t) = 𝒜W(t), dt

t > 0,

with p(⋅, t) ) v(⋅, t)

W(t) = ( and 𝒜 : 𝒟(𝒜) ⊂ ℋ 󳨀→ ℋ defined by 𝒜 := C(x)

c(x) d := ( 0 dx

0 d ) , −c(x) dx

(15.3.3)

with domain p(x) ) ∈ H 1 [0, L] × H 1 [0, L] such that p(0) = αv(0), p(L) = βv(L)} . v(x)

𝒟(𝒜) = {(

(15.3.4)

15.3.1 The resolvent expression of 𝒜 In this part, our goal is to give some important properties about a linear hyperbolic system to verify the assumptions of Theorem 5.2.5 in several steps. We shall assume without loss of generality that p(x) := p(x, t), v(x) := v(x, t), f (x) := f (x, t), and g(x) := g(x, t). In the following, we establish some preliminary results about the operator 𝒜. Proposition 15.3.1. Let 𝒜 be the operator given in (15.3.3) and (15.3.4). Then the following assertions are true:

332 | 15 Applications to mathematical models (i) 𝒟(𝒜) is dense in ℋ; and (ii) the canonical injection i from 𝒟(𝒜) into ℋ is compact. Proof. (i) It is seen that 𝒟(𝒜) is a closed subset of the Sobolev space H 1 [0, L]×H 1 [0, L], which is dense in ℋ. Assertion (i) follows immediately. (ii) From the Sobolev theorem (see [51, Theorem VIII.7, p. 129]), the injection from H 1 [0, L] into L2 [0, L] is compact, so the injection i from 𝒟(𝒜) into ℋ is too. Theorem 15.3.1. (i) The operator 𝒜 generates a C0 -semi-group on ℋ. (ii) The resolvent expression of 𝒜, R(λ, 𝒜), can be represented as x

f αv(0) f (s) R(λ, 𝒜) ( ) (x) = M(x, 0, λ) ( ) − ∫ M(x, s, λ)C −1 (s) ( ) ds, g v(0) g(s) 0

f ∀ ( ) ∈ ℋ, g

where e1 (x, s, λ) 0

M(x, s, λ) = (

ei (x, s, λ) = e(−1)

i+1

x

0 ), e2 (x, s, λ)

∫s c−1 (r)dr

,

i = 1, 2,

and L

{ −c−1 (s) 1 [ { { v(0) = − ∫⟨M(L, s, λ) ( 0 Δ(λ) { { [0 { {Δ(λ) = αe1 (L, 0, λ) − βe2 (L, 0, λ).

0 f (s) 1 )( ) , ( )⟩ds] , c−1 (s) g(s) −β ]

Proof. (i) Since the entries of the matrix C(x) are real-valued C 1 -functions in x, 𝒜 generates an evolution operator. (ii) First, we shall determine the resolvent of the operator 𝒜. To do this, we begin to solve the following equation: p(x) f (x) )=( ). v(x) g(x)

(λ − 𝒜) (

(15.3.5)

The expression (15.3.5) is equivalent to the following system: d p(x) = λc−1 (x)p(x) − c−1 (x)f (x), dx d v(x) = −λc−1 (x)v(x) + c−1 (x)g(x). dx

(15.3.6) (15.3.7)

To solve equation (15.3.6), we will first consider the following homogeneous equation: d p(x) = λc−1 (x)p(x). dx

15.3 Expansion of solution for a hyperbolic system

| 333

It is easy to see that p(x) = 0 is a trivial solution. Now, we shall consider the case where p(x) ≠ 0. Thus, dp(x) = λc−1 (x)dx. p(x) Solving this yields p(x) = Keλ ∫ c

−1

(r)dr

,

where K is a real constant. An elementary calculation by the variation of constants reveals that the general solution of equation (15.3.6) is given by x

p(x) = K(0)eλ ∫0 c

−1

x

t

− [∫ c−1 (t)f (t)e−λ ∫0 c

(r)dr

−1

(r)dr

x

dt]eλ ∫0 c

−1

(r)dr

.

0

Using the boundary condition, we get p(0) = K(0) = αv(0). Hence, we conclude that the general solution of equation (15.3.6) is given by x

p(x) = αv(0)eλ ∫0 c

−1

(r)dr

x

x

− ∫ c−1 (t)f (t)eλ ∫t

c−1 (r)dr

dt,

∀0 ≤ x ≤ L.

0

In what follows, the solution of equation (15.3.7) can be found in the same ways as equation (15.3.6). Then it is easy to check that x

v(x) = v(0)e−λ ∫0 c

−1

(r)dr

x

x

+ ∫ c−1 (t)g(t)e−λ ∫t

c−1 (r)dr

dt,

∀0 ≤ x ≤ L.

0

Hence, we write x

f (x) αv(0) f (s) ) = M(x, 0, λ) ( ) − ∫ M(x, s, λ)C −1 (s) ( ) ds, g(x) v(0) g(s)

R(λ, 𝒜) (

0

where M(x, s, λ) is defined as e1 (x, s, λ) 0

M(x, s, λ) = (

0 ) e2 (x, s, λ)

and ei (x, s, λ) = e(−1)

i+1

x

∫s c−1 (r)dr

i = 1, 2.

334 | 15 Applications to mathematical models As a preparation for attacking the expression of v(0), it is sufficient to solve p(L) = βv(L) by recalling the resolvent of 𝒜. We get L

[αe1 (L, 0, λ) − βe2 (L, 0, λ)]v(0) = −[∫ −e1 (L, s, λ)c−1 (s)f (s) − βe2 (L, s, λ)c−1 (s)g(s)ds]. 0

We denote by Δ(λ) the following expression: Δ(λ) = αe1 (L, 0, λ) − βe2 (L, 0, λ), which is an entire function on λ. Using the above equality with these notations, we get L

−c−1 (s) 0

Δ(λ)v(0) = − ∫⟨M(L, s, λ) ( 0

0 f (s) 1 )( ) , ( )⟩ds. c−1 (s) g(s) −β

A short computation shows that a complex number λ belongs to the resolvent set of 𝒜 if and only if Δ(λ) ≠ 0. In this case, L

−c−1 (s) 1 [ v(0) = − ∫⟨M(L, s, λ) ( 0 Δ(λ) [0

0 f (s) 1 )( ) , ( )⟩ds] , c−1 (s) g(s) −β ]

which completes the proof.

15.3.2 Spectrum of 𝒜 Theorem 15.3.2. The matrix 𝒜 has a discrete operator in ℋ, in other words, for any λ ∈ ρ(𝒜), R(λ, 𝒜) is compact on ℋ. Therefore, the spectrum, σ(𝒜), of 𝒜 consists only of isolated eigenvalues. Proof. It follows from Proposition 15.3.1 (ii), together with the fact that ρ(𝒜) ≠ 0, that 𝒜 has a compact resolvent on ℋ. Thus, its spectrum consists only of eigenvalues with non-zero imaginary parts which are determined by the zeros set of Δ(λ). Theorem 15.3.3. For each λ ∈ σ(𝒜), all eigenfunctions associated with λ can be represented as p(x, λ) αv(0) ) = M(x, 0, λ) ( ), v(x, λ) v(0)

( where Δ(λ)v(0) = 0.

15.3 Expansion of solution for a hyperbolic system

| 335

Proof. Consider the system p(x) p(x) ) = λ( ), v(x) v(x)

𝒜(

(15.3.8)

where we are going to determine the eigenvectors p(x, λ) ) v(x, λ)

(

associated with λ. The system (15.3.8) becomes d −1 { { { dx p(x) = λc (x)p(x), { { {d v(x) = −λc−1 (x)v(x). { dx The solution of the above system is exactly the solution of the homogeneous equation of the system (15.3.2). In this way, we express it as follows: p(x, λ) αv(0) ) = M(x, 0, λ) ( ). v(x, λ) v(0)

(

Moreover, to verify the boundary condition p(L) = βv(L), we solve it from the expression of the resolvent for x = L, which reveals that Δ(λ)v(0) = 0. The next theorem gives information about the distribution of the eigenvalues of the operator 𝒜. Theorem 15.3.4. Each eigenvalue of 𝒜 is simple and given by λn =

󵄨󵄨 β 󵄨󵄨 nπ 1 󵄨 󵄨 log󵄨󵄨󵄨 󵄨󵄨󵄨 + i , 󵄨󵄨 α 󵄨󵄨 2cL cL

n ∈ ℤ.

Proof. It is well known that the spectrum of 𝒜 is given by its eigenvalues, formulated by the zeros set of Δ(λ). In this case, we set Δ(λ) = 0, where λ ∈ ℂ. Define cL by L

cL := ∫ c−1 (r)dr. 0

336 | 15 Applications to mathematical models Since λ ∈ ℂ, there exist (a, b) ∈ ℝ2 such that λ = a + ib, so the zeros set of Δ(λ) is formulated by the solution of the following equation: αecL (a+ib) − βe−cL (a+ib) = 0. It is easy to see that the expression a is given by a=

󵄨󵄨 β 󵄨󵄨 1 󵄨 󵄨 log󵄨󵄨󵄨 󵄨󵄨󵄨. 󵄨󵄨 α 󵄨󵄨 2cL

If the expression a is determined, we express the constant b as a solution of the equation e2icL b = ±1, which is formulated as follows: cos(2cL b) = ±1, { sin(2cL b) = 0. The solution of the last system is given by b=

kπ , cL

k ∈ ℤ.

In what follows, the zeros set of Δ(λ) is {λn }, formulated by the following form: λn =

󵄨󵄨 β 󵄨󵄨 nπ 1 󵄨 󵄨 log󵄨󵄨󵄨 󵄨󵄨󵄨 + i , 󵄨󵄨 α 󵄨󵄨 2cL cL

n ∈ ℤ.

Obviously, each eigenvalue λn given by equation (15.3.9) of 𝒜 is simple. The corresponding eigenvector associated with λn is p(x, λn ) αe (x, 0, λn )v(0) )=( 1 ) v(x, λn ) e2 (x, 0, λn )v(0)

(

c [

1

β

log | |+i nπ ]

α cL v(0) αe x 2cL ), =( β −cx [ 2c1 log | α |+i nπ ] cL L e v(0)

where x

cx := ∫ c−1 (r)dr. 0

(15.3.9)

15.3 Expansion of solution for a hyperbolic system

| 337

Therefore, the eigenvector associated with λn has the following asymptotic form: cx

β

log | |

α p(x, λn ) αe 2cL v(0) )=( v(x, λn ) 0

(

e

i

0

cx

nπ

e cL ) ( −i cx nπ ) . β log | α | e cL v(0)

c

− 2cx

L

It is easily seen that cx

αe 2cL H(x) := (

β

log | α |

v(0)

0

e

0

c

β

− 2cx log | α | L

v(0)

)

is invertible in x and det H(x) ≠ 0. The motivation for our studies is to prove the Riesz basis property of {eλn t } in L2 [0, 2cL ]. To obtain such a property, explicit estimates for sine-type functions turn out to be suitable. Lemma 15.3.1. The system {eλn t } forms a Riesz basis for L2 [0, 2cL ]. Proof. It should be noted that the aim of our results is to provide a description for the Riesz basis property of {ei Im λn t }, since the real part of the eigenvalues {λn } are independent of n. To claim this, let g(μ) be the function defined by g(μ) := sin(cL μ). The zeros set {μn } of g(μ) is related by μn =

nπ . cL

Obviously, g(μ) is uniformly bounded on the real axis and, hence, g belongs to the Cartwright class. Thus, its indicator diagram is an interval. Furthermore, it is easy to show that g(μ) is of the exponential type with cL =

T . 2

In fact, let z := x + iy ∈ ℂ. Then 󵄨󵄨 󵄨 2 2 󵄨󵄨g(z)󵄨󵄨󵄨 = √sin2 (cL x) cosh (cL y) + cos2 (cL x) sinh (cL y) 󵄨 󵄨 ≤ 󵄨󵄨󵄨cosh(cL y)󵄨󵄨󵄨 ≤ ecL |y|

≤ ecL |z| .

338 | 15 Applications to mathematical models Moreover, from the last expression, we have 󵄨 󵄨 C1 ecL |y| ≤ 󵄨󵄨󵄨g(x + iy)󵄨󵄨󵄨 ≤ C2 ecL |y| , where 0 < C1 < 21 , C2 ≥ 1, and all y ∈ ℝ whenever y is sufficiently large. In what follows, the item (ii) of Definition 3.5.6 is satisfied. Moreover, it is seen that the zeros {μn } are separated. Indeed, let n ∈ ℤ. Then we have 󵄨󵄨 (n + 1)π nπ 󵄨󵄨 π 󵄨󵄨 󵄨 − |μn+1 − μn | = 󵄨󵄨󵄨 󵄨= . 󵄨󵄨 cL cL 󵄨󵄨󵄨 cL As a consequence from the above, we get the Riesz basis property by Theo-

rem 4.12.10 for {e

i nπ t c L

} in L2 [0, 2cL ] and, hence, for {eλn t } in L2 [0, 2cL ].

15.3.3 Spectral radius and essential spectral radius From the expression of the eigenvalues of the operator 𝒜, we will discuss its spectral radius r(⋅) and its essential spectral radius ress (⋅). We start by giving the following lemma. Lemma 15.3.2. If α0 is defined by α0 = sup{Re λ such that Δ(λ) = 0}, then r(T(t)) = ress (T(t)) = eα0 t ,

t ≥ 0.

Proof. First, we show that r(T(t)) ≤ eα0 t ,

t ≥ 0.

This is equivalent to saying that, for any β > α0 , there is a constant K (depending on β) for which 󵄩󵄩 󵄩 βt 󵄩󵄩T(t)󵄩󵄩󵄩 ≤ Ke ,

t ≥ 0.

Clearly, it suffices to show that 󵄩󵄩 󵄩 βt 󵄩󵄩T(t)f 󵄩󵄩󵄩 ≤ Ke ‖f ‖ for some K and for any f in a dense subset of L2 [0, L]. We know that, for any f ∈ 𝒟(A2 ), β+iy

1 T(t)f = lim ∫ eλt R(λ, A)fdλ, y→+∞ 2iπ β−iy

15.3 Expansion of solution for a hyperbolic system

| 339

where f will be taken smooth with compact support in [0, L]. The desired estimate will be obtained through this representation of T(t). We recall that the expression of R(λ, A) has two parts and we estimate the first only, because the second is much simpler. Using the expression of R(λ, A), we get β+iy

L

β−iy

0

1 1 T(t)f (x) = lim αe (x, 0, λ) ∫ e1 (L, s, λ)c−1 (s)f (s)ds]dλ. ∫ eλt [ y→+∞ 2iπ Δ(λ) 1 Since Δ(λ) is bounded away from zero on Re λ = β, it follows that ∞ 1 = ∑ bk eλγk , Δ(λ) k=1

with γk being real and ∞

∑ |bk |eβγk < ∞.

k=1

Next, we notice that, for λ = β + ir, we have to estimate the terms of the following type: y

L

−y

0

∞ 1 T(t)f (x) = lim ∫ e(β+ir)t [ ∑ bk e(β+ir)γk αe(β+ir)C0 ∫ e(β+ir)Cs c−1 (s)f (s)ds]dr, y󳨀→+∞ 2π k=1

where

x

C0 = ∫ c−1 (r)dr 0

and

L

Cs = ∫ c−1 (r)dr. s

Hence, y

L

1 ∞ T(t)f (x) = lim ∑ αbk eβγk eβt eβC0 ∫[ ∫ eirt eirγk eirC0 e(β+ir)Cs dr]c−1 (s)f (s)ds y󳨀→+∞ 2π k=1 0 −y

L

y

1 ∞ ∑ αbk eβ(γk +t+C0 ) ∫[ ∫ eir(t+γk +C0 +Cs ) dr]c−1 (s)eβCs f (s)ds. y󳨀→+∞ 2π k=1

= lim

0 −y

Since y

∫ eir(t+γk +C0 +Cs ) dr = 2 −y

sin(y(t + γk + C0 + Cs )) , t + γk + C0 + Cs

340 | 15 Applications to mathematical models we get L

sin(y(t + γk + C0 + Cs )) −1 1 ∞ c (s)eβCs f (s)ds. T(t)f (x) = lim ∑ αbk eβ(γk +t+C0 ) ∫ y→+∞ π t + γ + C + C k 0 s k=1 0

If we take f = 0 outside [0, L] and define the change of variables z = Cs , s = η(z), we get L

∫ 0

sin(y(t + γk + C0 + Cs )) −1 c (s)eβCs f (s)ds t + γk + C0 + Cs +∞

= ∫ −∞

sin(r(t + γk + C0 + z)) −1 c (η(z))eβz f (η(z))dz. t + γk + C0 + z

Since the limit of this expression at r → +∞ is πb1 (−(t + γk + C0 ))f (η(−(t + γk + C0 ))), where b1 (−(t + γk + C0 )) = c−1 (−(t + γk + C0 ))eβ(−(t+γk +C0 )) η󸀠 (−(t + γk + C0 )) (by the Fourier integral formula), we conclude that ∞

T(t)f (x) = ∑ αbk eβ(γk +t+C0 ) b1 (−(t + γk + C0 ))f (η(−(t + γk + C0 ))). k=1

Finally, the desired estimate of ‖T(t)f (x)‖ is bounded by ∞

∑ αbk eβt eβ(γk +C0 ) Mk ‖f ‖,

k=1

where Mk is a constant. This gives the desired estimate of ‖T(t)f (x)‖. Furthermore, if T(t) is the solution semi-group with infinitesimal generator 𝒜, then we may write α0 = sup{Re λ, λ ∈ σ(𝒜)} 1 󵄩 󵄩 ≤ lim log 󵄩󵄩󵄩T(t)󵄩󵄩󵄩. t→+∞ t Also, eα0 t ≤ ewt for t ≥ 0, so it follows that ress (T(t)) = eα0 t ≤ ewt

= r(T(t)), This proves the lemma.

t ≥ 0.

15.3 Expansion of solution for a hyperbolic system

| 341

15.3.4 Riesz basis Lemma 15.3.3. {e

inπt cL

} forms a Riesz basis for L2 [0, T], for some T > 0.

Proof. Let g(μ) be the function defined by g(μ) := sin(cL μ). The zeros set {μn } of g(⋅) is related through μn =

nπ . cL

Obviously, g(⋅) is uniformly bounded on the real axis and, hence, g(⋅) belongs to the Cartwright class. Thus, its indicator diagram is an interval. Furthermore, it is easy to show that g(⋅) is of the exponential type with cL =

T . 2

In fact, let z := x + iy ∈ ℂ. Then 󵄨󵄨 󵄨 2 2 󵄨󵄨g(z)󵄨󵄨󵄨 = √sin2 (cL x) cosh (cL y) + cos2 (cL x) sinh (cL y) 󵄨 󵄨 ≤ 󵄨󵄨󵄨cosh(cL y)󵄨󵄨󵄨 ≤ ecL |y|

≤ ecL |z| . Moreover, from the last expression, we have 󵄨 󵄨 C1 ecL |y| ≤ 󵄨󵄨󵄨g(x + iy)󵄨󵄨󵄨 ≤ C2 ecL |y| , where 0 < C1 < 21 , C2 ≥ 1, and all y∈ℝ, whenever y is sufficiently large. In what follows, the item (ii) of Definition 3.5.6 is satisfied. Moreover, it is seen that the zeros {μn } are separated. Indeed, let n ∈ ℤ. We have 󵄨󵄨 (n + 1)π nπ 󵄨󵄨 π 󵄨 󵄨󵄨 |μn+1 − μn | = 󵄨󵄨󵄨 − 󵄨= . 󵄨󵄨 cL cL 󵄨󵄨󵄨 cL In what follows, we show that g(⋅) is of the sine type and 󵄨 󵄨 hf (ϕ) = cL 󵄨󵄨󵄨sin(ϕ)󵄨󵄨󵄨, Gf = [−icL , icL ].

Therefore, the width of the indicator diagram of g(⋅) is equal to T = 2cL . This achieves the proof and {eiμn t } forms a Riesz basis for L2 [0, 2cL ].

342 | 15 Applications to mathematical models In view of the results cited above, we mention our main result. inπt

Corollary 15.3.1. If {e cL } forms a Riesz basis for L2 [0, T], then {eλn t } forms also a Riesz basis for L2 [0, T], for some T > 0. Proof. It is easy to check from the expression of λn , given in equation (15.3.9), that 1 󵄨2 󵄨 ∑ 󵄨󵄨󵄨⟨eλn t , U⟩󵄨󵄨󵄨 = e cL

β

log | α |t

n∈ℤ

󵄨 i nπ t 󵄨2 ∑ 󵄨󵄨󵄨⟨e cL , U⟩󵄨󵄨󵄨 .

n∈ℤ

Moreover, we infer from the definition of the Riesz basis of {e exist two constants Ki > 0, i = 1, 2, satisfying

inπt cL

} in L2 [0, T] that there

β

‖U‖2 ,

󵄨2 󵄨 i nπ t K1 ‖U‖2 ≤ ∑ 󵄨󵄨󵄨⟨e cL , U⟩󵄨󵄨󵄨 ≤ K2 ‖U‖2 , n∈ℤ

for all U ∈ L2 [0, T]. Hence, 1

K1 e cL

β

log | α |t

1 󵄨 󵄨2 ‖U‖2 ≤ ∑ 󵄨󵄨󵄨⟨eλn t , U⟩󵄨󵄨󵄨 ≤ K2 e cL

log | α |t

n∈ℤ

for all U ∈ L2 [0, T]. Therefore, we deduce from Propositioin 4.12.2 the desired result of {eλn t }. Let us summarize these results as follows. Lemma 15.3.4. (i) The operator 𝒜, given in (15.3.3), is dissipative. (ii) There exists δ > 0 such that the spectrum of 𝒜 is included in the sector {λ ∈ ℂ such that

π 3π + δ ≤ arg λ ≤ − δ} 2 2

and, on the complement of this sector, the resolvent of 𝒜 satisfies M 󵄩󵄩 −1 󵄩 , 󵄩󵄩(𝒜 − λ) 󵄩󵄩󵄩 ≤ |λ| where M is a positive constant. (iii) The operator 𝒜 generates a C0 -semi-group on ℋ. Proof. (i) Let U := ( p(x) v(x) ) ∈ 𝒟 (𝒜). Then we get ⟨𝒜U, U⟩ = −⟨(

dv c(x) dx

c(x) dp dx

p(x) ),( )⟩ ≤ 0, v(x)

since c(x) > 0 for all x ∈ [0, L]. Hence, we deduce that the operator 𝒜 is dissipative.

15.3 Expansion of solution for a hyperbolic system

| 343

(ii) The matrix operator 𝒜 is dissipative with compact resolvent. Consequently, there exists δ > 0 such that the spectrum of 𝒜 is included in the sector {λ ∈ ℂ such that

3π π + δ ≤ arg λ ≤ − δ}. 2 2

Moreover, let Sδ be the sector centered at 0, defined by the following form: Sδ = {λ ∈ ℂ such that

3π π + δ ≤ arg λ ≤ − δ} 2 2

for δ > 0. On the complement of this sector, the resolvent of 𝒜 satisfies 󵄩󵄩 󵄩 󵄩󵄩R(λ, 𝒜)󵄩󵄩󵄩 ≤

δ , dist(λ, W(𝒜))

where W(𝒜) designates the numerical range of the operator 𝒜. Hence, dist(λ, W(𝒜)) ≥ |λ| sin ϕ, with 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 π 3π 󵄨 󵄨󵄨 󵄨 ϕ = inf{󵄨󵄨󵄨arg λ − − δ󵄨󵄨󵄨, 󵄨󵄨󵄨arg λ − + δ󵄨󵄨󵄨}. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 2 Using this notation, we get 󵄩󵄩 󵄩 M , 󵄩󵄩R(λ, 𝒜)󵄩󵄩󵄩 ≤ |λ| where M=

1 . sin ϕ

(iii) The result follows from the Hille–Yosida theorem and item (ii). Theorem 15.3.5. The resolvent of the operator 𝒜 belongs to the Carleman class C1+ε , for every ε > 0. Proof. It is well known that the spectrum of 𝒜 consists of isolated eigenvalues {λn , n ∈ ℤ}. A short computation shows, from the expression of λn given in equation (15.3.9), that |λn | = √Re λn2 + Im λn2 =

󵄨󵄨 β 󵄨󵄨 nπ √ 1 󵄨 󵄨 1 + 2 2 log2 󵄨󵄨󵄨 󵄨󵄨󵄨, 󵄨󵄨 α 󵄨󵄨 |cL | 4n π

344 | 15 Applications to mathematical models so 󵄨󵄨 β 󵄨󵄨 |λn ||cL | 1 󵄨 󵄨 = lim √1 + 2 2 log2 󵄨󵄨󵄨 󵄨󵄨󵄨 = 1. n→+∞ n→+∞ 󵄨󵄨 α 󵄨󵄨 nπ 4n π lim

Therefore, |λn | ∼

+∞

nπ . |cL |

Obviously, 0 ∈ ρ(𝒜), so let (sn (𝒜−1 ))n be the eigenvalues of the operator √(𝒜−1 )∗ 𝒜−1 . It is clear that 1 0. Then 𝒜−1 belongs to the Carleman class C1+ε , for every ε > 0. As a consequence of Theorem 3.6.1, we have the following corollary. Corollary 15.3.2. The resolvent of 𝒜 can be represented as the ratio of two entire functions F(λ, 𝒜) and d(λ) by R(λ, 𝒜) =

F(λ, 𝒜) . d(λ)

Proof. Let λ ∈ ρ(𝒜). We have R(λ, 𝒜) = (λ − 𝒜)−1

−1

= ([λ𝒜−1 − I] 𝒜)

−1

−1

= 𝒜−1 (λ𝒜−1 − I) . Denoting K = 𝒜−1 , we have R(λ, 𝒜) = K(λK − I)−1 .

(15.3.10)

Since the resolvent of 𝒜 belongs to the class C1+ε for ε > 0, we infer from Theorem 3.6.1 and equation (15.3.10) that the resolvent of 𝒜 can be represented under the form F(λ, 𝒜) , d(λ) where F(λ, 𝒜) and d(λ) are two entire functions. Theorem 15.3.6. The system of the eigenvectors of the operator 𝒜 is dense in ℋ.

15.4 Frame of a one-dimensional wave control system

| 345

Proof. From Theorem 15.3.5, the resolvent of 𝒜 belongs to the Carleman class C1+ε , for ε > 0. The resolvent, R(λ, 𝒜), of 𝒜 is an entire function of λ and the orders of both entire functions of F(λ, 𝒜) and d(λ) are less than or equal to 1. That is, there is an ε > 0 such that 󵄩 󵄩󵄩 |λ|1+ε ), 󵄩󵄩R(λ, 𝒜)󵄩󵄩󵄩 = O(e

as |λ| → ∞.

As 𝒜 generates an analytic semi-group, it follows that ‖R(λ, 𝒜)‖ is uniformly bounded in both the real and the imaginary axis. We assume without loss of generality that ‖R(λ, 𝒜)‖ is uniformly bounded in the right complex plane, particularly on the imaginary axis. Set Sj = {λ ∈ ℂ such that arg γj ≤ arg λ ≤ arg γj+1 },

j = 1, 2.

By assumption, R(λ, 𝒜) is bounded on the boundary Sj and 󵄩󵄩 󵄩 |λ|1+ε ), 󵄩󵄩R(λ, 𝒜)󵄩󵄩󵄩 = O(e for all λ ∈ Sj , where ε > 0 is chosen so that 1 + ε < 2. Applying the Phragmén–Lindelöf theorem to R(λ, 𝒜) in each Sj (see Theorem 3.8.1), we know that R(λ, 𝒜) is uniformly bounded in Sj as well as in the whole complex plane. Therefore, all conditions of Theorem 3.10.1 are satisfied with p = 1 + ε, for ε > 0, m = 3, and γ2 = {λ ∈ ℂ such that arg λ = π}. This demonstrates the completeness of the generalized eigenvectors of 𝒜 on ℋ. Theorem 15.3.7. The eigenvectors associated with the operator 𝒜 form a Riesz basis in ℋ. Proof. It follows from Theorem 15.3.2, Corollary 15.3.1, Lemma 15.3.4, and Theorem 15.3.6 that all the hypotheses of Theorem 5.2.5 are satisfied. From this notion, we conclude that the system of eigenvectors of 𝒜 forms a Riesz basis in ℋ. Corollary 15.3.3. Our system satisfies the spectrum determined growth assumption, i. e., S(𝒜) = w(𝒜). Remark 15.3.1. The exponential family of the eigenvalues is then readily established from an expression of the eigenvalues which is also obtained in the process of verification of the Riesz basis property.

15.4 Frame of a one-dimensional wave control system 15.4.1 One-dimensional string equation In this section, we consider the following one-dimensional string equation: wtt (x, t) − wxx (x, t) = 0,

0 < x < 1, t > 0,

(15.4.1)

346 | 15 Applications to mathematical models where w(x, t) denotes the transversal displacement of the string departing from its equilibrium position at x and time t. The initial conditions are given by w(x, 0) = w0 (x),

wt (x, 0) = w1 (x),

whereas the Neumann boundary conditions are wx (0, t) = u1 (t),

{

wx (1, t) = u2 (t),

where uj (t), j = 1, 2, designates the control input. For instance, we adopt the following general linear controllers: u1 (t) = k1 wt (0, t) + γw(0, t),

{

(15.4.2)

u2 (t) = −k2 wt (1, t) − δw(1, t),

where γ ≥ 0, δ ≥ 0, k1 ≥ 0, k2 ≥ 0,

and k1 + k2 ≠ 0, γ + δ ≠ 0.

The abstract formulation of the problem is obtained by considering the following Hilbert space: 1

2

𝒳 := H (0, 1) × L (0, 1),

where H 1 (0, 1) is the usual Sobolev space of order 1 and is equipped with the inner product 1

(u, v)H 1 := ∫ u󸀠 (x)v󸀠 (x)dx + γu(0)v(0) + δu(1)v(1). 0

The inner product of two elements F = (f1 , f2 ), G = (g1 , g2 ) ∈ 𝒳 is defined by 1

1

0

0

⟨F, G⟩𝒳 := ∫ f1󸀠 (x)g1󸀠 (x)dx + γf1 (0)g1 (0) + δf1 (1)g1 (1) + ∫ f2 (x)g2 (x)dx. Here and hereafter, we use the notation u󸀠 (x) =

du = ux (x). dx

Define the operator A in 𝒳 by 2

1

󸀠

𝒟(A) := {(u, v) ∈ H (0, 1) × H (0, 1) such that u (0) = γu(0) + k1 v(0),

u󸀠 (1) = −δu(1) − k2 v(1)},

(15.4.3)

15.4 Frame of a one-dimensional wave control system

A(u, v) := (v, u󸀠󸀠 ),

(u, v) ∈ 𝒟(A).

| 347

(15.4.4)

With the help of these notations, we rewrite equation (15.4.1) into an evolutionary equation in 𝒳 . We have d { W(t) = AW(t), dt { { W(0) = W0 ,

t > 0,

where W(t) := (w(x, t), wt (x, t)) and W0 := (w0 (x), w1 (x)). It is interesting to note that, in control and transport theory, it is very difficult to show that a system satisfies the spectrum determined growth condition, so using the spectrum of the system operator to verify this property becomes an attractive alternative. However, the Riesz basis property is not always verified. For instance, in equations (15.4.2) and (15.4.3), if we assume that k1 = 1 and k2 ≥ 0, then the eigenvectors of the system operator (15.4.4) fail to form a basis since sup ‖Pn ‖ = ∞, n

where Pn is the Riesz projection corresponding to the isolated eigenvalue λn of A. Therefore, the question is whether we can extend the Riesz basis property to nonorthogonal fusion frame. We consider a controlled wave system given by wtt (x, t) = wxx (x, t), 0 < x < 1, t > 0, { { { { {wx (0, t) = wt (0, t) + γw(0, t), { { w (1, t) = −k2 wt (1, t) − δw(1, t), { { { x {w(x, 0) = w0 (x), wt (x, 0) = w1 (x), where γ, δ, k2 ≥ 0 and γ + δ ≠ 0. 15.4.2 Adjoint of A It is easy to see that the operator A is a densely defined closed operator in 𝒳 and its adjoint A∗ is given by A∗ (u, v) = −(v, u󸀠󸀠 ),

(u, v) ∈ 𝒟(A∗ ),

348 | 15 Applications to mathematical models where ∗

2

1

𝒟(A ) := {(u, v) ∈ H (0, 1) × H (0, 1) such that

u󸀠 (0) = −v(0) + γu(0), u󸀠 (1) = k2 v(1) − δu(1)}. 15.4.3 Spectrum of A Before going further, we recall the following result from [192]. Theorem 15.4.1 (G. Q. Xu and S. P. Yung [192, Theorems 3.1 and 3.2]). The operator A is with compact resolvent. Further, for k1 = 1 and k2 ≥ 0, we have σ(A) = {λ ∈ ℂ such that Γ(λ) = 0}, where Γ(λ) := [(1 + k2 )λ + δ](2λ + γ)eλ + γ[(1 − k2 )λ − δ]e−λ . Proof. Suppose that 0 ∈ σp (A). Then there exists (u, v) ≠ (0, 0) ∈ 𝒟(A) such that A(u, v) = (0, 0), i. e., v(x) = 0, { { { { {u󸀠󸀠 (x) = 0, { {u󸀠 (0) = γu(0), { { { 󸀠 {u (1) = −δu(1).

(15.4.5)

The solution of the system (15.4.5) is given by u(x) = a + bx. Substituting this into the boundary conditions, we find that the system (15.4.5) has only the zero solution. Hence, 0 is not an eigenvalue of A. Now, we consider the inhomogeneous equation AF = G, where F = (u, v) ∈ 𝒟(A) and G = (f , g) ∈ 𝒳 , i. e., v(x) = f (x), { { { { {u󸀠󸀠 (x) = g(x), { { u󸀠 (0) − γu(0) = f (0), { { { 󸀠 {u (1) + δu(1) = −k2 f (1).

15.4 Frame of a one-dimensional wave control system

| 349

It is easy to see that u1 (x) = 1 and u2 (x) = x are the solutions of the homogeneous equation u󸀠󸀠 (x) = 0. An elementary calculation by the variation of constants reveals that the general solution to the equation u󸀠󸀠 (x) = g(x) is given by x

u(x) = au1 (x) + bu2 (x) + ∫ 0

x

−u1 (x)u2 (t) + u2 (x)u1 (t) g(t)dt u1 (t)u󸀠2 (t) − u2 (t)u󸀠1 (t)

= a + bx + ∫(−t + x)g(t)dt. 0

Using the boundary conditions, we get 1

a=

(u󸀠1 (1) + δu1 (1))u2 (t) 1 g(t)dt [−f (0)(u󸀠2 (1) + δu2 (1)) − k2 f (1) + ∫ Δ u1 (t)u󸀠2 (t) − u2 (t)u󸀠1 (t) 0

1

−

(u󸀠2 (1) + δu2 (1))u1 (t) g(t)dt] ∫ u1 (t)u󸀠2 (t) − u2 (t)u󸀠1 (t) 0 1

=

1 [−f (0)(1 + δ) − k2 f (1) + ∫(δt − 1 − δ)g(t)dt] Δ 0

and 1

(u󸀠1 (1) + δu1 (1))u2 (t) 1 b = [f (0)(u󸀠1 (1) + δu1 (1)) − k2 f (1) + ∫ g(t)dt Δ u1 (t)u󸀠2 (t) − u2 (t)u󸀠1 (t) 1

−∫ 0

0

(u󸀠2 (1) + δu2 (1))u1 (t) g(t)dt] u1 (t)u󸀠2 (t) − u2 (t)u󸀠1 (t) 1

1 = [f (0)δ − k2 f (1) + ∫(δt − 1 − δ)g(t)dt], Δ 0

where Δ = (u󸀠1 (1) + δu1 (1)) + γ(u󸀠2 (1) + δu2 (1)) = δ + γ(1 + δ).

350 | 15 Applications to mathematical models Since 0 ∈ ̸ σp (A), Δ ≠ 0. Consequently, A−1 (f , g) = (u, v). Thus, we obtain 󵄩2 󵄩2 󵄩 󵄩󵄩 −1 󵄩󵄩A (f , g)󵄩󵄩󵄩 = 󵄩󵄩󵄩(u, v)󵄩󵄩󵄩

1

1

󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 = ∫ 󵄨󵄨󵄨u󸀠 (x)󵄨󵄨󵄨 dx + γ 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 + δ󵄨󵄨󵄨u(1)󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 dx 0

󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 = ∫󵄨󵄨b + ∫ g(t)dt 󵄨󵄨󵄨 dx + 󵄨󵄨 󵄨󵄨 󵄨 0󵄨 0 1󵄨

x

0

1 󵄨󵄨 󵄨2 󵄨b + f (0)󵄨󵄨󵄨 + γ󵄨

1 󵄨 󵄨󵄨2 󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨󵄨k2 f (1) − b − ∫ g(t)dt 󵄨󵄨󵄨 󵄨 󵄨󵄨 δ 󵄨󵄨 󵄨

1 󵄨󵄨 󵄨2 󵄨b + f (0)󵄨󵄨󵄨 + γ󵄨

1 󵄨 󵄨󵄨2 󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨󵄨k2 f (1) − b − ∫ g(t)dt 󵄨󵄨󵄨 󵄨󵄨 δ 󵄨󵄨󵄨 󵄨

0

1

󵄨 󵄨2 + ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx 0

x 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨 = ∫󵄨󵄨󵄨b + ∫ g(t)dt 󵄨󵄨󵄨 dx + 󵄨󵄨 󵄨󵄨 󵄨 0󵄨 0 1󵄨

2 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 + ∫󵄨󵄨∫ f (t)dt 󵄨󵄨 dx 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨0

0

1󵄨 x

1

2

1

2 1 2 󵄨 󵄨 󵄨 󵄨 2 2k 󵄨 󵄨2 ≤ ∫[( + 1)(b + ∫ 󵄨󵄨󵄨g(t)󵄨󵄨󵄨dt) + (b + 󵄨󵄨󵄨f (0)󵄨󵄨󵄨) + 2 󵄨󵄨󵄨f (1)󵄨󵄨󵄨 . δ γ δ 0

2

1

0

󵄨 󵄨 + (∫ 󵄨󵄨󵄨f 󸀠 (t)󵄨󵄨󵄨dt) ]dx 0

1

1

1 2

2

2 1 2 󵄨 󵄨 󵄨2 󵄨 2 2k 󵄨 󵄨2 ≤ ∫[( + 1)(b + (∫ 󵄨󵄨󵄨g(t)󵄨󵄨󵄨 dt) ) + (b + 󵄨󵄨󵄨f (0)󵄨󵄨󵄨) + 2 󵄨󵄨󵄨f (1)󵄨󵄨󵄨 δ γ δ 0

0

1

󵄨 󵄨2 + ∫ 󵄨󵄨󵄨f 󸀠 (t)󵄨󵄨󵄨 dt]dx. 0

Hence, we have 1

1

1

0

0

0

󵄩󵄩 −1 󵄩2 󵄨 󸀠 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄩󵄩A (f , g)󵄩󵄩󵄩 ≤ C ∫[∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 dt + γ 󵄨󵄨󵄨f (0)󵄨󵄨󵄨 + δ󵄨󵄨󵄨f (1)󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨g(t)󵄨󵄨󵄨 dt]dx 󵄩 󵄩2 = 󵄩󵄩󵄩(f , g)󵄩󵄩󵄩 , where C is a positive constant. Then we get 󵄩󵄩 −1 󵄩󵄩 󵄩󵄩A 󵄩󵄩 ≤ C,

15.4 Frame of a one-dimensional wave control system

| 351

so 0 ∈ ρ(A). Further, it follows from the Sobolev embedding theorem that A−1 is compact. Thus, the resolvent set of A is compact. On the other hand, let λ ∈ ℂ and consider the following eigenvalue problem: {

(A − λ)F = 0, F = (u, v) ∈ 𝒟(A).

(15.4.6)

The system (15.4.6) is equivalent to v(x) − λu(x) = 0, { { { { {u󸀠󸀠 (x) − λv(x) = 0, { 󸀠 { {u (0) = γu(0) + v(0), { { 󸀠 {u (1) = −δu(1) − k2 v(1).

(15.4.7)

Clearly, we have v(x) = λu(x). Substituting this into the system (15.4.7), we get u󸀠󸀠 (x) − λ2 u(x) = 0, { { { 󸀠 u (0) = γu(0) + λu(0), { { { 󸀠 {u (1) = −δu(1) − k2 λu(1).

(15.4.8)

The solution of the system (15.4.8) is formally given by u(x) = aeλx + be−λx and satisfies the following boundary conditions: u󸀠 (0) = aλ − bλ = (γ + λ)(a + b),

{

u󸀠 (1) = aλeλ − bλe−λ = (−δ − k2 λ)(aeλ + be−λ ).

(15.4.9)

Hence, the system (15.4.9) can be written as γa = −(2λ + γ)b

{

a[(1 + k2 )λ + δ]eλ − b[(1 − k2 )λ − δ]e−λ = 0.

We deduce that the necessary and sufficient condition to obtain a non-zero solution is [(1 + k2 )λ + δ](2λ + γ)eλ + γ[(1 − k2 )λ − δ]e−λ = 0.

352 | 15 Applications to mathematical models 15.4.4 Non-orthogonal fusion frame Let σ(A) = (λn )∞ n=1 . For each eigenvalue λn ∈ σ(A), the corresponding eigenvector of A is φn = (λn−1 [eλn x −

γ γ e−λn x ], [eλn x − e−λn x ]) 2λn + γ 2λn + γ

and the eigenvector of A∗ corresponding to λn is φ∗n = ξn (λn−1 [eλn x −

γ

2λn + γ

e−λn x ], −[eλn x −

γ

2λn + γ

e−λn x ]),

where ξn−1 =

2

2

4γ γ γ γ δ + 2 [1 − ] + 2 [eλn − e−λn ] . 2λn + γ λn 2λn + γ 2λn + γ λn

Evidently, ⟨φn , φ∗n ⟩𝒳 = 1,

⟨φm , φ∗n ⟩𝒳 = 0,

m ≠ n.

For each λn ∈ σ(A) and for F ∈ 𝒳 , the corresponding Riesz projection Pn is Pn F = ⟨F, φ∗n ⟩𝒳 φn and 󵄩 󵄩 ‖Pn ‖ = 󵄩󵄩󵄩φ∗n 󵄩󵄩󵄩‖φn ‖ = |ξn |‖φn ‖2 .

(15.4.10)

Proposition 15.4.1 (G. Q. Xu and S. P. Yung [192, p. 253]). The eigenvectors of A fail to be a basis for 𝒳 . Furthermore, we have ‖Pn ‖ ≈

|λn |2 M e−4 Re(λn ) , ≤ { 1 −2 Re(λn ) 2|Re(λn )|δ M1 e ,

as k2 ≠ 1, as k2 = 1,

(15.4.11)

where M1 is a positive constant. Proof. An elementary calculation reveals that 󵄨 󵄨 lim 󵄨󵄨λ−4 e−2λn ξn 󵄨󵄨󵄨 n→∞ 󵄨 n

=

4 γ2 δ

(15.4.12)

and γ2 󵄨 󵄨 lim 󵄨󵄨󵄨λn2 Re λn e2 Re λn 󵄨󵄨󵄨‖φn ‖2 = . n→∞ 8

(15.4.13)

15.4 Frame of a one-dimensional wave control system

| 353

Then, combining equations (15.4.12) and (15.4.13), we get 󵄨 󵄨 lim 󵄨󵄨λ−2 Re λn 󵄨󵄨󵄨|ξn |‖φn ‖2 n→∞ 󵄨 n

=

1 . 2δ

(15.4.14)

Hence, equations (15.4.10) and (15.4.14) imply that sup ‖Pn ‖ = ∞ n

and, consequently, the eigenvectors of A fail to be a basis for 𝒳 . On the other hand, Γ(λ) = 0 yields e2λ =

γ(k2 − 1)λ + γδ , 2λ2 (1 + k2 ) + [γ(1 + k2 ) + 2δ]λ + γδ

so, for k2 ≠ 1, we obtain |γ(k2 − 1)λ + γδ| 󵄨󵄨 2λ 󵄨󵄨 2 Re λ = 󵄨󵄨e 󵄨󵄨 = e 2 |2λ (1 + k2 ) + [γ(1 + k2 ) + 2δ]λ + γδ| |γ(k2 − 1)λ + γδ| ≤ |2λ2 (1 + k2 )| ≤

|γ(k2 − 1)| +

γδ |λ|

2|λ|(1 + k2 )

(15.4.15)

.

Thus, it follows from equation (15.4.15) that |λ| ≤ D1 e−2 Re λ ,

(15.4.16)

where D1 is a positive constant. Further, for k2 = 1, we have γδ 󵄨󵄨 2λ 󵄨󵄨 2 Re λ . ≤ 󵄨󵄨e 󵄨󵄨 = e 4|λ2 | Hence, we get 󵄨󵄨 2 󵄨󵄨 γδ −2 Re λ e . 󵄨󵄨λ 󵄨󵄨 ≤ 4

(15.4.17)

Consequently, equations (15.4.10), (15.4.14), (15.4.16), and (15.4.17) imply that ‖Pn ‖ ≈

|λn |2 M e−4 Re(λn ) , ≤ { 1 −2 Re(λn ) 2|Re(λn )|δ , M1 e

where M1 is a positive constant. The following results hold.

as k2 ≠ 1, as k2 = 1,

354 | 15 Applications to mathematical models Proposition 15.4.2. The family {Pn , vn }∞ n=1 forms a non-orthogonal fusion frame for 𝒳 , where vn =

1

5

ξ

|λn | 2 + 2

,

ξ > 0.

Proof. It is clear here that vn > 0. Now, let f ∈ 𝒳 . In view of equation (15.4.11), we have ∞

∞

n=1

n=1

󵄩2 󵄩 ∑ vn2 󵄩󵄩󵄩Pn (f )󵄩󵄩󵄩 ≤ ‖f ‖2 ∑ vn2 ‖Pn ‖2 ∞

≤ ‖f ‖2 ∑ [ n=1

|λn |2 1 ] 2|Re(λn )|δ |λ | 21 (5+ξ ) n

2

2

∞ 1 1 ] . ≤ ‖f ‖2 ∑ [ 1 2|Re(λ )|δ 2 n |λn | (1+ξ ) n=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(15.4.18)

0. Consequently, we have 2

∞

∑[

n=1

1 1 ] < ∞. 2|Re(λn )|δ |λ | 21 (1+ξ ) n

On the other hand, we know Pn is a bounded operator with closed range. Hence, in view of equation (2.18.3), Pn admits a pseudo-inverse Pn† and inf{‖Pn g‖ such that ‖g‖ = 1, g ∈ N(Pn )⊥ } =

1 > 0. ‖Pn† ‖

Hence, ∞

∞

󵄩 󵄩2 󵄩 󵄩−2 ∑ vn2 󵄩󵄩󵄩Pn (f )󵄩󵄩󵄩 ≥ ∑ ‖fn ‖2 vn2 󵄩󵄩󵄩Pn† 󵄩󵄩󵄩 , n=1 n=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

fn ∈ N(Pn )⊥

>0

∞

󵄩 󵄩−2 = ∑ (‖f ‖2 − ‖gn ‖2 )vn2 󵄩󵄩󵄩Pn† 󵄩󵄩󵄩 , n=1

2

∞

gn ∈ N(Pn )

‖g ‖ 󵄩 󵄩−2 ≥ ‖f ‖2 inf(1 − n 2 ) ∑ vn2 󵄩󵄩󵄩Pn† 󵄩󵄩󵄩 . n≥1 ‖f ‖ n=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(15.4.19)

>0

As a consequence, equations (15.4.18) and (15.4.19) imply that the family {Pn , vn }∞ n=1 forms a non-orthogonal fusion frame for 𝒳 .

15.4 Frame of a one-dimensional wave control system

| 355

Now, let us consider the following operator: Ak (u, v) := (−1)k (v, u󸀠 ),

(u, v) ∈ 𝒟(Ak ),

where 1

2

𝒟(Ak ) := H (0, 1) × L (0, 1).

Let (u, v) ∈ 𝒟(A). We have 1

1

0

0

󵄨 󸀠 󵄨2 󵄨 󸀠 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄩󵄩 󵄩2 󵄩󵄩Ak (u, v)󵄩󵄩󵄩 = ∫ 󵄨󵄨󵄨v (x)󵄨󵄨󵄨 dx + γ 󵄨󵄨󵄨v(0)󵄨󵄨󵄨 + δ󵄨󵄨󵄨v(1)󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨u (x)󵄨󵄨󵄨 dx 1

1

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 ≤ ∫ 󵄨󵄨󵄨v󸀠 (x)󵄨󵄨󵄨 dx + γ 󵄨󵄨󵄨v(0)󵄨󵄨󵄨 + δ󵄨󵄨󵄨v(1)󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨u󸀠󸀠 (x)󵄨󵄨󵄨 dx 0

0

1

1

󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 + ∫ 󵄨󵄨󵄨u󸀠 (x)󵄨󵄨󵄨 dx + γ 󵄨󵄨󵄨u(0)󵄨󵄨󵄨 + δ󵄨󵄨󵄨u(1)󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨v(x)󵄨󵄨󵄨 dx 0

0

󵄩 󵄩2 󵄩 󵄩2 = 󵄩󵄩󵄩A(u, v)󵄩󵄩󵄩 + 󵄩󵄩󵄩(u, v)󵄩󵄩󵄩 . Consequently, 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩Ak (u, v)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩A(u, v)󵄩󵄩󵄩 + 󵄩󵄩󵄩(u, v)󵄩󵄩󵄩.

(15.4.20)

Using the results described above, we can now prove the objective of this part. Proposition 15.4.3. For |ε| < 1, the series A + ∑ εk Ak F k≥1

converges for all F = (u, v) ∈ 𝒟(A). If we designate its sum by A(ε)F, we define a linear operator A(ε) with domain 𝒟(A). For |ε| < 21 , the operator A(ε) is closed. Proof. The proof follows immediately from Theorem 11.1.1 and equation (15.4.20). Theorem 15.4.2. For |εn | small enough and |ε| small enough, there exist two sequences ∞ of projections {Pn (εn )}∞ n=1 and {Pn (ε)}n=1 of T(ε), having the following form: Pn (εn ) = Pn + εn Pn,1 + εn2 Pn,2 + ⋅ ⋅ ⋅ , Pn (ε) = Pn + εPn,1 + ε2 Pn,2 + ⋅ ⋅ ⋅ ,

such that the systems ∞

N

∞

{Pn (εn ), vn󸀠 }n=1 , {Pn (ε), vn󸀠 }1 ∪ {Pn (εn ), vn󸀠 }N+1 ,

and

N

∞

{Pn (ε), vn󸀠 }1 ∪ {Pn , vn󸀠 }N+1

form non-orthogonal fusion frames for 𝒳 , where {vn󸀠 }∞ n=1 is a family of weights satisfying vn ≤ vn󸀠 ≤ √2vn .

356 | 15 Applications to mathematical models Proof. The proof is a direct implication from Theorems 13.2.1, 13.2.2, and 15.4.1, Proposition 15.4.2, and equation (15.4.20). Theorem 15.4.3. For |ε| small enough, there exists a sequence of projections {Pn (ε)}∞ n=1 having the following form: Pn (ε) = Pn + εPn,1 + ε2 Pn,2 + ⋅ ⋅ ⋅ such that the family {Pn (ε), vn󸀠 }∞ n=1 forms a non-orthogonal fusion frame for 𝒳 , where {vn󸀠 }∞ is a family of weights satisfying n=1 vn ≤ vn󸀠 ≤ √2vn . Proof. Let n ∈ ℕ∗ , let λn be the nth eigenvalue of A, and let rn = min{

||λn | − π3 |λn−1 || ||λn+1 | − π3 |λn || , }. 2 2

As {z ∈ ℂ, |z − λn | ≤ rn } ∩ σ(A) = {λn }, let 𝒞n = 𝒞 (λn , rn ) be the closed circle with center λn , radius rn , and z ∈ 𝒞n . It is easy to verify that the operator A is normal. Hence, it follows from equation (2.18.1) that 󵄩 󵄩 ‖Rz ‖ = 󵄩󵄩󵄩(A − zI)−1 󵄩󵄩󵄩 =

1 . dist(z, σ(A))

Consequently, we obtain αn = aMn + bNn

= a max ‖Rz ‖ + b max ‖ARz ‖ z∈𝒞n

z∈𝒞n

a + b max ‖I + zRz ‖ = z∈𝒞n rn a ≤ + b max(1 + |z|‖Rz ‖). z∈𝒞n rn Thus, we get αn ≤ If rn =

||λn |− π3 |λn−1 || , 2

|λ | a + b(2 + n ). rn rn

then equation (15.4.21) yields

αn ≤ (

2|λn | 2a + b(2 + )) ||λn | − π3 |λn−1 || ||λn | − π3 |λn−1 ||

(15.4.21)

15.4 Frame of a one-dimensional wave control system

≤

2a 2 + b(2 + ). π |λn−1 | ||λn | − π3 |λn−1 || |1 − 3 |λ | |

| 357

(15.4.22)

n

As Γ(λ) is an entire function of finite order at most 1, Theorem 3.5.2 implies n(r) ≤ Cr κ for some constant C and all values of r, where 1 < κ < 1 + ξ . If we choose r = |λn |, then n(r) ≥ n and, hence, 1

1

n κ ≤ C κ |λn |. Then it follows from equation (15.4.22) that 1

sup αn ≤ 2aC κ + b(2 + n≥1

For rn =

||λn+1 |− π3 |λn || , 2

2 ) < ∞. |1 − π3 |

we show in a similar way as above that

sup αn ≤ sup n≥1

n≥1

2 2a + b sup(2 + |λ | ) n+1 ||λn+1 | − π3 |λn || n≥1 | |λ | − π3 | n

2 ≤ 2aC + b(2 + ) < ∞. |1 − π3 | 1 κ

On the other hand, we have ∞

2 ∞

2

∑ (vn󸀠 rn Mn αn ) ≤ (sup αn ) ∑ (vn󸀠 )

n=1

n≥1

2

n=1 2 ∞

≤ (sup αn ) ∑ 2vn2 n≥1

n=1 2 ∞

2 < ∞. (5+ξ ) |λ | n=1 n

≤ (sup αn ) ∑ n≥1

Consequently, the family {Pn (ε), vn󸀠 }∞ n=1 forms a non-orthogonal fusion frame for 𝒳 .

16 Reggeon field theory 16.1 Gribov operator in Bargmann space We will discuss a family of operators used in theoretical physics of the following form: n

n

n

n−1

j=1

j=1

j=1

j=1

2 ∗ ∗ ∗ ∗ ∗ Hλ󸀠 ,μ,α,λ = λ󸀠 ∑ A∗2 j Aj + μ ∑ Aj Aj + iλ ∑ Aj (Aj + Aj )Aj + α ∑ (Aj+1 Aj + Aj Aj+1 ),

where λ󸀠 , μ, λ, and α are real parameters on the physical meaning of which we do not expand and Aj , A∗j are the annihilation and creation operators acting on a Hilbert space E and checking the following commutation relations: I [Aj , A∗k ] = { 0

if j = k, if not,

[Aj , Ak ] = [A∗j , A∗k ] = 0, where j, k = 1, . . . , n. We generally study these operators on the Bargmann space E defined by 2

E = {φ : ℂn 󳨀→ ℂ holomorphic such that ∫ e−|z| |φ(z)|2 dμ(z) < ∞}, ℂn

where z = (z1 , . . . , zn ), zi = xi + iyi , and dμ(z) =

1 dx ⋅ ⋅ ⋅ dxn dy1 ⋅ ⋅ ⋅ dyn . (2π)n 1

E is a Hilbert space with the inner product 2

⟨φ, ψ⟩ = ∫ e−|z| φ(z)ψ(z)dμ(z). ℂn

In this space, the operators Aj , A∗j are expressed by Aj φ =

𝜕φ 𝜕zj

and A∗j φ = zj φ. https://doi.org/10.1515/9783110493863-016

360 | 16 Reggeon field theory 16.1.1 General results Lemma 16.1.1 (A. Intissar [109, Lemma 1]). (i) The polynomial space on ℂn is dense in E. k k k (ii) Let (k1 , . . . , kn ) ∈ ℕn and let ek (z) = √z , where z k = z1 1 ⋅ ⋅ ⋅ znn and k! = k1 ! ⋅ ⋅ ⋅ kn !. k! Then {ek }k∈ℕn is a Hilbert basis of E. (iii) There is an isometric from E into ln2 (ℂ) given by the Hilbert basis, where ln2 (ℂ) denotes the set of all applications from ℕn into ℂ, so, for φ(z) = ∑ ak z k , the range of φ by this isometry is (√k!ak )k∈ℕn . The space E is isometric to the space Es , where Es is the set of the sequence a = (ak )k such that ‖a‖ = ∑ k!|ak |2 < ∞. k≥0

The isometry above is given by φ(z) = ∑ ak z k 󳨀→ a = (ak )k . k≥0

Remark 16.1.1. We consider the operators Aj , A∗j , A2j , and A∗2 j acting on the Hilbert basis (ek )k by 0 Aj ek = { √kj ek−εj A∗j ek = √kj + 1ek+εj

if k = 0,

if not

for j ∈ [1, n],

for j ∈ [1, n],

A2j ek = √kj (kj − 1)ek−2εj

for j ∈ [1, n],

and A∗2 j ek = √(kj + 1)(kj + 2)ek+2εj

for j ∈ [1, n],

where (εj )1≤j≤n is the canonical basis of ℕn , so we have Aj [(ak )k∈ℕn ] = (√kj + 1ak+εj )k∈ℕn , A∗j [(ak )k∈ℕn ] = (√kj ak−εj )k∈ℕn . This new representation allows one to show the following. Lemma 16.1.2 (A. Intissar [109, Lemma 2]). (i) We have 2

𝒟(Aj ) = {(ak ) such that ∑(kj + 1)|ak+εj | < ∞} = 𝒟(Aj ). k

∗

16.1 Gribov operator in Bargmann space

| 361

(ii) The subspaces D := ⋂1≤j≤n 𝒟(Aj ) and D2 := ⋂1≤j≤n 𝒟(A2j ) are dense in E and injected into E compactly. Moreover, D and D2 are, respectively, provided with the norms n

1 2

‖φ‖D = (∑ ‖Aj φ‖2 ) , j=1

‖φ‖D2 =

n

󵄩 󵄩2 (∑ 󵄩󵄩󵄩A2j φ󵄩󵄩󵄩 ) j=1

1 2

.

(iii) The operators Aj and A∗j are associated with each other. Now, we will consider the operators Hλ󸀠 ,μ,α,λ acting on the closed invariant subspace E0 := {φ ∈ E such that φ(0) = 0} of E. The operator Hλ󸀠 ,μ,α,λ is unbounded and non-symmetric. Let S, H0 , H, and Hλ󸀠 be the operators defined by S : 𝒟(S) ⊂ E0 󳨀→ E0 , { { n { { 2 { φ 󳨀→ Sφ(z) = ∑ A∗2 j Aj , { { j=1 { { { {𝒟(S) = {φ ∈ E0 such that Sφ ∈ E0 }, H0 : 𝒟(H0 ) ⊂ E0 󳨀→ E0 , { { n { { { φ 󳨀→ H0 φ(z) = ∑ A∗j Aj , { { j=1 { { { {𝒟(H0 ) = {φ ∈ E0 such that H0 φ ∈ E0 }, H : 𝒟(H) ⊂ E0 󳨀→ E0 , { { n n−1 { { { φ 󳨀→ Hφ(z) = μH0 + iλ ∑ A∗j (Aj + A∗j )Aj + α ∑ (A∗j+1 Aj + A∗j Aj+1 ), { { j=1 j=1 { { { {𝒟(H) = {φ ∈ E0 such that Hφ ∈ E0 }, and Hλ󸀠 : 𝒟(Hλ󸀠 ) ⊂ E0 󳨀→ E0 , { { φ 󳨀→ Hλ󸀠 φ(z) = λ󸀠 S + H, { { {𝒟(Hλ󸀠 ) = {φ ∈ E0 such that Hλ󸀠 φ ∈ E0 }, with λ󸀠 ≠ 0. The closeness of these operators follows from the fact that the Bargmann space is injected continuously into the distribution space 𝒟(ℝ2n ). Their domains are dense (since they contain P0 , the set of polynomials on ℂn which are zero at the origin) and it can be shown that, if we equip them for their graph norm, their injections in E0 are compact. The minimum domain of Hλ󸀠 is 𝒟min (Hλ󸀠 ) = {φ ∈ E0 such that there is pn ∈ P0 and ψ ∈ E0 such that

pn → φ and Hλ󸀠 pn → ψ}.

Let Hλmin be the operator Hλ󸀠 with domain 𝒟min (Hλ󸀠 ). 󸀠

362 | 16 Reggeon field theory 16.1.2 Auxiliary results Lemma 16.1.3 (A. Intissar [109, Lemma 4]). Let j ∈ [1, n] and let k be a non-zero integer. Then: (i) for all polynomials p in Aj and A∗j of degree r < 2k, we know, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(A2k j ),

󵄨 󵄨󵄨 ∗k k 2 ∗ 󵄨󵄨⟨p(Aj , Aj )φ, φ⟩󵄨󵄨󵄨 ≤ ε⟨Aj Aj φ, φ⟩ + cε ‖φ‖ ;

and

(ii) for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(Ak+1 j ), 󵄩󵄩 k 󵄩󵄩2 󵄩 k+1 󵄩2 2 󵄩󵄩Aj φ󵄩󵄩 ≤ ε󵄩󵄩󵄩Aj φ󵄩󵄩󵄩 + cε ‖φ‖ . Proof. (i) Let φ(z) = ∑k ak ek (z). Then we have l A∗m j Aj φ = ∑ kj (kj − 1) ⋅ ⋅ ⋅ (kj − l + 1) k

ak k1 ⋅⋅⋅kj +m−l⋅⋅⋅kn z √k!

= ∑(kj + l − m)(kj + l − m − 1) ⋅ ⋅ ⋅ (kj − m + 1)

ak1 ⋅⋅⋅(kj +l−m)⋅⋅⋅kn

k

√k!

z k1 ⋅⋅⋅kj +m−l⋅⋅⋅kn .

If l ≥ m, we have √(kj + l − m)! = √kj !√kj + 1 ⋅ ⋅ ⋅ √kj + l − m. This implies that l A∗m j Aj φ = ∑ √kj + l − m√kj + l − m − 1√kj + 1 ⋅ ⋅ ⋅ (kj − m + 1)ak1 ⋅⋅⋅(kj +l−m)⋅⋅⋅kn ek (z). k

Let u(kj ) = √kj + l − m√kj + l − m − 1 ⋅ ⋅ ⋅ √kj + 1 ⋅ ⋅ ⋅ (kj − m + 1). Then we have u(kj ) ∼ kj1/2(l−m) kjm = kj1/2(l+m) . Hence, u(kj ) = O(kj1/2(l+m) ).

(16.1.1)

If l ≤ m, then, by a similar reasoning as above, we deduce that u(kj ) = O(kj1/2(l+m) ).

(16.1.2)

16.1 Gribov operator in Bargmann space

| 363

l Now, calculate the expression of ⟨A∗m j Aj φ, φ⟩. We have

󵄨󵄨 󵄨 󵄨 󵄨󵄨 ∗m l 󵄨 󵄨󵄨 󵄨󵄨⟨Aj Aj φ, φ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨∑ u(kj )ak1 ⋅⋅⋅(kj +l−m)⋅⋅⋅kn ak1 ⋅⋅⋅kn 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 k ≤ ∑ u(kj )|ak1 ⋅⋅⋅(kj +l−m)⋅⋅⋅kn ak1 ⋅⋅⋅kn | k

1 ≤ ∑[u(kj ) + u(kj + l − m)]|ak |2 . 2 k Using equations (16.1.1) and (16.1.2), there is c0 > 0 and c1 > 0 such that u(kj ) ≤ c0 + c1 kj1/2(l+m) . For m + l < 2k, applying the inequality of Young, we know, for all δ > 0, there is cδ > 0 such that kj1/2(l+m) ≤ δkjk + cδ . Hence, 󵄨󵄨 ∗m l 󵄨 k 2 2 󵄨󵄨⟨Aj Aj φ, φ⟩󵄨󵄨󵄨 ≤ c1 δ ∑ kj |ak | + (cδ + c0 ) ∑ |ak | , so, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(A2k j ), we have 󵄨󵄨 ∗m l 󵄨 ∗k k 2 󵄨󵄨⟨Aj Aj φ, φ⟩󵄨󵄨󵄨 ≤ ε⟨Aj Aj φ, φ⟩ + cε ‖φ‖ . (ii) This assertion follows from (i). Lemma 16.1.4 (A. Intissar [109, Lemma 5]). Let j ∈ [1, n] and let k be an integer. For all polynomials p in Aj and A∗j of degree r < 2k, we have:

(i) for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(A2(k+1) ), j

󵄨󵄨 󵄨 ∗ ∗k k 2 󵄨󵄨⟨p(Aj , Aj )φ, φ⟩𝒟(Aj ) 󵄨󵄨󵄨 ≤ ε⟨Aj Aj φ, φ⟩𝒟(Aj ) + cε ‖φ‖𝒟(Aj ) ;

and

(ii) for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(A2(k+2) ), j 󵄨󵄨 󵄨 ∗ ∗k k 2 󵄨󵄨⟨p(Aj , Aj )φ, φ⟩𝒟(A2 ) 󵄨󵄨󵄨 ≤ ε⟨Aj Aj φ, φ⟩𝒟(A2 ) + cε ‖φ‖𝒟(A2 ) . j j j Proof. (i) Using Lemma 16.1.3 (i), we know, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(A2(k+1) ), j 󵄨󵄨 󵄨 ∗ ∗(k+1) (k+1) Aj φ, φ⟩ + cε ‖φ‖2 . 󵄨󵄨⟨p(Aj , Aj )φ, φ⟩𝒟(Aj ) 󵄨󵄨󵄨 ≤ ε⟨Aj Since Aj A∗j − A∗j Aj = I,

364 | 16 Reggeon field theory we infer that, for all integer k, ∗k ∗(k−1) A∗k . j Aj = Aj Aj − kAj

Hence, k ⟨A∗(k+1) A(k+1) φ, φ⟩ = ⟨A∗k j Aj Aj φ, Aj φ⟩ j j

∗(k−1) k = ⟨(Aj A∗k )Aj φ, Aj φ⟩ j − kAj

k ∗k k = ⟨Aj A∗k j Aj φ, Aj φ⟩ − k⟨Aj Aj φ, φ⟩.

Thus, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(A2(k+1) ), j 󵄨󵄨 󵄨 ∗ ∗k k 2 󵄨󵄨⟨p(Aj , Aj )φ, φ⟩𝒟(Aj ) 󵄨󵄨󵄨 ≤ ε⟨Aj Aj φ, φ⟩𝒟(Aj ) + cε ‖φ‖𝒟(Aj ) . (ii) The proof of (ii) is similar to the previous one. We have the following. Lemma 16.1.5 (A. Intissar [109, Lemma 6]). (i) For all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(S), ‖Hφ‖ ≤ ε‖Sφ‖ + cε ‖φ‖. (ii) For all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(S), 󵄨󵄨 󵄨 2 󵄨󵄨⟨Hφ, φ⟩󵄨󵄨󵄨 ≤ ε⟨Sφ, φ⟩ + cε ‖φ‖ . (iii) For all ε > 0, there is cε > 0 such that, for all φ ∈ ⋂nj=1 𝒟(A6j ), 󵄨󵄨 󵄨 2 󵄨󵄨⟨Hφ, φ⟩D 󵄨󵄨󵄨 ≤ ε⟨Sφ, φ⟩D + cε ‖φ‖D , where D is given in Lemma 16.1.2. (iv) For all ε > 0, there is cε > 0 such that, for all φ ∈ ⋂nj=1 𝒟(A8j ), 󵄨󵄨 󵄨 2 󵄨󵄨⟨Hφ, φ⟩D2 󵄨󵄨󵄨 ≤ ε⟨Sφ, φ⟩D2 + cε ‖φ‖D2 , where D2 is given in Lemma 16.1.2. Proof. (i) Let φ ∈ 𝒟(S). For φ(z) = ∑|k|≥1 ak ek , we have ∑ [k1 + k2 + ⋅ ⋅ ⋅ + kn ]|ak |2 < ∞, |k|≥1

∑ [k12 + k22 + ⋅ ⋅ ⋅ + kn2 ]|ak |2 < ∞,

|k|≥1

∑ [k13 + k23 + ⋅ ⋅ ⋅ + kn3 ]|ak |2 < ∞,

|k|≥1

16.1 Gribov operator in Bargmann space

| 365

and ∑ [k12 (k1 − 1)2 + k22 (k2 − 1)2 + ⋅ ⋅ ⋅ + kn2 (kn − 1)2 ]|ak |2 < ∞. |k|≥1

Hence, ∑ [k14 + k24 + ⋅ ⋅ ⋅ + kn4 ]|ak |2 < ∞. |k|≥1

Since n

n

j=1

j=1

Hφ(z) = ∑ {μ ∑ kj ak + iλ ∑[(kj − 1)√kj ak−εj + √kj + 1kj ak+εj ] |k|≥1 n−1

+ α ∑ [√kj √kj+1 + 1ak−εj +εj +1 + √kj + 1√kj + 1ak+εj −εj +1 ]}ek (z), j=1

combining the Cauchy and the Hölder inequality, we obtain n

n

j=1

j=1

‖Hφ‖2 ≤ ∑[c1 ∑ kj2 + c2 ∑ kj3 ]|ak |2 , k

where c1 and c2 are positive constants. Hence, ‖Hφ‖2 ≤ c ∑[k13 + k23 + ⋅ ⋅ ⋅ + kn3 ]|ak |2 , k

where c is a constant. Apply, again, Hölder’s inequality to the pair (kj3 , 1), for all j∈[1, n]. We know, for all δ > 0, there is cδ > 0 such that kj3 ≤ δkj4 + cδ . Hence, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(S), ‖Hφ‖ ≤ ε‖Sφ‖ + cε ‖φ‖. Statements (ii), (iii), and (iv) follow from Lemmas 16.1.3 and 16.1.4.

16.1.3 Spectral properties of Hλ󸀠 Proposition 16.1.1 (A. Intissar [109, Propositions 1–3]). (i) The operator H is S-compact. (ii) We have 𝒟(Hλmin ) = 𝒟(S) for λ󸀠 ≠ 0. 󸀠 󸀠 (iii) For λ ≠ 0, there is β0 ∈ ℝ such that Hλmin + β0 is one-to-one and onto. 󸀠 (iv) The maximum domain, 𝒟(Hλ󸀠 ), of Hλ󸀠 is equal to the minimal domain, 𝒟min (Hλ󸀠 ).

366 | 16 Reggeon field theory Proof. (i) Since S has compact resolvent, by applying Lemma 16.1.5 (i), we infer that H is S-compact. (ii) The result follows from (i) and Theorem 2.8.1. β (iii) Let β > 0. If λ󸀠 > 0, then − λ󸀠 ∈ ρ(S) and β 1 + H) λ󸀠 λ󸀠

Hλ󸀠 + β = λ󸀠 (S + = λ󸀠 [I +

β β 1 H(S + 󸀠 ) ](S + 󸀠 ). 󸀠 λ λ λ −1

(16.1.3)

By using Lemma 16.1.5 (i), we know, for all ε > 0, there is cε > 0 such that, for all φ ∈ E, −1 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 β β β 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󸀠 H(S + 󸀠 ) φ󵄩󵄩󵄩 ≤ ε󵄩󵄩󵄩S(S + 󸀠 ) φ󵄩󵄩󵄩 + cε 󵄩󵄩󵄩(S + 󸀠 ) φ󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 λ 󵄩 󵄩 λ λ λ 󵄩 󵄩 󵄩 󵄩 −1 −1 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 β β β β 󵄩 󵄩 󵄩 = ε󵄩󵄩󵄩(S + 󸀠 − 󸀠 )(S + 󸀠 ) φ󵄩󵄩󵄩 + cε 󵄩󵄩󵄩(S + 󸀠 ) φ󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 λ λ λ λ 󵄩󵄩 −1 󵄩 󵄩 β β 󵄩󵄩 󵄩󵄩 ≤ ε‖φ‖ + (ε 󸀠 + cε )󵄩󵄩󵄩(S + 󸀠 ) φ󵄩󵄩󵄩. (16.1.4) 󵄩󵄩 󵄩󵄩 λ λ

From equation (16.1.4), using the fact that S is a positive self-adjoint operator and satisfies −1 󵄩 󵄩󵄩 󵄩󵄩 λ󸀠 β 󵄩󵄩 󵄩󵄩(S + 󸀠 ) 󵄩󵄩󵄩 ≤ , 󵄩󵄩 β 󵄩󵄩 λ

we have −1 󵄩 󵄩󵄩 1 󵄩󵄩 λ 󸀠 cε β 󵄩󵄩 )‖φ‖. 󵄩󵄩 󸀠 H(S + 󸀠 ) φ󵄩󵄩󵄩 ≤ (2ε + 󵄩󵄩 λ β 󵄩󵄩 λ

If we take ε


λ󸀠 cε , 1−2ε

we obtain −1 󵄩 󵄩󵄩 1 󵄩󵄩 β 󵄩󵄩 󵄩󵄩 󸀠 H(S + 󸀠 ) 󵄩󵄩󵄩 < 1. 󵄩󵄩 λ 󵄩󵄩 λ

The result of (iii) follows from (16.1.3). (iv) It is easy to see that 𝒟min (Hλ󸀠 ) ⊂ 𝒟(Hλ󸀠 ).

Moreover, we claim the opposite inclusion. Let φ∈ 𝒟(Hλ󸀠 ). We have φ∈E0 and Hλ󸀠 φ∈E0 . Using (iii), there is β0 ∈ ℝ such that Hλmin + β0 is one-to-one and onto from 𝒟min (Hλ󸀠 ) 󸀠 into E0 , so there is φ1 ∈ 𝒟min (Hλ󸀠 ) such that (Hλ󸀠 + β0 )φ = (Hλmin + β0 )φ1 . 󸀠

16.1 Gribov operator in Bargmann space

| 367

In particular, (Hλ󸀠 + β0 )(φ − φ1 ) = 0.

(16.1.5)

We must demonstrate that Hλ󸀠 + β0 is one-to-one. For this purpose, let λ󸀠 > 0. We have Re⟨Hλmin φ, φ⟩ = λ󸀠 ⟨Sφ, φ⟩ + Re⟨Hφ, φ⟩. 󸀠 By using Lemma 16.1.5 (ii), we infer that there is c > 0 such that, for all φ ∈ 𝒟min (Hλ󸀠 ), we have Re⟨Hλmin φ, φ⟩ ≥ (λ󸀠 − ε)⟨Sφ, φ⟩ − c‖φ‖2 . 󸀠 If we take ε ≤ λ󸀠 , we infer that, for all φ ∈ 𝒟min (Hλ󸀠 ), we have Re⟨Hλmin φ, φ⟩ ≥ −c‖φ‖2 . 󸀠

(16.1.6)

Let λ󸀠 < 0. By using a similar reasoning as before, we show that there exists c > 0 such that, for all φ ∈ 𝒟min (Hλ󸀠 ), we have Re⟨Hλmin φ, φ⟩ ≤ c‖φ‖2 . 󸀠

(16.1.7)

From equations (16.1.6) and (16.1.7), we deduce that Hλmin + β0 is an operator with 󸀠 closed range. Moreover, Hλ†󸀠 ,μ,λ,α = Hλ󸀠 ,μ,−λ,α , where Hλ†󸀠 ,μ,λ,α denotes the formal adjoint of Hλ󸀠 (since the adjointness of the minimal operator is the formal adjoint of the maximal operator). It follows from Proposition 16.1.1 (iii) that Hλ†󸀠 + β0 is one-to-one and onto. This implies that Hλ󸀠 + β0 is oneto-one. Hence, by using equation (16.1.5), we have φ = φ1 and φ ∈ 𝒟min (Hλ󸀠 ), which proves our claim and the proof is completed. Proposition 16.1.2 (A. Intissar [109, Proposition 4]). For λ󸀠 ≠ 0, we have: (i) Hλ󸀠 , with domain 𝒟(S), is an operator with compact resolvent; (ii) the spectrum of Hλ󸀠 is a sequence of complex numbers (σk )k (we may order with ascending module such that |σk | → ∞ as k → ∞); (iii) σk is an eigenvalue of Hλ󸀠 for all k ≥ 0; and (iv) R(Hλ󸀠 − σk ) is closed. Proof. (i) Since the canonical injection from 𝒟(S) into E0 is compact and ρ(Hλ󸀠 ) ≠ 0, Hλ󸀠 is an operator with compact resolvent. Statements (ii), (iii), and (iv) are classical results.

368 | 16 Reggeon field theory Proposition 16.1.3. For all λ󸀠 ≠ 0, the resolvent of Hλ󸀠 is in Cp (E0 ) for p > n2 . In fact, this resolvent is in C n ,∞ (E0 ), where 2

2

C n ,∞ (E0 ) = {A ∈ ℒ(E0 ) such that (p n sp (A))p is bounded}. 2

Proof. For some β, as was done in the proof of Proposition 16.1.1 (iii), we have (Hλ󸀠 + β)−1 =

β β 1 1 (S + 󸀠 ) [I + 󸀠 H(S + 󸀠 ) ] . 󸀠 λ λ λ λ −1

−1 −1

Since [I +

β 1 H(S + 󸀠 ) ] λ󸀠 λ

−1 −1

∈ ℒ(E0 ),

for p > n2 , we have (S +

β ) λ󸀠

−1

∈ Cp (E0 ),

which implies that (Hλ󸀠 + β)−1 ∈ Cp (E0 ). 16.1.4 Cauchy problem We consider the Cauchy problem 𝜕u { (t) = −Hλ󸀠 u(t) { 𝜕t {u(0) = φ,

t > 0,

(16.1.8)

where φ ∈ E0 . Lemma 16.1.6 (A. Intissar [109, Lemma 7]). Consider the Cauchy problem (16.1.8) and let λ󸀠 > 0 and u(0) ∈ D := ⋂nj=1 𝒟(Aj ). Then: (i) there is c > 0 such that ‖u(t)‖D ≤ ect ‖u(0)‖D ; (ii) e−tHλ󸀠 φ ∈ L2 ([0, T]) for all φ ∈ E0 ; and (iii) e−tHλ󸀠 φ ∈ D for all t > 0 and all φ ∈ E0 . Proof. (i) We have Re⟨u󸀠 (t), u(t)⟩D = −Re⟨Hλ󸀠 u(t), u(t)⟩D . By using Lemma 16.1.5 (iii), we infer that there is c > 0 such that 1 d 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩u(t)󵄩󵄩󵄩D ≤ c󵄩󵄩󵄩u(t)󵄩󵄩󵄩D . 2 dt 󵄩

16.1 Gribov operator in Bargmann space

| 369

By using the Grönwall lemma, we deduce that 󵄩 󵄩 󵄩󵄩 ct 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩D ≤ e 󵄩󵄩󵄩u(0)󵄩󵄩󵄩D ,

(16.1.9)

where u(0) ∈ 𝒟(Hλ2󸀠 ). Since 𝒟(Hλ2󸀠 ) is dense in D, for all u(0) ∈ D, there is um (0) ∈ 𝒟(Hλ2󸀠 ) such that um (0) → u(0) as m → ∞. Equation (16.1.9) gives 󵄩 󵄩 󵄩󵄩 ct 󵄩 󵄩󵄩u(t)󵄩󵄩󵄩D ≤ e 󵄩󵄩󵄩u(0)󵄩󵄩󵄩D , where u(0) ∈ D. (ii) Since Aj + A∗j is a symmetric operator, we have 󵄨󵄨 󵄨󵄨 n−1 n 󵄨󵄨 󵄨 󵄨󵄨Re⟨ ∑ (A∗ Aj + A∗ Aj+1 )u(t), u(t)⟩󵄨󵄨󵄨 ≤ 2 ∑ 󵄩󵄩󵄩Aj u(t)󵄩󵄩󵄩2 . j+1 j 󵄨󵄨 󵄨󵄨 󵄩 󵄩 󵄨󵄨 󵄨󵄨 j=1 j=1 Hence, n n 1 d 󵄩󵄩 󵄩2 󵄩 2 󵄩2 󵄩 󵄩2 󸀠 󵄩󵄩u(t)󵄩󵄩󵄩 + λ ∑ 󵄩󵄩󵄩Aj u(t)󵄩󵄩󵄩 − (|μ| + 2α) ∑ 󵄩󵄩󵄩Aj u(t)󵄩󵄩󵄩 ≤ 0. 2 dt j=1 j=1

By using Lemma 16.1.3, for all ε > 0, there is cε > 0 such that 󵄩 󵄩2 󵄩󵄩 󵄩2 󵄩 2 󵄩2 󵄩󵄩Aj u(t)󵄩󵄩󵄩 ≤ ε󵄩󵄩󵄩Aj u(t)󵄩󵄩󵄩 + cε 󵄩󵄩󵄩u(t)󵄩󵄩󵄩 . Hence, for λ󸀠 > 0, we get n 1 d 󵄩󵄩 󵄩2 󵄩 2 󵄩2 󵄩 󵄩2 󸀠 󵄩󵄩u(t)󵄩󵄩󵄩 + (λ − ε(|μ| + 2α)) ∑ 󵄩󵄩󵄩Aj u(t)󵄩󵄩󵄩 − cε (|μ| + 2α)󵄩󵄩󵄩u(t)󵄩󵄩󵄩 ≤ 0. 2 dt j=1

Choosing ε
0. Proposition 16.1.4 (A. Intissar [109, Proposition 5]). For λ󸀠 > 0, we have: (i) −Hλ󸀠 is an infinitesimal generator of a strongly continuous semi-group e−tHλ󸀠 and there is β0 > 0 such that 󵄩󵄩 −tHλ󸀠 󵄩󵄩 β t 󵄩󵄩 ≤ e 0 󵄩󵄩e

for all t ≥ 0;

is compact for all t > 0; (ii) e −tHλ󸀠 ) = e−tσ(Hλ󸀠 ) ∪ {0}. (iii) σ(e −t(Hλ󸀠 +β0 )

Proof. (i) By using equations (16.1.6) and (16.1.7) and Proposition 16.1.1 (iv), we have the existence of β0 > 0 such that, for all φ ∈ 𝒟(Hλ󸀠 ), we have Re⟨Hλ󸀠 φ, φ⟩ ≥ −β0 ‖φ‖2 . This implies that Hλ󸀠 + β0 is an accretive operator. Hence, for all β > β0 and for all φ ∈ 𝒟(Hλ󸀠 ), ‖φ‖ ≤

1 󵄩󵄩 󵄩 󵄩(H 󸀠 + β)φ󵄩󵄩󵄩. β − β0 󵄩 λ

Thus, by using Proposition 16.1.2, we have, for all β > β0 , 1 󵄩󵄩 󸀠 −1 󵄩 . 󵄩󵄩(Hλ + β) 󵄩󵄩󵄩 ≤ β − β0

The result follows from the Hille–Yosida theorem (see Theorem 5.1.1). (ii) Using Lemma 16.1.6, we deduce that e−tHλ󸀠 +β0 is an operator strongly continuous on D. Since the canonical injection from D into E0 is compact, e−tHλ󸀠 +β0 is compact for all t > 0. (iii) The result follows from Theorem 5.1.3. 16.1.5 Denseness of the generalized eigenvectors of the operator Hλ󸀠 for λ󸀠 ≠ 0 By using Proposition 16.1.3, the resolvent of Hλ󸀠 belongs to Cp (E0 ) for p > easy to see that sk ((λ − Hλ󸀠 )−1 ) = o(k as k → ∞ for λ ∈ ρ(Hλ󸀠 ) and p > n2 .

− p1

)

n , 2

so it is

16.1 Gribov operator in Bargmann space

| 371

Proposition 16.1.5. Let λ󸀠 > 0. For all p > 0, there is β0 > 0 such that the numerical range of Hλ󸀠 + β0 is contained in the sector 󵄨 π 󵄨 {z ∈ ℂ such that 󵄨󵄨󵄨arg(z)󵄨󵄨󵄨 ≤ − ε} p with

π p

> ε > 0.

Proof. Let φ ∈ 𝒟(Hλ󸀠 ) = 𝒟(S). We have 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨Im⟨Hλ󸀠 φ, φ⟩󵄨󵄨󵄨 = 󵄨󵄨󵄨Im⟨Hφ, φ⟩󵄨󵄨󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨⟨Hφ, φ⟩󵄨󵄨󵄨. By Lemma 16.1.5 (ii), we know, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(S), 󵄨󵄨 󵄨 2 󵄨󵄨Im⟨Hλ󸀠 φ, φ⟩󵄨󵄨󵄨 ≤ ε⟨Sφ, φ⟩ + cε ‖φ‖ and similarly, for all β > 0, we have Im⟨(Hλ󸀠 + β)φ, φ⟩ = Im⟨Hλ󸀠 φ, φ⟩. Similarly, for all ε > 0, there is cε > 0 such that, for all φ ∈ 𝒟(S), 󵄨󵄨 󵄨 󸀠 2 󵄨󵄨Re⟨Hλ󸀠 φ, φ⟩󵄨󵄨󵄨 ≥ (λ − ε)⟨Sφ, φ⟩ − cε ‖φ‖ . Hence, if β ∈ ]cε , ∞[, then, for all φ ∈ 𝒟(S), 󵄨󵄨 󵄨 󸀠 2 󵄨󵄨Re⟨(Hλ󸀠 + β)φ, φ⟩󵄨󵄨󵄨 ≥ (λ − ε)⟨Sφ, φ⟩ + (β − cε )‖φ‖ . Let a > 0 and let ε > 0 (respectively β0 ) such that have

ε λ󸀠 −ε


n2 . The case of Hλ󸀠 is deduced immediately since Hλ󸀠 and Hλ󸀠 + β0 have the same generalized eigenvectors. Similarly, the case where λ󸀠 < 0 is deduced by multiplying by −1. We have the following theorem.

372 | 16 Reggeon field theory Theorem 16.1.1. For λ󸀠 ≠ 0, the subspace spanned by the generalized eigenvectors of Hλ󸀠 is dense in E0 .

16.2 The case of null transverse dimension (n = 1) The main purpose of this section is to study the existence of some bases of generalized eigenvectors of a class of operators arising from the Reggeon field theory. This theory was invented by the theoretical physicist Vladimir Naumovich Gribov in 1967 and constitutes one of the most important attempts to understand strong interactions, i. e., the interaction between protons and neutrons among other less stable particles. It is indeed governed by the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ = λ󸀠󸀠 G + λ󸀠 S + μH0 + iλH1 ,

(16.2.1)

where G = A∗3 A3 ,

S = A∗2 A2 ,

H0 = A∗ A,

and H1 = A∗ (A∗ + A)A,

and, in Reggeon field theory, the real parameter λ󸀠󸀠 is the magic coupling of pomerons, λ󸀠 is the quadruple coupling of pomerons, μ is the pomeron intercept, λ is the triple coupling of pomerons, and i2 = 1. It is convenient to regard the above operators as acting on the Bargmann space 2 󵄨 󵄨2 E0 = {φ : ℂ 󳨀→ ℂ entirely such that ∫ e−|z| 󵄨󵄨󵄨φ(z)󵄨󵄨󵄨 dxdy < ∞ and φ(0) = 0}.

ℂ

The Bargmann space E0 with the paring ⟨φ, ψ⟩ =

2 1 ∫ e−|z| φ(z)ψ(z)dxdy π

ℂ

is a Hilbert space and en (z) =

zn , √n!

n = 0, 1, . . . ,

is an orthonormal basis in E0 . The space E0 was used in [38] as representation space for the canonical computation rules of quantum mechanics. Since then it has occurred in many different contents, e.g., in representation theory of nilpotent Lie groups and as a class of symbols in the theory of Hankel and Toeplitz operators. In this representation, the standard annihilation and creation operators are defined by A : 𝒟(A) ⊂ E0 󳨀→ E0 , { { { { dφ φ 󳨀→ Aφ(z) = (z), { dz { { { {𝒟(A) = {φ ∈ E0 such that Aφ ∈ E0 }

16.2 The case of null transverse dimension (n = 1)

| 373

and A∗ : 𝒟(A∗ ) ⊂ E0 󳨀→ E0 , { { φ 󳨀→ A∗ φ(z) = zφ(z), { { ∗ ∗ {𝒟(A ) = {φ ∈ E0 such that A φ ∈ E0 }. 16.2.1 Annihilation and creation operators Let G, S, H0 , and H1 be the operators defined by G : 𝒟(G) ⊂ E0 󳨀→ E0 , { { φ 󳨀→ Gφ(z) = A∗3 A3 φ(z), { { {𝒟(G) = {φ ∈ E0 such that Gφ ∈ E0 }, S : 𝒟(S) ⊂ E0 󳨀→ E0 , { { φ 󳨀→ Sφ(z) = A∗2 A2 φ(z), { { {𝒟(S) = {φ ∈ E0 such that Sφ ∈ E0 }, H0 : 𝒟(H0 ) ⊂ E0 󳨀→ E0 , { { φ 󳨀→ H0 φ(z) = A∗ Aφ(z), { { {𝒟(H0 ) = {φ ∈ E0 such that H0 φ ∈ E0 }, and H1 : 𝒟(H1 ) ⊂ E0 󳨀→ E0 , { { φ 󳨀→ H1 φ(z) = A∗ (A + A∗ )Aφ(z), { { {𝒟(H1 ) = {φ ∈ E0 such that H1 φ ∈ E0 }. Remark 16.2.1. The operators A, A∗ , A2 , and A∗2 act on the Hilbert basis (ek )k by Aek = {

0 √kek−1

if k = 0, if not,

A∗ ek = √k + 1ek+1 , A2 ek = √k(k − 1)ek−2 , and A∗2 ek = √(k + 1)(k + 2)ek+2 . Hence, we have A[(ak )k∈ℕ ] = (√k + 1ak+1 )k∈ℕ , A∗ [(ak )k∈ℕ ] = (√kak−1 )k∈ℕ .

374 | 16 Reggeon field theory Now, we recall the following result from [110]. Proposition 16.2.1. (i) The set of all polynomials P0 , which vanish at zero, is dense in E0 . n (ii) The family {en := √zn! }n≥1 is an orthonormal basis of E0 . (iii) The operators G and H0 are self-adjoint and with compact resolvent. n (iv) The system {en := √zn! }n≥1 is a system of eigenvectors for H0 associated with the eigenvalues {n}n≥1 . n (v) The system {en := √zn! }n≥2 is a system of eigenvectors for G associated with the eigenvalues {n(n − 1)(n − 2)}n≥2 . Using Proposition 16.2.1 (iii), H0 is self-adjoint with compact resolvent. Moreover, {en (z) =

zn } √n! 1

∞

is a system of eigenvectors associated with the eigenvalues {n}, so the spectral decomposition of H0 is given by ∞

H0 φ := ∑ n⟨φ, en ⟩en , n=1

where n is the eigenvalue associated with the eigenvector en := introduce the operator β

β H0 .

We have β

zn . √n!

H0 : 𝒟(H0 ) ⊂ E0 󳨀→ E0 , { { { ∞ { { β { { φ 󳨀→ H0 φ = ∑ nβ ⟨φ, en ⟩en , { n=1 { { { { ∞ { 2 { { {𝒟(H0β ) = {φ ∈ E such that ∑ n2β 󵄨󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨󵄨 < ∞}. n=1 { 16.2.2 Subordinate and boundedness Lemma 16.2.1. H0,0,μ,λ is 21 -subordinate to G. Proof. Let U = A∗ A be the harmonic oscillator with domain 𝒟(U) = {φ ∈ E0 such that Uφ ∈ E0 }

and ̃ = μU, U

For β > 0, we

16.2 The case of null transverse dimension (n = 1)

| 375

̃ is a self-adjoint operator with compact resolvent where μ is the intercept pomeron, U ∗ in the Bargmann space E0 , and A A is a generator of an analytic semi-group. Let ∞

̃ = ∑ λk ⟨Φ, ek ⟩ek UΦ k=1

̃ associated be its spectral decomposition, where λk = μk is the kth eigenvalue of U with the eigenvector ek (z) =

zk . √k!

̃ β by For β > 0, we define the operator U ∞

̃ β Φ = ∑ λβ ⟨Φ, ek ⟩ek U k k=1

with ∞

̃ β ) = {Φ ∈ E0 such that ∑ λ2β 󵄨󵄨󵄨⟨Φ, ek ⟩󵄨󵄨󵄨2 < ∞}. 𝒟 (U 󵄨 k 󵄨 k=1

From both 3

𝒟(G) = 𝒟(U )

and 󵄩 󵄩 ‖GΦ‖ ≈ 󵄩󵄩󵄩U 3 Φ󵄩󵄩󵄩, it is easy to deduce that 3

𝒟(U 2 ) ⊂ 𝒟(H0,0,μ,λ )

and there exists Cμ,λ > 0 such that 󵄩 3 󵄩 ‖H0,0,μ,λ Φ‖ ≤ Cμ,λ 󵄩󵄩󵄩U 2 Φ󵄩󵄩󵄩 3

for any Φ in 𝒟(U 2 ). However, 3 3 󵄩󵄩 32 󵄩󵄩2 󵄩󵄩U Φ󵄩󵄩 = ⟨U 2 Φ, U 2 Φ⟩ = ⟨U 3 Φ, Φ⟩ 󵄩 󵄩 ≤ 󵄩󵄩󵄩U 3 Φ󵄩󵄩󵄩‖Φ‖.

Consequently, for every Φ in 𝒟(U 3 ), we get 1 󵄩 󵄩1 ‖H0,0,μ,λ Φ‖ ≤ Cμ,λ 󵄩󵄩󵄩U 3 Φ󵄩󵄩󵄩 2 ‖Φ‖ 2

and, for every Φ in 𝒟(G), we have 1

1

‖H0,0,μ,λ Φ‖ ≤ Cμ,λ ‖GΦ‖ 2 ‖Φ‖ 2 .

376 | 16 Reggeon field theory Lemma 16.2.2. Let (λk )k be the eigenvalues of G. Then, for each λ ≠ λk , k ∈ ℕ∗ , we have 1 ⟨Φ, ek ⟩ek . λ −λ k=1 k ∞

(G − λ)−1 Φ = ∑

Moreover, if Im λ ≠ 0 and if λ belongs to a ray with origin zero and of angle θ with θ ≠ 0 and θ ≠ π, we have 1 c(θ) 󵄩󵄩 −1 󵄩 = . 󵄩󵄩(G − λ) 󵄩󵄩󵄩 ≤ |Im λ| |λ| Proof. This follows immediately from the fact that G is a self-adjoint operator with compact resolvent. Lemma 16.2.3. There exists a sequence of circles C(0, rk ), k = 1, 2, . . . , ∞, with radius rk going to infinity such that 2 󵄩󵄩 󵄩−1 󵄩󵄩(G − λ)󵄩󵄩󵄩 ≤ 2 |λ| β for any β ≥ 3 and |λ| = rk . Proof. First, we remark that the eigenvalues of the operator G are given by λk = k(k − 1)(k − 2) + 1 (see Proposition 16.2.1 (v)) and that λk+1 − λk ≥ k 2 for k ≥ 2. We set rk =

λ − λk λk+1 + λk = λk + k+1 . 2 2

Then, from the equality 1 󵄩󵄩 󵄩−1 , 󵄩󵄩(G − λ)󵄩󵄩󵄩 = dist(λ, σ(G)) we obtain 1 󵄩󵄩 󵄩−1 󵄩󵄩(G − λ)󵄩󵄩󵄩 = dist(λ, σ(G)) 2 ≤ λk+1 − λk 2 ≤ 2, k

16.2 The case of null transverse dimension (n = 1)

| 377

where |λ| = rk . From λk+1 + λk 2 = k 3 − 2k 2 + k

rk =

≤ k3

≤ kβ for any β ≥ 3, we deduce that −2

k −2 ≤ rk β and − β2 2 󵄩󵄩 󵄩−1 , 󵄩󵄩(G − λ)󵄩󵄩󵄩 ≤ 2rk = 2 |λ| β

for any β ≥ 3. Lemma 16.2.4. Let 3 ≤ β < 4. Then, for each ε > 0, there exists Cε > 0 such that 2

1− β2

‖H0,0,μ,λ Φ‖ ≤ ε‖GΦ‖ β ‖Φ‖

+ Cε ‖Φ‖,

for any Φ ∈ 𝒟(G). Proof. Using the interpolation inequality (see Lemma 3.11.4), for each ε > 0, there exists Cε1 > 0 such that aα bγ−α ≤ ε1 aγ + Cε1 bγ . 1

1

Writing ‖GΦ‖ 2 ‖Φ‖ 2 as 1

1− β2

1

‖GΦ‖ 2 ‖Φ‖ 2 = ‖Φ‖

1

2

‖GΦ‖ 2 ‖Φ‖ β

− 21

and because β ≤ 4, we apply the foregoing interpolation inequality (see Lemma 3.11.4) 1

2

to ‖GΦ‖ 2 and ‖Φ‖ β

− 21

1 2

with α =

and γ = 2

1

‖GΦ‖ 2 ‖Φ‖ β

− 21

2 β

to deduce that 2

2

≤ ε1 ‖GΦ‖ β + Cε1 ‖Φ‖ β ,

for every Φ ∈ 𝒟(G), and 1

1

2

1− β2

‖GΦ‖ 2 ‖Φ‖ 2 ≤ ε1 ‖GΦ‖ β ‖Φ‖

+ Cε1 ‖Φ‖,

(16.2.2)

for every Φ ∈ 𝒟(G). Finally, from both inequality (16.2.2) and Lemma 16.2.1, we obtain 2

1− β2

‖H0,0,μ,λ Φ‖ ≤ Cμ,λ ε1 ‖GΦ‖ β ‖Φ‖

+ Cμ,λ Cε1 ‖Φ‖

378 | 16 Reggeon field theory for every Φ ∈ 𝒟(G), that is, 1− β2

2

‖H0,0,μ,λ Φ‖ ≤ ε‖GΦ‖ β ‖Φ‖

+ Cε ‖Φ‖

for every Φ ∈ 𝒟(G). Now, we state a straightforward but useful result. We include a proof for the sake of completeness. Lemma 16.2.5. (i) There exists c1 > 0 such that ∞

󵄨2 ‖H1 φ‖ ≤ c1 ( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) 3 󵄨󵄨

1 2

for all φ ∈ 𝒟(H1 ).

n=1

(ii) The following inequality holds: ∞

1 3

∞

󵄨 󵄨2 󵄨2 ‖Sφ‖ ≤ ( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) ( ∑ 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) 6 󵄨󵄨

n=1

n=1

1 6

for all φ ∈ 𝒟(S).

Proof. (i) Let φ ∈ 𝒟(H1 ). Then, according to Remark 16.2.1, we have 1 2

∞

󵄨2 󵄨 ‖H1 φ‖ = ( ∑ 󵄨󵄨󵄨n√n + 1⟨φ, en+1 ⟩ + (n − 1)√n⟨φ, en−1 ⟩󵄨󵄨󵄨 ) . n=1

Using Minkowski’s inequality (see Lemma 3.11.3), we obtain ∞

󵄨2

1 2

∞

󵄨2

󵄨 󵄨 ‖H1 φ‖ ≤ ( ∑ 󵄨󵄨󵄨n√n + 1⟨φ, en+1 ⟩󵄨󵄨󵄨 ) + ( ∑ 󵄨󵄨󵄨(n − 1)√n⟨φ, en−1 ⟩󵄨󵄨󵄨 ) n=1

n=1

1 2

∞

∞

1 2

1 2

󵄨 󵄨 󵄨2 󵄨2 ≤ ( ∑ n2 (n + 1)󵄨󵄨󵄨⟨φ, en+1 ⟩󵄨󵄨󵄨 ) + ( ∑ (n − 1)2 n󵄨󵄨󵄨⟨φ, en−1 ⟩󵄨󵄨󵄨 ) . n=1

n=1

∙ On the one hand, we have ∞

∞

n=1

n=2 ∞

󵄨 󵄨2 󵄨 󵄨2 ∑ n2 (n + 1)󵄨󵄨󵄨⟨φ, en+1 ⟩󵄨󵄨󵄨 = ∑ (n − 1)2 n󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 󵄨 󵄨2 = ∑ (n − 1)2 n󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 . n=1

Since, for all n ≥ 1, (n − 1)2 n ≤ n3 , we have ∞

∞

n=1

n=1

󵄨 󵄨2 󵄨 󵄨2 ∑ (n − 1)2 n󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ≤ ∑ n3 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 .

16.2 The case of null transverse dimension (n = 1)

Thereby, ∞

1 2

1 2

∞

󵄨2 󵄨 󵄨2 󵄨 ( ∑ (n − 1) n󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) ≤ ( ∑ n3 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) . 2

n=1

n=1

∙ On the other hand, we have ∞

∞

n=1

n=2 ∞

󵄨2 󵄨2 󵄨 󵄨 ∑ n(n − 1)2 󵄨󵄨󵄨⟨φ, en−1 ⟩󵄨󵄨󵄨 = ∑ n(n − 1)2 󵄨󵄨󵄨⟨φ, en−1 ⟩󵄨󵄨󵄨 󵄨 󵄨2 = ∑ (n + 1)n2 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 . n=1

It is easy to see that, for all n ≥ 1, (n + 1)n2 ≤ (n + 1)3 and (n + 1)3 = n3 + 1 + 3n2 + 3n ≤ 8n3 . Thereby, for all n ≥ 1, (n + 1)n2 ≤ 8n3 , so ∞

∞

n=1

n=1

󵄨 󵄨2 󵄨 󵄨2 ∑ (n + 1)n2 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ≤ 8 ∑ n3 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 .

Hence, ∞

1 2

1 2

∞

󵄨 󵄨2 󵄨2 ( ∑ (n + 1)n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) ≤ 2√2( ∑ n3 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) . 2 󵄨󵄨

n=1

n=1

Consequently, ∞

󵄨2 ‖H1 φ‖ ≤ c1 ( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) 3 󵄨󵄨

n=1

for all φ ∈ 𝒟(H1 ), where c1 = 1 + 2√2. (ii) Let φ ∈ 𝒟(S). Using Remark 16.2.1, we have ∞

󵄨 󵄨2 ‖Sφ‖ = ( ∑ 󵄨󵄨󵄨n(n − 1)⟨φ, en ⟩󵄨󵄨󵄨 ) n=1

1 2

1 2

| 379

380 | 16 Reggeon field theory

∞

󵄨2 󵄨 ≤ ( ∑ n4 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 )

1 2

n=1 ∞

1 2

󵄨2 󵄨4 󵄨 ≤ ( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 3 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 3 ) . 4 󵄨󵄨

n=1

Applying Hölder’s inequality (see Lemma 3.11.2), we obtain ∞

1 3

∞

1 6

󵄨2 󵄨 󵄨2 ‖Sφ‖ ≤ ( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) ( ∑ 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) , 6 󵄨󵄨

n=1

n=1

for all φ ∈ 𝒟(S). This achieves the proof of the lemma. Proposition 16.2.2. The operator H0,λ󸀠 ,μ,λ is G-bounded with order 32 . Proof. It follows that 3

𝒟(H02 ) ⊂ 𝒟(H0,0,μ,λ ).

Moreover, using Lemma 16.2.5 (i), we infer that there exists c > 0 such that, for all 3

φ ∈ 𝒟(H02 ), we have 3

󵄩 󵄩 ‖H0,0,μ,λ φ‖ ≤ c󵄩󵄩󵄩H02 φ󵄩󵄩󵄩.

(16.2.3)

Let φ ∈ 𝒟(H03 ). Taking account of the fact that 3

3

𝒟(H0 ) ⊂ 𝒟(H02 ),

using the Cauchy–Schwarz inequality, we obtain 3 3 󵄩󵄩 32 󵄩󵄩2 󵄩 3 󵄩 3 󵄩󵄩H0 φ󵄩󵄩 = ⟨H02 φ, H02 φ⟩ = ⟨H0 φ, φ⟩ ≤ 󵄩󵄩󵄩H0 φ󵄩󵄩󵄩‖φ‖.

(16.2.4)

Hence, it follows from both (16.2.3) and (16.2.4) that 1 󵄩 󵄩1 ‖H0,0,μ,λ φ‖ ≤ c󵄩󵄩󵄩H03 φ󵄩󵄩󵄩 2 ‖φ‖ 2 .

Combining the last inequality with the fact that 0 ∈ ρ(H0 ), we get 1 3 󵄩 󵄩1 󵄩 󵄩1 ‖H0,0,μ,λ φ‖ ≤ c󵄩󵄩󵄩(H0−1 ) H03 φ󵄩󵄩󵄩 6 󵄩󵄩󵄩H03 φ󵄩󵄩󵄩 2 ‖φ‖ 3 1 3󵄩 1 󵄩 󵄩 󵄩2 ≤ c󵄩󵄩󵄩(H0−1 ) 󵄩󵄩󵄩 6 󵄩󵄩󵄩H03 φ󵄩󵄩󵄩 3 ‖φ‖ 3 .

(16.2.5)

Thus, H0,0,μ,λ is 32 -subordinate to H03 . Furthermore, Lemma 16.2.5 (ii) allows us to write 1 󵄩 󵄩2 ‖Sφ‖ ≤ 󵄩󵄩󵄩H03 φ󵄩󵄩󵄩 3 ‖φ‖ 3 .

(16.2.6)

16.2 The case of null transverse dimension (n = 1)

| 381

Thus, S is also 32 -subordinate to H03 . Now, by Proposition 16.2.1 (v), we have 󵄩󵄩 3 󵄩󵄩 󵄩󵄩H0 φ󵄩󵄩 ≤ d0 ‖Gφ‖ + d1 ‖φ‖

(16.2.7)

for some positive constants d0 and d1 . Finally, equations (16.2.5)–(16.2.7) guarantee the existence of positive constants a, b > 0 satisfying 2

2

‖H0,λ󸀠 ,μ,λ φ‖ ≤ b‖Gφ‖ 3 ‖φ‖1− 3 + a‖φ‖. Lemma 16.2.6. For λ󸀠󸀠 ≠ 0, the resolvent of Hλ󸀠󸀠 ,λ󸀠 ,μ,λ belongs to the Carleman class Cp for any p > 31 . Proof. Because the eigenvalues of the self-adjoint operator G are λn = n(n − 1)(n − 2), the resolvent of G belongs to the Carleman class Cp for any p > 31 . Now, as (λ󸀠󸀠 G + H0,0,μ,λ − σ)

−1

= (λ󸀠󸀠 G − σ)

−1

∞

−1 k

∑ (−H0,0,μ,λ (λ󸀠󸀠 G − σ) ) ,

k=0

by applying the minimax theorem, it follows that the resolvent operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ belongs to the Carleman class Cp for any p > 31 . Lemma 16.2.7. For λ󸀠󸀠 > 0 and μ > 0, there exist β0 > 0 and δ > 0 such that the values of the quadratic form ⟨(Hλ󸀠󸀠 ,λ󸀠 ,μ,λ + β0 )φ, φ⟩ lie in the sector of the complex z-plane. We have S = {z ∈ ℂ such that −

π π + δ ≤ arg(z) ≤ − δ}, 2 2

for any φ ∈ 𝒟(Hλ󸀠󸀠 ,λ󸀠 ,μ,λ ). Proof. We remark that 󵄨󵄨 󵄨 󵄨 󵄨 ∗ ∗ ∗ 󵄨󵄨⟨iλA (A + A )Aφ, φ⟩󵄨󵄨󵄨 ≤ |λ|󵄨󵄨󵄨⟨(A + A )Aφ, Aφ⟩󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ |λ|[󵄨󵄨󵄨⟨A2 φ, Aφ⟩󵄨󵄨󵄨 + 󵄨󵄨󵄨⟨Aφ, A2 φ⟩󵄨󵄨󵄨] 󵄩 󵄩 ≤ 2|λ|󵄩󵄩󵄩A2 φ󵄩󵄩󵄩‖Aφ‖. Using the estimate ‖φ‖ ≤ ‖Aφ‖, we get 󵄨󵄨 󵄨 󵄩 3 󵄩 ∗ ∗ 2 󵄨󵄨⟨iλA (A + A )Aφ, φ⟩󵄨󵄨󵄨 ≤ 2|λ|󵄩󵄩󵄩A φ󵄩󵄩󵄩‖Aφ‖ + β0 ‖φ‖

382 | 16 Reggeon field theory 󵄩2 󵄩 ≤ |λ|[󵄩󵄩󵄩A3 φ󵄩󵄩󵄩 + ‖Aφ‖2 ] + β0 ‖φ‖2 ≤ C⟨(λ󸀠󸀠 G + μA∗ A + β0 )φ, φ⟩. The last inequality implies that tan(

|Im⟨Hλ󸀠󸀠 ,λ󸀠 ,μ,λ φ, φ⟩|

|Re⟨(Hλ󸀠󸀠 ,λ󸀠 ,μ,λ + β0 )φ, φ⟩|

)

is bounded. Hence, there exists δ > 0 such that −

π π + δ ≤ arg⟨(Hλ󸀠󸀠 ,λ󸀠 ,μ,λ + β0 )φ, φ⟩ ≤ − δ 2 2

for any φ ∈ 𝒟(Hλ󸀠󸀠 ,λ󸀠 ,μ,λ ). 16.2.3 Unconditional basis in Bargmann space In this section, we prove that the system of generalized eigenvectors of the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ forms an unconditional basis in the Bargmann space E0 . Theorem 16.2.1. If λ󸀠󸀠 ≠ 0 and λ󸀠 ≠ 0, then the spectrum of the operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ is discrete and, for any Φ∈ 𝒟(Hλ2󸀠󸀠 ,λ󸀠 ,μ,λ ), there exists a subsequence of partial sums of series ∞

∑ Pn Φ

n=1

converging to Φ in E0 . Proof. Let λ ∈ ρ(G). From the equality G(G − λ)−1 = I + λ(G − λ)−1 and Lemma 16.2.3, we obtain 1− 2 󵄩󵄩 −1 󵄩 󵄩󵄩G(G − λ) 󵄩󵄩󵄩 ≤ 1 + 2|λ| β

for any β ≥ 3 and |λ| = rk → ∞. Therefore, for β ≥ 3 and |λ| = rk sufficiently large, we have 1− 2 󵄩󵄩 −1 󵄩 󵄩󵄩G(G − λ) 󵄩󵄩󵄩 ≤ 3|λ| β .

that

(16.2.8)

Now, it follows from Lemma 16.2.4 that, for each ε > 0, there exists Cε > 0 such

2 2 󵄩󵄩 󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩 −1 󵄩1− −1 󵄩 󵄩󵄩H0,0,μ,λ (G − λ) Φ󵄩󵄩󵄩 ≤ ε󵄩󵄩󵄩G(G − λ) Φ󵄩󵄩󵄩 β 󵄩󵄩󵄩(G − λ) Φ󵄩󵄩󵄩 β + Cε 󵄩󵄩󵄩(G − λ) Φ󵄩󵄩󵄩.

(16.2.9)

16.2 The case of null transverse dimension (n = 1)

that

| 383

From equations (16.2.8) and (16.2.9) and the estimate of Lemma 16.2.3, it follows −2 󵄩󵄩 −1 󵄩 󵄩󵄩H0,0,μ,λ (G − λ) Φ󵄩󵄩󵄩 ≤ 3ε‖Φ‖ + 2Cε |λ| β ‖Φ‖ − β2

= (3ε + 2Cε |λ| Hence, for ε
0.

Theorem 16.2.3. For λ󸀠󸀠 > 0 and μ > 0, we have the following statements: (i) the system of generalized eigenvectors of the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ is complete in Bargmann space E0 ; and (ii) the corresponding Fourier series with respect to the system of generalized eigenvectors of the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ is summable by Abel’s method of unit order and the series ∞

∑e

−tHλ󸀠󸀠 ,λ󸀠 ,μ,λ

n=1

Pn

converges strongly to e−tHλ󸀠󸀠 ,λ󸀠 ,μ,λ for any t > 0. Proof. By means of Lemmas 16.2.6 and 16.2.7, the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ satisfies the assumptions of Lidskii’s theorem (see Theorem 8.4.1). Therefore, the former theorem follows directly from the Lidskii theorem. In particular, if we set u(t) = e−tHλ󸀠󸀠 ,λ󸀠 ,μ,λ Φ, then u(t) is the solution of the following Cauchy problem: {

u󸀠 (t) = −Hλ󸀠󸀠 ,λ󸀠 ,μ,λ u(t), u(0) = Φ.

Also, the series ∞

∑e

−tHλ󸀠󸀠 ,λ󸀠 ,μ,λ

n=1

converges strongly to e−tHλ󸀠󸀠 ,λ󸀠 ,μ,λ for any t > 0.

Pn

16.2 The case of null transverse dimension (n = 1)

|

385

From the previous theorem and Krein’s theorem [142, Chap. I, Theorem x. 5.4], one can define a square root √Hλ󸀠󸀠 ,λ󸀠 ,μ,λ such that −√Hλ󸀠󸀠 ,λ󸀠 ,μ,λ generates an analytic semi1

group e

−tH 2󸀠󸀠

λ ,λ󸀠 ,μ,λ

. The function 1

u(t) = e

−tH 2󸀠󸀠

λ ,λ󸀠 ,μ,λ

Φ

is a solution of the abstract elliptic problem u󸀠󸀠 (t) = Hλ󸀠󸀠 ,λ󸀠 ,μ,λ u(t), t > 0, with the boundary condition u(0) = Φ. Because the operator Hλ−1󸀠󸀠 ,λ󸀠 ,μ,λ belongs to the Carleman class Cp for all p >

follows from Corollary 3.7.1 that p>

2 . 3

√Hλ−1󸀠󸀠 ,λ󸀠 ,μ,λ

1 , 3

it

belongs to the Carleman class Cp for any

Thus, by Lidskii’s theorem, we get the following result.

Theorem 16.2.4. The series ∞

∑e

1

−tH 2󸀠󸀠

λ ,λ󸀠 ,μ,λ

n=1

Pn

1

converges strongly to e

−tH 2󸀠󸀠

λ ,λ󸀠 ,μ,λ

for any t > 0. In particular, if we set 1

u(t) = e

−tH 2󸀠󸀠

λ ,λ󸀠 ,μ,λ

Φ

3

with Φ ∈ 𝒟(G 2 ), then u(t) is the solution of the following second-order problem: {

u󸀠󸀠 (t) = Hλ󸀠󸀠 ,λ󸀠 ,μ,λ u(t), u(0) = Φ.

Open problem. For λ󸀠󸀠 = 0 and λ󸀠 ≠ 0, the resolvent of H0,λ󸀠 ,μ,λ belongs to the Carleman class Cp for any p > 21 and we can apply the Lidskii theorem to give a generalized diagonalization of the semi-group e−tHλ󸀠󸀠 ,λ󸀠 ,μ,λ for any t > 0. However, the problem of 1

diagonalization of e

−tH 2󸀠󸀠

λ ,λ󸀠 ,μ,λ

is open.

Open problem. For λ󸀠󸀠 = 0, λ󸀠 = 0, and μ ≠ 0, the resolvent of H0,0,μ,λ belongs to the Carleman class C1+ε for any ε > 0. However, the completeness of the system of generalized eigenvectors of H0,0,μ,λ and the diagonalization of the semi-group e−tH0,0,μ,λ are open problems.

386 | 16 Reggeon field theory Open problem. For λ󸀠󸀠 = 0, λ󸀠 = 0, and μ = 0, in [111], the boundary conditions at infinity are used in a description of all maximal dissipative extensions in Bargmann space of the minimal operator A∗ (A + A∗ )A. The characteristic functions of the dissipative extensions are computed and completeness theorems are obtained for the system of generalized eigenvectors.

16.2.5 Riesz basis of finite-dimensional Hλ󸀠󸀠 ,λ󸀠 ,μ,λ -invariant subspaces Consider the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ whose expression is given by equation (16.2.1) and with minimal domain 𝒟min (Hλ󸀠󸀠 ,λ󸀠 ,μ,λ ) = {φ ∈ E0 , ∃pn ∈ P0 and ψ ∈ E0 such that pn → φ and Hλ󸀠󸀠 ,λ󸀠 ,μ,λ pn → ψ},

where P0 is the set of all polynomials which vanish at zero. We have the following result. Theorem 16.2.5. For λ󸀠󸀠 ≠ 0, the Gribov operator Hλ󸀠󸀠 ,λ󸀠 ,μ,λ admits a Riesz basis of finitedimensional Hλ󸀠󸀠 ,λ󸀠 ,μ,λ -invariant subspaces in E0 . Proof. Let λn be the nth eigenvalue of G and let n(𝔹(0, rn ), G) be the sum of multiplicities of the eigenvalues of the operator G which are included in the circle 𝒞 (O, rn ) with radius rn =

λn + λn+1 . 2

Using Proposition 16.2.1 (v), λn = n(n − 1)(n − 2). Hence, n(𝔹(0, rn ), G) 1− 32

rn

1

=

23 n

2

(n(n − 1)(2n − 1))1− 3

.

(16.2.11)

Equation (16.2.11) shows that lim inf rn →∞

n(𝔹(0, rn ), G) 1− 32

rn

< ∞.

Thus, combining Theorem 4.15.2, Proposition 16.2.1 (iii), and Proposition 16.2.2, the result follows.

16.3 Schauder basis of Gribov operator in the Bargmann space

| 387

16.3 Schauder basis of Gribov operator in the Bargmann space Let us consider the following Gribov operator: 3

3u2

(A∗ A) + εA∗ (A + A∗ )A + ε2 (A∗ A)

3uk

+ ⋅ ⋅ ⋅ + εk (A∗ A)

+ ⋅⋅⋅,

(16.3.1)

where ε ∈ ℂ and (uk )k∈ℕ is a strictly decreasing sequence with strictly positive terms such that u0 = 1 and u1 = 21 . Let T0 = H03 , so T0 is defined by T0 : 𝒟(T0 ) ⊂ E0 󳨀→ E0 , { { { ∞ { { { { φ 󳨀→ T0 φ = ∑ n3 ⟨φ, en ⟩en , { n=1 { { { { ∞ { { 󵄨2 6󵄨 { {𝒟(T0 ) = {φ ∈ E0 such that ∑ n 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 < ∞}. n=1 { We see the following. Proposition 16.3.1. We have the following assertions: (i) T0 is a closed linear operator with dense domain; (ii) the resolvent set of T0 is not empty (in fact, 0 ∈ ρ(T0 )); (iii) T0 is a self-adjoint operator with compact resolvent; (iv) the eigenvalues of T0 are simple and isolated; (v) the eigenvectors of T0 form a Riesz basis in E0 ; n 3 (vi) {en (z) = √zn! }∞ 1 is a system of eigenvectors associated with the eigenvalues {n }n≥1 of T0 . Let n ∈ ℕ∗ and let λn be the nth eigenvalue of T0 . If rn =

λn+1 −λn , 2

then

{z ∈ ℂ, |z − λn | ≤ rn } ∩ σ(T0 ) = {λn }. Let 𝒞n = 𝒞 (λn , rn ) be the closed circle with center λn and radius rn and let z ∈ 𝒞n . Since T0 is with compact resolvent, we have 󵄩 󵄩 ‖Rz ‖ = 󵄩󵄩󵄩(T0 − zI)−1 󵄩󵄩󵄩 1 = dist(z, σ(T0 )) 1 = . rn Thus, Mn = max ‖Rz ‖ z∈𝒞n

= max n≥1

1 . rn

388 | 16 Reggeon field theory We have ωn rn Mn = ωn rn max ≤ ωn .

n≥1

1 rn

Thus, (ωn rn Mn )2 ≤ ωn . Since ∞

∞

n=1

n=1 ∞

∑ ω2n = ∑ (

2

1 + α) √n!

4 < ∞, n! n=1

2 2 >n .

rn =

Thus, 1 1 < . rn n2 Hence, β

1−β

λ a 1 + b(2 + n ) ( ) rn rn rn

β

Then αn ≤ max( n≥1

a + 3b ). n2−3β

Since β ∈ [ 21 , 32 ], we have αn < d, where d = a + 3b, so q + αn + ω2n rn Mn αn ≤ 1 + d + ≤ 1 + 2d.

d n!

Consequently, sup(q + αn + ω2n rn Mn ) < 1 + 2d < ∞. n≥1

1−β

n3 1 a + b(2 + 2 ) ( 2 ) 2 n n n a ≤ 2 + b(3n)β n2(β−1) n a 3b ≤ 2 + 2−3β n n a + 3b ≤ 2−3β . n ≤

| 389

390 | 16 Reggeon field theory u

For the operators (T0 k )k≥0 , let u

u

T0 k : 𝒟(T0 k ) ⊂ E0 󳨀→ E0 , { { { ∞ { { u { { φ 󳨀→ T0 k φ = ∑ n3uk ⟨φ, en ⟩en , { n=1 { { { { ∞ { { u 󵄨2 6u 󵄨 { {𝒟(T0 k ) = {φ ∈ E0 such that ∑ n k 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 < ∞}. n=1 { We easily see that, for all k ≥ 0, u

u

𝒟(T0 k ) ⊂ 𝒟(T0 k+1 ),

so u

u

⋂ 𝒟(T0 k ) = 𝒟(T0 2 ).

k≥2 u

Let 𝒟 = 𝒟(T0 2 )∩𝒟(H1 ) and T1 (respectively (Tk )k≥2 ) be the restriction of H1 (respectively u T0 k ) to 𝒟. Hence, the operators (Tk )k≥1 have the same domain 𝒟 and we have 𝒟(T0 ) ⊂ 𝒟. Let us denote 3

3u2

T(ε) := (A∗ A) + εA∗ (A + A∗ )A + ε2 (A∗ A)

3uk

+ ⋅ ⋅ ⋅ + εk (A∗ A)

+ ⋅⋅⋅,

where ε ∈ ℂ and (uk )k∈ℕ is a strictly decreasing sequence with strictly positive terms such that u0 = 1 and u1 = 21 . Theorem 16.3.1. There exist constants a, b, q > 0, and β ∈ [ 21 , 1] such that ‖Tk φ‖ ≤ qk−1 (a‖φ‖ + b‖T0 φ‖β ‖φ‖1−β ) for all k ≥ 1 and for all φ ∈ 𝒟(T0 ). Proof. First, we claim that there exists c > 0 such that ‖Tk φ‖ ≤ c‖T0 φ‖uk ‖φ‖1−uk for all k ≥ 1 and for all φ ∈ 𝒟(T0 ). ∙ For k = 1, let φ ∈ 𝒟(T0 ). Due to Remark 16.2.1, we have ∞

T1 φ = ∑ (n√n + 1⟨φ, en+1 ⟩ + (n − 1)√n⟨φ, en−1 ⟩)en . n=1

Thus, ∞

1 2

󵄨 󵄨2 ‖T1 φ‖ = ( ∑ 󵄨󵄨󵄨n√n + 1⟨φ, en+1 ⟩ + (n − 1)√n⟨φ, en−1 ⟩󵄨󵄨󵄨 ) . n=1

16.3 Schauder basis of Gribov operator in the Bargmann space

| 391

Using Minkowski’s inequality (see Lemma 3.11.3), we obtain 1 2

∞

∞

󵄨 󵄨2 󵄨 󵄨2 ‖T1 φ‖ ≤ ( ∑ 󵄨󵄨󵄨n√n + 1⟨φ, en+1 ⟩󵄨󵄨󵄨 ) + ( ∑ 󵄨󵄨󵄨(n − 1)√n⟨φ, en−1 ⟩󵄨󵄨󵄨 ) n=1 ∞

n=1

1 2

1 2

1 2

∞

󵄨2 󵄨2 󵄨 󵄨 ≤ ( ∑ n (n + 1)󵄨󵄨󵄨⟨φ, en+1 ⟩󵄨󵄨󵄨 ) + ( ∑ (n − 1)2 n󵄨󵄨󵄨⟨φ, en−1 ⟩󵄨󵄨󵄨 ) . 2

n=1

n=1

Consequently, ∞

1 2

󵄨2 ‖T1 φ‖ ≤ c( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) , 3 󵄨󵄨

n=1

for all φ ∈ 𝒟(T0 ), where c = 1 + 2√2, so, for all φ ∈ 𝒟(T0 ), we have 󵄩 1 󵄩2 ‖T1 φ‖2 ≤ c2 󵄩󵄩󵄩T02 φ󵄩󵄩󵄩 .

(16.3.2)

On the other hand, we have 1 1 󵄩󵄩 21 󵄩󵄩2 󵄩󵄩T0 φ󵄩󵄩 = ⟨T02 φ, T02 φ⟩ 1

1

= ⟨T02 T02 φ, φ⟩ = ⟨T0 φ, φ⟩

≤ ‖T0 φ‖‖φ‖. Thus, equation (16.3.2) implies, for all φ ∈ 𝒟(T0 ), 1

1

‖T1 φ‖ ≤ c‖T0 φ‖ 2 ‖φ‖ 2 .

(16.3.3)

∙ Now, let k ≥ 2 and φ ∈ 𝒟(T0 ). We have ∞

Tk φ = ∑ n3uk ⟨φ, en ⟩en , n=1

so ∞

󵄨 󵄨2 ‖Tk φ‖2 = ∑ n6uk 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 n=1 ∞

󵄨 󵄨2u 󵄨 󵄨2(1−uk ) = ∑ n6uk 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 k 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 . n=1

Using Hölder’s inequality, we obtain ∞

uk

∞

1−uk

󵄨2 󵄨 󵄨2 ‖Tk φ‖ ≤ ( ∑ n 󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) ( ∑ 󵄨󵄨󵄨⟨φ, en ⟩󵄨󵄨󵄨 ) 2

n=1

6 󵄨󵄨

n=1

392 | 16 Reggeon field theory ≤ ‖T0 φ‖2uk ‖φ‖2(1−uk ) . Thus, for all φ ∈ 𝒟(T0 ), ‖Tk φ‖ ≤ ‖T0 φ‖uk ‖φ‖1−uk .

(16.3.4)

Since c > 1, equations (16.3.3) and (16.3.4) imply that, for all k ≥ 1, ‖Tk φ‖ ≤ c‖T0 φ‖uk ‖φ‖1−uk ,

for all φ ∈ 𝒟(T0 ).

(16.3.5)

This ends the proof of the claim. Second, let β ∈ [u1 , u0 ] and φ ∈ 𝒟(T0 ). Writing ‖T0 φ‖uk ‖φ‖1−uk as ‖T0 φ‖uk ‖φ‖1−uk = ‖φ‖1−β ‖T0 φ‖uk ‖φ‖β−uk and keeping in mind that, for all k ≥ 1, 0 < uk < β, we apply the foregoing interpolation inequality (see Lemma 3.11.4) to ‖T0 φ‖uk and ‖φ‖β−uk with θ = β and δ = uk , to deduce that ‖T0 φ‖uk ‖φ‖β−uk ≤ ‖T0 φ‖β + ‖φ‖β and ‖T0 φ‖uk ‖φ‖1−uk ≤ ‖φ‖ + ‖T0 φ‖β ‖φ‖1−β . Hence, equation (16.3.5) implies that, for all k ≥ 1, ‖Tk φ‖ ≤ c‖φ‖ + c‖T0 φ‖β ‖φ‖1−β for all φ ∈ 𝒟(T0 ), so it suffices to choose a = b = c and q = 1. This achieves the proof of the theorem. The objective of this subsection is formulated as follows. ∞ Theorem 16.3.2. There exists a sequence of complex (εn )∞ n=1 and a sequence (φn (εn ))n=1 of eigenvectors of T(εn ), given in (16.3.1), which can be developed as an entire series of εn , such that the system (φn (εn ))∞ n=1 forms a Schauder basis in E0 .

Proof. We have already proved that the hypotheses (H1) and (H2) are fulfilled. Moreover, the eigenvectors (en )∞ n=1 of T0 form an orthonormal basis in E0 , so the result follows from Theorem 12.2.1, which ends the proof of the theorem.

Bibliography [1]

[2]

[3]

[4]

[5]

[6]

[7] [8] [9]

[10] [11] [12] [13]

[14]

[15]

[16] [17] [18]

B. Abdelmoumen, A. Dehici, A. Jeribi and M. Mnif, Some new properties of Fredholm theory, Schechter essential spectrum, and application to transport theory, J. Inequal. Appl., Art. ID 852676, 1–14 (2008). B. Abdelmoumen, A. Jeribi and M. Mnif, Invariance of the Schechter essential spectrum under polynomially compact operators perturabation, Extracta Math., 26 (1), 61–73 (2011). B. Abdelmoumen, A. Jeribi and M. Mnif, On graph measures in Banach spaces and description of essential spectra of multidimensional transport equation, Acta Math. Sci. Ser. B Engl. Ed., 32 (5), 2050–2064 (2012). B. Abdelmoumen, A. Jeribi and M. Mnif, Measure of weak noncompactness, some new properties in Fredholm theory, characterization of the Schechter essential spectrum and application to transport operators, Ric. Mat., 61, 321–340 (2012). F. Abdmouleh and A. Jeribi, Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators, J. Math. Anal. Appl., 364, 414–423 (2010). F. Abdmouleh and A. Jeribi, Gustafson, Weidmann, Kato, Wolf, Schechter, Browder, Rakoc̆ević and Schmoger essential spectra of the sum of two bounded operators and application to a transport operator, Math. Nachr., 284 (2–3), 166–176 (2011). F. Abdmouleh, S. Charfi and A. Jeribi, On a characterization of the essential spectra of the sum and the product of two operators, J. Math. Anal. Appl., 386 (1), 83–90 (2012). F. Abdmouleh, A. Ammar and A. Jeribi, Stability of the S-essential spectra on a Banach space, Math. Slovaca, 63 (2), 299–320 (2013). F. Abdmouleh, A. Ammar and A. Jeribi, A characterization of the pseudo-Browder essential spectra of linear operators and application to a transport equation, J. Comput. Theor. Transp., 44 (3), 141–153 (2015). Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Grad. Stud. Math., vol. 50 (2002). M. Ahues, A. Largillier and B. V. Limaye, Spectral computations for bounded operators, Appl. Math. (Boca Raton), vol. 18, Chapman Hall/CRC, Boca Raton, FL (2001). P. Aiena, Fredholm and local spectral theory, with applications to multipliers, Kluwer Academic Publishers, Dordrecht (2004). M. T. Aimar, A. Intissar and J. M. Paoli, Densité des vecteurs propres généralisés d’une classe d’opérateurs compacts non auto-adjoints et applications, Comm. Math. Phys., 156, 169–177 (1993). M. T. Aimar, A. Intissar and J. M. Paoli, Critères de complétude des vecteurs propres généralisés d’une classe d’opérateurs non auto-adjoints compacts ou à résolvante compacte et applications, Publ. Res. Inst. Math. Sci., 32 (2) (1996). M. T. Aimar, A. Intissar and A. Jeribi, On an unconditional basis of generalized eigenvectors of the nonself-adjoint Gribov operator in Bargmann space, J. Math. Anal. Appl., 231, 588–602 (1999). A. Ammar, Some properties of the Wolf and Weyl essential spectra of a sequence of linear operators ν-convergent, Indag. Math., 28 (2), 424–435 (2017). A. Ammar and A. Jeribi, A characterization of the essential pseudospectra on a Banach space, Arab. J. Math., 2 (2), 139–145 (2013). A. Ammar and A. Jeribi, A characterization of the essential pseudospectra and application to a transport equation, Extracta Math., 28, 95–112 (2013).

https://doi.org/10.1515/9783110493863-017

394 | Bibliography

[19] [20] [21]

[22] [23] [24] [25] [26] [27] [28]

[29] [30] [31] [32] [33] [34] [35]

[36] [37] [38] [39] [40] [41]

A. Ammar and A. Jeribi, Measures of noncompactness and essential pseudo-spectra on Banach Space, Math. Methods Appl. Sci., 37 (3), 447–452 (2014). A. Ammar and A. Jeribi, The Weyl essential spectrum of a sequence of linear operators in Banach spaces, Indag. Math., 27, 282–295 (2016). A. Ammar, A. Jeribi and N. Moalla, On a characterization of the essential spectra of a 3 × 3 operator matrix and application to three-group transport operators, Ann. Funct. Anal. 4 (2), 153–170 (2013) (electronic only). A. Ammar, B. Boukettaya and A. Jeribi, Stability of the S-left and S-right essential spectra of a linear operator, Acta Math. Sci. Ser. B Engl. Ed., 34 (6), 1922–1934 (2014). A. Ammar, M. Z. Dhahri and A. Jeribi, Stability of essential approximate point spectrum and essential defect spectrum of linear operator, Filomat, 29 (9), 1983–1994 (2015). A. Ammar, M. Z. Dhahri and A. Jeribi, Some properties of the M-essential spectra of closed linear operator on a Banach space, Funct. Anal. Approx. Comput., 7 (1), 15–28 (2015). A. Ammar, M. Z. Dhahri and A. Jeribi, A characterization of S-essential spectrum by mean of measure of non-strict-singularity and application, Azerb. J. Math., 5 (1), 2218–6816 (2015). A. Ammar, M. Z. Dhahri and A. Jeribi, Some properties of upper triangular 3 × 3-block matrices of linear relations, Boll. Unione Mat. Ital., 8 (3), 189–204 (2015). A. Ammar, B. Boukettaya and A. Jeribi, A note on the essential pseudospectra and application, Linear Multilinear Algebra, 64 (8), 1474–1483 (2016). A. Ammar, F. Bouzayeni and A. Jeribi, Perturbation of unbounded linear operators by γ-relative boundedness, Ricerche Mat. (2017), DOI https://doi.org/10.1007/s11587-017-0341-0, in press. A. Ammar, N. Djaidja and A. Jeribi, The essential spectrum of a sequence of linear operators in Banach spaces, Int. J. Anal. Appl., 1–7 (2017). A. Ammar, A. Jeribi and K. Mahfoudhi, A characterization of the essential approximation pseudospectrum on a Banach space, Filomat, 31 (11), 3599–3610 (2017). A. Ammar, A. Jeribi and K. Mahfoudhi, A characterization of the condition pseudospectrum, preprint (2017). A. Ammar, A. Jeribi and B. Saadaoui, Frobenius–Schur factorization for multivalued 2 × 2 matrices linear operator, Mediterr. J. Math. 14 (1), Paper No. 29 (2017), 29 pp. S. I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal., 128, 374–383 (1995). M. S. Asgari, New characterizations of fusion frames (frames of subspaces), Proc. Indian Acad. Sci. Math. Sci., 119 (3), 369–382 (2009). S. A. Avdonin and S. A. Ivanov, Families of exponentials: the method of moments in controllability problems for distributed systems, Cambridge University Press, Cambridge, UK (1995). H. Baloudi and A. Jeribi, Left–right Fredholm and Weyl spectra of the sum of two bounded operators and application, Mediterr. J. Math., 11, 939–953 (2014). H. Baloudi and A. Jeribi, Holomorphically Weyl-decomposably regular, Funct. Anal. Approx. Comput., 8 (2), 13–22 (2016) (in English). V. Bargmann, On a Hilbert space of analytic functions and associated integral transform, Comm. Pure Appl. Math., 14, 187–214 (1961). B. A. Barnes, Common operator properties of the linear operators RS and SR, Proc. Amer. Math. Soc., 126, 1055–1061 (1998). B. Barnes, Restrictions of bounded linear operators: closed range, Proc. Amer. Math. Soc., 135, 1735–1740 (2007). F. Bayart and É. Matheron, Dynamics of linear operators, Cambridge University Press, Cambridge (2009).

Bibliography | 395

[42]

[43] [44]

[45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

[55] [56]

[57] [58]

[59]

[60]

[61] [62] [63] [64]

N. Ben Ali and A. Jeribi, On the Riesz basis of a family of analytic operators in the sens of Kato and application to the problem of radiation of a vibriting structure in a light fluid, J. Math. Anal. Appl., 320 (1), 78–94 (2006). N. Ben Ali, A. Jeribi and N. Moalla, Essential spectra of some matrix operators, Math. Nachr., 283 (9), 1245–1256 (2010). A. Ben Amar, A. Jeribi and B. Krichen, Essential spectra of a 3 × 3 operator matrix and application to three-group transport equation, Integral Equations Operator Theory, 68, 1–21 (2010). A. Ben Amar, A. Jeribi and M. Mnif, Some applications of the regularity and irreducibility on transport theory, Acta Appl. Math., 110, 431–448 (2010). A. Ben Amar, A. Jeribi and M. Mnif, Some results on Fredholm and semi-Fredholm perturbations, Arab. J. Math., 3 (3), 313–323 (2014). A. Ben Amar, A. Jeribi, B. Krichen and El. H. Zerouali, Gamma-hypercyclic set of a bounded linear operator, preprint (2015). M. Benharrat, A. Ammar, A. Jeribi and B. Messirdi, On the Kato, semi-regular and essentially semi-regular spectra, Funct. Anal. Approx. Comput., 6 (2), 9–22 (2014). R. Ph. Boas, Entire functions, 2nd edn, Academic Press, New-York (1968). P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J., 44, 345–353 (1997). H. Brezis, Analyse fonctionnelle théorie et applications, Masson, Paris (1983). J. Cahill, P. G. Cassaza and S. Li, Non-orthogonal fusion frames and the sparsity of fusion frame operators, J. Fourier Anal. Appl., 18, 278–308 (2012). S. R. Caradus, W. E. Pfaffenberger and B. Yood, Calkin algebras and algebras of operators on Banach spaces, Marcel Dekker, Inc, New-York (1974). P. G. Cassaza and G. Kutyniok, Frames of subspaces, in: Wavelets, frames and operator theory (College Park, MD, 2003) Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 87–113 (2004). W. Chaker, A. Jeribi and B. Krichen, Demicompact linear operators, essential spectrum and some perturbation results, Math. Nachr., 288 (13), 1476–1486 (2015). S. Charfi and A. Jeribi, On a characterization of the essential spectra of some matrix operators and applications to two-group transport operators, Math. Z., 262 (4), 775–794 (2009). S. Charfi, A. Jeribi and I. Walha, Essential spectra, matrix operator and applications, Acta Appl. Math., 111 (3), 319–337 (2010). S. Charfi, A. Jeribi and I. Walha, Riesz basis property of families of nonharmonic exponentials and application to a problem of a radiation of a vibrating structure in a light fluid, Numer. Funct. Anal. Optim., 32 (54), 370–382 (2011). S. Charfi, A. Damergi and A. Jeribi, On a Riesz basis of finite-dimensional invariant subspaces and application to Gribov operator in Bargmann space, Linear Multilinear Algebra, 61 (11), 1577–1591 (2013). S. Charfi, A. Jeribi and N. Moalla, Time asymptotic behavior of the solution of an abstract Cauchy problem given by a one-velocity transport operator with Maxwell boundary condition, Collect. Math., 64, 97–109 (2013). S. Charfi, A. Jeribi and R. Moalla, Essential spectra of operator matrices and applications, Methods Appl. Sci., 37 (4), 597–608 (2014). O. Christensen, Frames and the projection method, Appl. Comput. Harmon. Anal., 1, 50–53 (1993). O. Christensen, Frame perturbations, Proc. Amer. Math. Soc., 123, 1217–1220 (1995). O. Christensen, Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods, J. Math. Anal. Appl., 199, 256–270 (1996).

396 | Bibliography

[65] [66] [67] [68] [69] [70] [71] [72]

[73] [74] [75] [76] [77] [78] [79] [80] [81]

[82]

[83] [84] [85] [86]

[87] [88]

O. Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. (N. S.), 38, 273–291 (2001) (electronic). O. Christensen, An introduction to frames and Riesz bases, Birkhäuser, Boston (2003). J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity, Integral Equations Operator Theory, 2, 174–198 (1979). M. Damak and A. Jeribi, On the essential spectra of some matrix operators and applications, Electron. J. Differential Equations, 11, 1–16 (2007). I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 24, 1271–1283 (1986). E. B. Davies, One parameter semigroups, Academic Press, San Diego (1980). E. B. Davies, Linear operators and their spectra, Cambridge University Press, Cambridge (2007). J. Diestel, A survey of results related to Dunford–Pettis property, in: Conf. on integration, topology and geometry in linear spaces, Contemp. Math., vol. 2, Amer. Math. Soc., Providence, RI, 15–60 (1980). J. Ding, On the perturbation of the reduced minimum modulus of bounded linear operators, Appl. Math. Comput., 140, 69–75 (2003). R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72, 341–366 (1952). N. Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc., 47, 323–392 (1940). N. Dunford and J. T. Schwartz, Linear operators, I, Pure and Appl. Math., vol. 7, Interscience, New-York (1958). N. Dunford and J. T. Schwartz, Linear operators, II, Pure and Appl. Math., vol. 7, Interscience, New-York (1963). N. Dunford and J. T. Schwartz, Linear operators, Part III, Wiley–Interscience, New-York (1971). A. Elleuch and A. Jeribi, On a characterization of the structured Wolf, Schechter and Browder essential pseudospectra, Indag. Math., 27, 212–224 (2016). A. Elleuch and A. Jeribi, New description of the structured essential pseudospectra, Indag. Math., 27, 368–382 (2016). H. Ellouz, I. Feki and A. Jeribi, On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-self-adjoint problem deduced from a perturbation method, J. Math. Phys., 54, 112101 (2013). H. Ellouz, I. Feki and A. Jeribi, Frames of eigenvectors related to an analytic operator and an application to a problem of radiation of a vibrating structure in a light fluid, preprint (2016). H. Ellouz, I. Feki and A. Jeribi, On the basis property of root vectors related to a non-self-adjoint analytic operator and applications, preprint (2016). H. Ellouz, I. Feki and A. Jeribi, Non-orthogonal fusion frames of an analytic operator and application to a one-dimensional wave control system, preprint (2016). P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math., 130, 309–317 (1973). I. Feki, A. Jeribi and R. Sfaxi, On an unconditional basis of generalized eigenvectors of an analytic operator and application to a problem of radiation of a vibrating structure in a light fluid, J. Math. Anal. Appl., 375, 261–269 (2011). I. Feki, A. Jeribi and R. Sfaxi, On a Schauder basis related to the eigenvectors of a family of non-self-adjoint analytic operators and applications, Anal. Math. Phys., 3 (4), 311–331 (2013). I. Feki, A. Jeribi and R. Sfaxi, On a Riesz basis of eigenvectors of a nonself-adjoint analytic operator and applications, Linear Multilinear Algebra, 62 (8), 1049–1068 (2014).

Bibliography | 397

[89] [90]

[91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114]

P. J. T. Filippi, Rayonnement d’une structure vibrante en fluide léger: application des techniques classiques des pertubations, J. Acoust., 2, 39–45 (1989). P. J. T. Filippi, O. Lagarrigue and P. O. Mattei, Perturbation method for sound radiation by a vibrating plate in a light fluid: comparison with the exact solution, J. Sound Vib., 177, 259–275 (1994). P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl., 333, 871–879 (2007). G. Geymonat and P. Grisvard, Expansions on generalized eigenvectors of operators arising in the theory of elasticity, Differential Integral Equations, 4 (3), 459–481 (1991). G. Godefroy and J. H. Shapiro, Operators with dense invariant cyclic manifolds, J. Funct. Anal., 98, 229–269 (1991). I. C. Gohberg and M. G. Krein, Introduction to the theory of linear non-self-adjoint operators in Hilbert space, Amer. Math. Soc., Providence (1969). I. C. Gohberg, A. S. Markus and I. A. Feldman, Normally solvable operators and ideals associated with them, Amer. Math. Soc. Transl. Ser., 261, 63–84 (1967). S. Goldberg, Unbounded linear operators, McGraw–Hill, New-York (1966). J. A. Goldstein, Semigroups of operators and applications, Oxford University PressOxford (1985). V. N. Gribov, A Reggon diagram technique, Sov. Phys. JETP, 26, 414–423 (1968). S. Grivaux, Sums of hypercyclic operators, J. Funct. Anal., 202, 486–503 (2003). K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc., 36, 345–381 (1996). A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16, 13–30 (1955). C. Heil, Wiener amalgam spaces in generalized harmonic analysis and wavelet theory, Thesis, University of Maryland (1990). C. Heil, A Basis theory primer, Springer, New-York, Dordrecht, Heidelberg, London (1998). D. A. Herrero and Z. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J., 39, 819–830 (1990). H. Heuser, Functional analysis, Wiley, New-York (1982). E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, RI (1957). F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1), 43–56 (1985). A. E. Ingham, Some trigonometrical equalities with applications to the theory of series, Math. Z., 41, 367–399 (1936). A. Intissar, Etude spectrale d’une famille d’opérateurs non-symétriques intervenant dans la théorie des champs de Reggeons, Comm. Math. Phys., 113, 263–297 (1987). A. Intissar, Analyse fonctionnelle et théorie spectrale pour les opérateurs compacts non-autoadjoints, Editions CEPADUES, Toulouse (1997). A. Intissar, Analyse de Scattering d’un opérateur cubique de Heun dans l’espace de Bargmann, Comm. Math. Phys., 199, 243–256 (1998). A. Intissar, Diagonalization of non-self-adjoint analytic semigroups and application to the shape memory alloys operator, J. Math. Anal. Appl., 257, 1–20 (2001). A. Intissar, A. Jeribi and I. Walha, Riez basis of exponential family for a hyperbolic system, preprint (2017). S. Ja. Jakubov and K. S. Mamedov, The multiple completeness of the system of eigen- and associated elements of a polynomial operator pencil and multiple expansions in that system, Funktsional. Anal. i Prilozhen., 9 (1), 91–93 (1975).

398 | Bibliography

[115] A. Jeribi, Quelques remarques sur les opérateurs de Fredholm et application à l’équation de transport, C. R. Acad. Sci. Paris Sér. I 325, 43–48 (1997). [116] A. Jeribi, Quelques remarques sur le spectre de Weyl et applications, C. R. Acad. Sci. Paris Sér. I Math. 327, 485–490 (1998). [117] A. Jeribi, Une nouvelle caractérisation du spectre essentiel et application, C. R. Acad. Sci. Paris Sér. I Math., 331, 525–530 (2000). [118] A. Jeribi, A characterization of the essential spectrum and applications, Boll. Unione Mat. Ital., 8 (B-5), 805–825 (2002). [119] A. Jeribi, A characterization of the Schechter essential spectrum on Banach spaces and applications, J. Math. Anal. Appl., 271, 343–358 (2002). [120] A. Jeribi, Some remarks on the Schechter essential spectrum and applications to transport equations, J. Math. Anal. Appl., 275, 222–237 (2002). [121] A. Jeribi, On the Schechter essential spectrum on Banach spaces and applications, Ser. Math. Inform., 17, 35–55 (2002). [122] A. Jeribi, Fredholm operators and essential spectra, Arch. Inequal. Appl., 2 (2–3), 123–140 (2004). [123] A. Jeribi, Spectral theory and applications of linear operators and block operator matrices, Springer-Verlag, New-York (2015). [124] A. Jeribi, Spectral theory of linear non-self-adjoint operators in Hilbert and Banach spaces: Denseness and bases with applications, preprint (2018). [125] A. Jeribi, Linear operators and their essential pseudospectra, CRC Press, Boca Raton (2018). [126] A. Jeribi, Développement de certaines propriétés fines de la théorie spectrale et applications à des modèles monocinétiques et à des modèles de Reggeons, Thesis of Mathematics, University of Corsica, Frensh (16 Janvier 1998). [127] A. Jeribi and A. Intissar, On an Riesz basis of generalized eigenvectors of the non-self-adjoint problem deduced from a perturbation method for sound radiation by a vibrating plate in a light fluid, J. Math. Anal. Appl., 292, 1–16 (2004). [128] A. Jeribi, K. Latrach, Quelques remarques sur le spectre essentiel et application à l’équation de transport, C. R. Acad. Sci. Paris Sér. I, 323, 469–474 (1996). [129] A. Jeribi and M. Mnif, Fredholm operators, essential spectra and application to transport equation, Acta Appl. Math., 89, 155–176 (2005). [130] A. Jeribi and N. Moalla, Fredholm operators and Riesz theory for polynomially compact operators, Acta Appl. Math., 90 (3), 227–245 (2006). [131] A. Jeribi and N. Moalla, A characterization of some subsets of Schechter’s essential spectrum and application to singular transport equation, J. Math. Anal. Appl., 358, 434–444 (2009). [132] A. Jeribi and I. Walha, Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of some matrix operator and application to two-group transport equation, Math. Nachr., 284 (1), 67–86 (2011). [133] A. Jeribi, N. Moalla and I. Walha, Spectra of some block operator matrices and application to transport operators, J. Math. Anal. Appl., 351 (1), 315–325 (2009). [134] A. Jeribi, N. Moalla and S. Yengui, S-essential spectra and application to an example of transport operators, Math. Methods Appl. Sci., 37 (16), 2341–2353 (2014). [135] A. Jeribi, M. A. Hammami and A. Masmoudi (eds.), Applied mathematics in Tunisia. International conference on advances in applied mathematics (ICAAM), Hammamet, Tunisia, December 16–19, 2013, Springer Proc. Math. Stat., vol. 131, Springer, Cham (2015) (ISBN 978-3-319-18040-3/hbk; 978-3-319-18041-0/ebook), xix+397 pp. [136] A. Jeribi, B. Krichen and M. Zarai Dhahri, Essential spectra of some of matrix operators involving γ-relatively bounded entries and an applications, Linear Multilinear Algebra, 1654–1668 (2016).

Bibliography | 399

[137] M. I. Kadeč, The exact value of the Paley Wiener constant, Dokl. Akad. Nauk SSSR, 155, 1253–1254 (1964) (in Russian); English transl: Soviet Math. Dokl., 5, 559–561 (1964). [138] S. Kaniel and M. Schechter, Spectral theory for Fredholm operators, Comm. Pure Appl. Math., XVI, 423–448 (1963). [139] T. Kato, Perturbation theory for linear operators, 2nd edn, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin, New-York (1976). [140] J. Kergomard and V. Debut, Resonance modes in a one-dimensional medium with two purely resistive boundaries, calculation methods, orthogonality, and completeness, J. Acoust. Soc. Amer., 1356–1367 (2005). [141] C. Kitai, Invariant closed sets for linear operators, Ph.D. Thesis, Univ. of Toronto (1982). [142] S G. Krein, Linear differential equations in Banach space, Amer. Math. Soc., Providence, RI (1971). [143] N. Kunimatsu and H. Sano, Stability analysis of heat-exchanger equations with boundary feedbacks, IMA J. Math. Control Inform., 15, 317–330 (1998). [144] P. Lang and J. Locker, Spectral decomposition of a Hilbert space by a Fredholm operators, J. Funct. Anal., 79, 9–17 (1988). [145] P. Lang and J. Locker, Spectral representation of the resolvent of a discrete operator, J. Funct. Anal., 79, 18–31 (1988). [146] P. Lang and J. Locker, Denseness of the generalized eigenvectors of an H–S discrete operator, J. Funct. Anal., 82, 316–329 (1989). [147] K. Latrach and A. Jeribi, On the essential spectrum of transport operators on L1 -spaces, J. Math. Phys., 37 (12), 6486–6494 (1996). [148] K. Latrach and A. Jeribi, Sur une équation de transport intervenant en dynamique des populations, C. R. Acad. Sci. Paris Sér. I, 325, 1087–1090 (1997). [149] K. Latrach and A. Jeribi, Some results on Fredholm operators, essential spectra and application, J. Math. Anal. Appl., 225 (2), 461–485 (1998). [150] A. Lebow and M. Schechter, Semigroups of operators and measures of noncompactness, J. Funct. Anal. 7, 1–26 (1971). [151] F. Leon-Saavedra, V. Müller, Rotations of hypercyclic operators, Integral Equations Operator Theory, 50, 385–391 (2004). [152] B. Ya. Levin, On bases of exponential functions in L2 , Zap. Mekh.-Mat. Fak. Khar’kov. Gos. Univ. Khar’kov. Mat. Obshch., 27, 39–48 (1961) (in Russian). [153] V. B. Lidskii, Summability of series in the principal vectors of non-self-adjoint operators, Amer. Math. Soc. Transl. (2), vol. 40 (1964). [154] V. I. Macaev and Ju. A. Palant, O Stepenjah ogranicennogo dissipativnogo operatora, Ukrainian Math. J. 14 (3) (1962). [155] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, vol. 71, Amer. Math. Soc., Providence, RI (1988). [156] V. Matache, Notes on hypercyclic operators, Acta Sci. Math., 58, 401–410 (1993). [157] V. Matache, Spectral properties of operators having dense orbits, in: Topics in operator theory, operator algebras and applications, vol. 32, 221–237 (1994). [158] T. L. Miller and V. G. Miller, Local spectral theory and orbits of operators, Proc. Amer. Math. Soc., 127, 1029–1037 (1999). [159] N. Moalla, M. Damak and A. Jeribi, Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions, J. Math. Anal. Appl., 323 (2), 1071–1090 (2006). [160] P. M. Morse and H. Feshbach, Methods of theoretical physics, Book co., Inc., New-York, Toronto, London (1953).

400 | Bibliography

[161] B. Sz. Nagy, Perturbations des transformations linéaires fermées, Acta Sci. Math., 14, 125–137 (1951). [162] J. C. Nedelec, Acoustic and electromagnetic equations, integral representations for Harmonic problems, Springer-Verlag, New-York (2001). [163] J. D. Newburgh, The variation of spectra, Duke Math. J., 18, 165–176 (1951). [164] B. Noble, Methods based on the Wiener–Hopf Technique for the solution of partial differential equations, Pergamon Press, New-York (1958). [165] K. K. Oberai, On the Weyl spectrum, Illinois J. Math. 18, 208–212 (1974). [166] R. Paley and N. Wiener, Fourier transform in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., New-York (1934). [167] J. M. Paoli, Conditions suffisantes de densité du sous-espace engendré par les vecteurs propres généralisés d’un opérateur compact ou à résolvante compacte sur un espace de Hilbert et applications aux opérateurs de Gribov, Thesis of Mathematics, University of Corsica, Frensh (29 October 1993). [168] B. S. Pavlov, Basicity of an exponential systems and Muckenhoupt’s condition, Sov. Math., Dokl., 20, 655–659 (1979). [169] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New-York (1983). [170] A. Pelczynski, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(X )-spaces, Bull. Acad. Pol. Sci., 13, 13–36 (1965). [171] A. Pelczynski, On strictly singular and strictly cosingular operators. II. Strictly singular and strictly cosingular operators in L(μ)-spaces, Bull. Acad. Pol. Sci., 13, 37–41 (1965). [172] F. Rellich, Storungstheorie der Spektral zerlegung, I. Mitt., Math. Ann. 113, 600–619 (1937). [173] F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New-York (1955). [174] S. Rolewicz, On orbits of elements, Stud. Math., 32, 17–22 (1969). [175] B. Russo, On the Hausdorff–Young theorem for integral operators, Pacific J. Math., 68, 241–253 (1977). [176] H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347, 993–1004 (1995). [177] S. Sánchez-Perales and S. V. Djordjevic̀ Spectral continuity using ν-convergence, J. Math. Anal. Appl., 433, 405–415 (2016). [178] A. Schechter, On the essential spectrum of an arbitrary operator, J. Math. Anal. Appl., 13, 205–215 (1966). [179] M. Schechter, Principles of functional analysis, Grad. Stud. Math., vol. 36, Amer. Math. Soc., Providence, RI (2002). [180] C. Schmoeger, On the operator equations ABA = A2 and BAB = B2 , Publ. Inst. Math. (N. S.), 78 (92), 127–133 (2005). [181] A. Schueller, Uniqueness for near-constant data in fourth-order inverse eigenvalue problems, J. Math. Anal. Appl., 258, 658–670 (2001). [182] R. D. Spies, A state-space approach to a one-dimensional mathematical model for the dynamics of phase transitions in pseudoelastic materials, J. Math. Anal. Appl., 190, 58–100 (1995). [183] W. S. Tang, Oblique projections, biorthogonal Riesz bases and multiwavelets in Hilbert spaces, Proc. Amer. Math. Soc. 128, 463–473 (1999). [184] A. E. Taylor, Introduction to functional analysis, Wiley, New-York (1958). [185] A. E. Taylor and D. C. Lay, Introduction to functional analysis, 2nd edn, Wiley, New-York (1980). [186] H. Triebel, Interpolation theory, function spaces, differential operators, vol. 18, North-Holland, Amsterdam, New-York, Oxford (1978). [187] H. Von Koch, Sur quelques points de la théorie des déterminants infinis, Acta Math., 24, 89–122 (1900).

Bibliography | 401

[188] H. Von Koch, Sur la convergence des déterminants infinis, Rend. Palermo 28, 255–266 (1909). [189] C. Wyss, Riesz bases for p-subordinate perturbations of normal operator, J. Funct. Anal., 258, 208–240 (2010). [190] G. Q. Xu, S. H. Wang, The completeness of systems of generalized eigenfunctions of generators of Riesz semigroups, Acta Math. Sin., 39 (2) 263–267 (1996) (in Chinese). [191] G. Q. Xu and S. P. Yung, The expansion of a semigroup and a Riesz basis criterion, J. Differential Equations, 210, 1–24 (2005). [192] G. Q. Xu and S. P. Yung, Properties of a class of C0 -semigroups on Banach spaces and their applications, J. Math. Anal. Appl., 328, 245–256 (2007). [193] S. Yakubov, Completeness of root functions of regular differential operators, Longman, Harlow (1994). [194] R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, London (1980).

Index [2p] 81 α(A) 22 (β) 235 β(A) 22 χ(−a,x−ε) (⋅) 291 δmn 21 Γ-hypercyclic set 48 Γhyp (A) 48 Γhyp (T ) 4 γ(A) 44 ν-continuous 42 ν-convergent 41 ν-lower semi-continuous 42 ν-upper semi-continuous 41 ω-linearly independent 21 Ω(φ) 68 Ω(φ− , φ+ ) 68 Φ 33, 145 Φ+ (X , Y ) 22 Φ− (X , Y ) 22 Φ± (X , Y ) 22 ΦA 22 ΦbA 22 Φb+ (X , Y ) 22 Φb− (X , Y ) 22 Φb (X , Y ) 22 πV 66 ρ+s−F (T ) 201 ρ−s−F (T ) 201 ρks−F (T ) 201 ρ(A) 28 σap (A) 45 󸀠 σap (S) 53 σf (A) 45 σj (A) 44 σp (A) 28 σr (A) 28 σw (A) 45 σ(A) 28 σc(A) 28 ‖ ⋅ ‖D 361 ‖ ⋅ ‖D2 361 𝒜 95 A-bounded 25 A-bounded with order p ∈ [0, 1] 25

A-compact 26 A-compact of order p ∈ [0, 1] 25 A-convergent 19 A-invariant 130 𝔸-tight fusion frame 172 A∗ = A 28 A† 43 A ⊂ A∗ 28 Abel’s method of order α 193 adjoint operator 27 algebraic complement of M 22 algebraic multiplicity 32, 80, 147 algebraically simple 32 alternate dual frame 185 analysis operator 163, 175 annihilation operator 359, 372 annihilator 28 asc(A) 34, 35 ascent 34, 35 associated orthonormal basis 174 associated vector of the kth rank to the eigenvector x0 31 atomic bounds 170 backward shift 236 Banach–Steinhaus theorem 37 Bargmann space 359 basis 21 basis constant 87 basis of subspaces 127 Bessel bound 55 Bessel fusion sequence with Bessel fusion bound 𝔹 172 Bessel inequality 58 Bessel sequence 55 biorthogonal 21 biorthogonal system 21 Bishop’s property (β) 235 𝔹(λ, ε) 22 bounded 19 c0 48 C0 -semi-group 133 Cp 76 Cp (X ) 77 (0) 𝒞p,∞ (X ) 80

404 | Index

Carleman class 76 Cartwright class 73 characteristic function 291 characteristic numbers 191 closable 19 closed 19 closed graph theorem 20 coefficient functionals 86 compact 25 compact resolvent 33 complete 21, 80 completely continuous 25 conditionally convergent 128 convergent series 37 core 20 creation operator 359, 372 𝒞(X , Y ) 20 𝒟(A) 19 desc(A) 34, 35 descent 34, 35 det(I − T ) 138 direct sum 22 discrete operator 33 dissipative 134 dist(λ, σ(A)) 29, 43 d(λ) 79 dual Riesz basis 114 dual system 21 Dunford–Pettis property 44 eigenmode 291 eigenspace 31 eigenvalue 31, 291 eigenvector 31 eigenvector of the rth rank 31 E(λ, A) 63 equivalent 20 equivalent basis 107, 108 essential spectral radius 74 essentially quasi-nilpotent 48 exact frames 8 exponential type 71 family of weights 172 finite order 70 finite-dimensional A-invariant 130 Fourier coefficients 192 Fourier series 191, 192

frame 159 frame of subspaces 172 frame operator 163 frame sequence 172 Fredholm determinant 79, 80 Fredholm operator 22 Fredholm perturbation 25 Fredholm set 22 fusion frame bounds 172 fusion frame sequence 178 ℱ (X , Y ) 25 Gδ -set 47 G(A) 19 generalized eigenspace 80 generalized eigenvector 32 generating function of exponential family 73 geometric multiplicity 32, 80 geometrically simple 32 Gram matrix 104, 176 graph 19 Gribov operator 359, 372 growth indicator of an exponential-type function 72 (H1) 241 (H2) 241 (H3) 259 (H3’) 266 (H4) 263 (H4’) 266 (H5) 271 (H6) 275 H-Lipschitz function 266 Hλ󸀠 ,μ,α,λ 359 Hλ󸀠󸀠 ,λ󸀠 ,μ,λ 372 Hamel basis 85 Hankel function 292 Hankel operator 291 HC(X ) 47 Hilbert basis 22 Hilbert–Schmidt discrete operator 149 Hilbert–Schmidt operator 77, 105 Hille–Yosida theorem 133 Hölder’s inequality 82 HS discrete operator 149 hypercyclic 4, 47 hypernormal operator 204 hyponormal operator 43

Index | 405

i(A) 46 independent 21 indicator diagram 72 infinitesimal generator 133 interpolation inequality 83 invariant 48 invertible 32 isometric 19 isometric isomorphism 19

Neumann theorem 32 non-orthogonal fusion frame 188 non-orthogonal fusion frame bounds 188 norm-preserving 19 normal operator 43, 204 normal point 73 n(𝔹(0, r), L) 129 nuclear operator 77 numerical range 75, 76

Jeribi essential spectrum 44, 45 Jordan basis 191 Jordan chain 31, 191

o(1) 71 O(1) 71 o(ϕ(x)) 71 O(ϕ(x)) 71 operator 19 optimal lower frame bound 160 optimal upper frame bound 160 Orb(A, x) 47 orb(T , x) 4 order of f 70 orthogonal 20 orthogonal projection 62 orthogonal sequence 20 orthonormal basis 99 orthonormal fusion basis 172 orthonormal sequence 21 orthonormal system 99

Kronecker delta 21 𝒦(X , Y ) 25 ℒA (λ) 32 l2 (ℕ) 21 l2 (X , I) 174 length of the chain 31 lim inf En 39 limit inferior 39 limit superior 39 lim sup En 39 linear 19 linear span 21 Lipschitz function 262 lower frame bounds 160 lower semi-continuous 40 lower semi-Fredholm 22 ℒ(X , Y ) 20 |M| 231 ma (λ) 32 mg (λ0 ) 31 M∞ 2, 145 M ⊕ N 22 maximal dissipative 134 minimum type 71 Minkowski’s inequality 82 Muckenhoupt condition 127 multiplicity 32 p

p

n+ (rk − qrk , rk + qrk , L) 69 N⊥ 28 n(Δ, A) 131, 272 N(A) 20 natural projections associated with (xn )n 86

𝒫 49 p-subordinate 68 P0 361 Paley–Wiener criterion 125 Parseval fusion frame 172 Parseval’s equation 101 partial sum operators 86 Phragmén–Lindelöf theorem 78 Plancherel equality 159 pointwise 37 pre-frame operator 163 principal vectors 191 projection 62 projection method works 167 pseudo-convergence 208 pseudo-inverse operator 43 Pythagorean theorem 58 quadratically close 119 quasi-affinities 239

406 | Index

quasi-nilpotent operator 105 quasi-similar 239 R 28 ress (T ) 74 R(A) 20 r(A) 73 reduced minimum modulus 44 resolvent 29 resolvent set 28 Riesz basis 110 Riesz basis with parentheses 129 Riesz operator 35 Riesz projection 63 Riesz representation theorem 61 R(λ, A) 28, 33 root subspaces 32, 192 root vectors 32, 192 r(T ) 234 ℛ(X ) 35 ⊥

s-numbers 74 𝕊ℂ (α, r) 231 S∞ 2, 145 sn (K) 74 𝕊X 76 𝒮(X , Y ) 26 S(A) 143 Schauder basis 85 self-adjoint 28 separated 127 set of atoms 170 sine type 73 singular value of compact operator 74 span 21 span{en }∞ n=1 21 spectral inclusion property 76 spectral projection 63 spectral radius 73, 234 spectral space 63 spectrum 28 strictly singular 26 strongly continuous semi-group 133 strongly convergent 40

subnormal operator 43, 204 subordinate to G 68 subordinate to G with order p 68 summable 193 symbol of Kronecker 21 symmetric 28 synthesis operator 163, 175 √T 115 T -hypercyclic 4 T -orbit 4 T|W 48 topological isomorphism 62 total 22 tr(A) 105 trace class operator 77 trace of A 105 type 71 Un →U 40 Un converges to U in the strong operator topology 37 ν

Un → U 41 unconditional 127, 129 unconditional basis with parentheses 129 unconditionally convergent 56, 128 uniform boundedness principle 37 unitary operator 19 upper frame bounds 160 upper semi-continuous 3, 40 upper semi-Fredholm 22 Von Koch determinant 138 w-uniform 172 W (A) 75, 76 w(A) 143 𝒲(X , Y ) 26 𝒲∗ (X ) 44, 45 weakly compact 26 Weyl essential spectrum 45 Wolf essential spectrum 45 Young’s inequality 81