A Hilbert Space Problem Book (Graduate Texts in Mathematics, 19) [2nd rev. and enlarged ed. 1982] 0387906851, 9780387906850

Table of contents :
Preface
Contents
PROBLEMS
1. Vectors (P1-12)
P1. Limits of quadratic forms
Hint
Solution
P2. Schwarz inequality
Hint
Solution
P3. Representation of linear functionals
Hint
Solution
P4. Strict convexity
Hint
Solution
P5. Continuous curves
Hint
Solution
P6. Uniqueness of crinkled arcs
Hint
Solution
P7. Linear dimension
Hint
Solution
P8. Total sets
Hint
Solution
P9. Infinitely total sets
Hint
Solution
P10. Infinite Vandermondes
Hint
Solution
P11. T-total sets
Hint
Solution
P12. Approximate bases
Hint
Solution
2. Spaces (P13-18)
P13. Vector sums
Hint
Solution
P14. Lattice of subspaces
Hint
Solution
P15. Vector sums and the modular law
Hint
Solution
P16. Local compactness and dimension
Hint
Solution
P17. Separability and dimension
Hint
Solution
P18. Measure in Hilbert space
Hint
Solution
3. Weak Topology (P19-30)
P19. Weak closure of subspaces
Hint
Solution
P20. Weak continuity of norm and inner product
Hint
Solution
P21. Semicontinuity of norm
Hint
Solution
P22. Weak separability
Hint
Solution
P23. Weak compactness of the unit ball
Hint
Solution
P24. Weak metrizability of the unit ball
Hint
Solution
P25. Weak closure of the unit sphere
Hint
Solution
P26. Weak metrizability and separability
Hint
Solution
P27. Uniform boundedness
Hint
Solution
P28. Weak metrizability of Hilbert space
Hint
Solution
P29. Linear functional on l^2
Hint
Solution
P30. Weak completeness
Hint
Solution
4. Analytic Functions (P31-43)
P31. Analytic Hilbert spaces
Hint
Solution
P32. Basis for A^2
Hint
Solution
P33. Real functions in H^2
Hint
Solution
P34. Products in H^2
Hint
Solution
P35. Analytic characterization of H^2
Hint
Solution
P36. Functional Hilbert spaces
Hint
Solution
P37. Kernel functions
Hint
Solution
P38. Conjugation in functional Hilbert spaces
Hint
Solution
P39. Continuity of extension
Hint
Solution
P40. Radial limits
Hint
Solution
P41. Bounded approximation
Hint
Solution
P42. Multiplicativity of extension
Hint
Solution
P43. Dirichlet problem
Hint
Solution
5. Infinite Matrices (P44-49)
P44. Column-finite matrices
Hint
Solution
P45. Schur test
Hint
Solution
P46. Hilbert matrix
Hint
Solution
P47. Exponential Hilbert matrix
Hint
Solution
P48. Positivity of the Hilbert matrix
Hint
Solution
P49. Series of vectors
Hint
Solution
6. Boundedness and Invertibility (P50-60)
P50. Boundedness on bases
Hint
Solution
P51. Uniform boundedness of linear transformations
Hint
Solution
P52. Invertible transformations
Hint
Solution
P53. Diminishable complements
Hint
Solution
P54. Dimension in inner-product spaces
Hint
Solution
P55. Total orthonormal sets
Hint
Solution
P56. Preservation of dimension
Hint
Solution
P57. Projections of equal rank
Hint
Solution
P58. Closed graph theorem
Hint
Solution
P59. Range inclusion and factorization
Hint
Solution
P60. Unbounded symmetric transformations
Hint
Solution
7. Multiplication Operators (P61-69)
P61. Diagonal operators
Hint
Solution
P62. Multiplications on l^2
Hint
Solution
P63. Spectrum of a diagonal operator
Hint
Solution
P64. Norm of a multiplication
Hint
Solution
P65. Boundedness of multipliers
Hint
Solution
P66. Boundedness of multiplications
Hint
Solution
P67. Spectrum of a multiplication
Hint
Solution
P68. Multiplications on functional Hilbert spaces
Hint
Solution
P69. Multipliers of functional Hilbert spaces
Hint
Solution
8. Operator Matrices (P70-72)
P70. Commutative operator determinants
Hint
Solution
P71. Operator determinants
Hint
Solution
P72. Operator determinants with a finite entry
Hint
Solution
9. Properties of Spectra (P73-78)
P73. Spectra and conjugation
Hint
Solution
P74. Spectral mapping theorem
Hint
Solution
P75. Similarity and spectrum
Hint
Solution
P76. Spectrum of a product
Hint
Solution
P77. Closure of approximate point spectrum
Hint
Solution
P78. Boundary of spectrum
Hint
Solution
10. Examples of Spectra (P79-85)
P79. Residual spectrum of a normal operator
Hint
Solution
P80. Spectral parts of a diagonal operator
Hint
Solution
P81. Spectral parts of a multiplication
Hint
Solution
P82. Unilateral shift
Hint
Solution
P83. Structure of the set of eigenvectors
Hint
Solution
P84. Bilateral shift
Hint
Solution
P85. Spectrum of a functional multiplication
Hint
Solution
11. Spectral Radius (P86-98)
P86. Analyticity of resolvents
Hint
Solution
P87. Non-emptiness of spectra
Hint
Solution
P88. Spectral radius
Hint
Solution
P89. Weighted shifts
Hint
Solution
P90. Similarity of weighted shifts
Hint
Solution
P91. Norm and spectral radius of a weighted shift
Hint
Solution
P92. Power norms
Hint
Solution
P93. Eigenvalues of weighted shifts
Hint
Solution
P94. Approximate point spectrum of a weighted shift
Hint
Solution
P95. Weighted sequence spaces
Hint
Solution
P96. One-point spectrum
Hint
Solution
P97. Analytic quasinilpotents
Hint
Solution
P98. Spectrum of a direct sum
Hint
Solution
12. Norm Topology (P99-106)
P99. Metric space of operators
Hint
Solution
P100. Continuity of inversion
Hint
Solution
P101. Interior of conjugate class
Hint
Solution
P102. Continuity of spectrum
Hint
Solution
P103. Semicontinuity of spectrum
Hint
Solution
P104. Continuity of spectral radius
Hint
Solution
P105. Normal continuity of spectrum
Hint
Solution
P106. Quasinilpotent perturbations of spectra
Hint
Solution
13. Operator Topologies (P107-115)
P107. Topologies for operators
Hint
Solution
P108. Continuity of norm
Hint
Solution
P109. Semicontinuity of operator norm
Hint
Solution
P110. Continuity of adjoint
Hint
Solution
P111. Continuity of multiplication
Hint
Solution
P112. Separate continuity of multiplication
Hint
Solution
P113. Sequential continuity of multiplication
Hint
Solution
P114. Weak sequential continuity of squaring
Hint
Solution
P115. Weak convergence of projections
Hint
Solution
14. Strong Operator Topology (P116-122)
P116. Strong normal continuity of adjoint
Hint
Solution
P117. Strong bounded continuity of multiplication
Hint
Solution
P118. Strong operator versus weak vector convergence
Hint
Solution
P119. Strong semicontinuity of spectrum
Hint
Solution
P120. Increasing sequences of Hermitian operators
Hint
Solution
P121. Square roots
Hint
Solution
P122. Infimum of two projections
Hint
Solution
15. Partial Isometries (P123-133)
P123. Spectral mapping theorem for normal operators
Hint
Solution
P124. Decreasing squares
Hint
Solution
P125. Polynomially diagonal operators
Hint
Solution
P126. Continuity of the functional calculus
Hint
Solution
P127. Partial isometries
Hint
Solution
P128. Maximal partial isometries
Hint
Solution
P129. Closure and connectedness of partial isometries
Hint
Solution
P130. Rank, co-rank, and nullity
Hint
Solution
P131. Components of the space of partial isometries
Hint
Solution
P132. Unitary equivalence for partial isometries
Hint
Solution
P133. Spectrum of a partial isometry
Hint
Solution
16. Polar Decomposition (P134-141)
P134. Polar decomposition
Hint
Solution
P135. Maximal polar representation
Hint
Solution
P136. Extreme points
Hint
Solution
P137. Quasinormal operators
Hint
Solution
P138. Mixed Schwarz inequality
Hint
Solution
P139. Quasinormal weighted shifts
Hint
Solution
P140. Density of invertible operators
Hint
Solution
P141. Connectedness of invertible operators
Hint
Solution
17. Unilateral Shift (P142-159)
P142. Reducing subspaces of normal operators
Hint
Solution
P143. Products of symmetries
Hint
Solution
P144. Unilateral shift versus normal operators
Hint
Solution
P145. Square root of shift
Hint
Solution
P146. Commutant of the bilateral shift
Hint
Solution
Pi47. Commutant of the unilateral shift
Hint
Solution
P148. Commutant of the unilateral shift as limit
Hint
Solution
P149. Characterization of isometries
Hint
Solution
P150. Distance from shift to unitary operators
Hint
Solution
P151. Square roots of shifts
Hint
Solution
P152. Shifts as universal operators
Hint
Solution
P153. Similarity to parts of shifts
Hint
Solution
P154. Similarity to contractions
Hint
Solution
P155. Wandering subspaces
Hint
Solution
P156. Special invariant subspaces of the shift
Hint
Solution
P157. Invariant subspaces of the shift
Hint
Solution
P158. F. and M. Riesz theorem
Hint
Solution
P159. Reducible weighted shifts
Hint
Solution
18. Cyclic Vectors (P160-168)
P160. Cyclic vectors
Hint
Solution
P161. Density of cyclic operators
Hint
Solution
P162. Density of non-cyclic operators
Hint
Solution
P163. Cyclicity of a direct sum
Hint
Solution
P164. Cyclic vectors of adjoints
Hint
Solution
P165. Cyclic vectors of a position operator
Hint
Solution
P166. Totality of cyclic vectors
Hint
Solution
P167. Cyclic operators and matrices
Hint
Solution
P168. Dense orbits
Hint
Solution
19. Properties of Compactness (P169-185)
P169. Mixed continuity
Hint
Solution
P170. Compact operators
Hint
Solution
P171. Diagonal compact operators
Hint
Solution
P172. Normal compact operators
Hint
Solution
P173. Hilbert-Schmidt operators
Hint
Solution
P174. Compact versus Hilbert-Schmidt
Hint
Solution
P175. Limits of operators of finite rank
Hint
Solution
P176. Ideals of operators
Hint
Solution
P177. Compactness on bases
Hint
Solution
P178. Square root of a compact operator
Hint
Solution
P179. Fredholm alternative
Hint
Solution
P180. Range of a compact operator
Hint
Solution
P181. Atkinson's theorem
Hint
Solution
P182. Weyl's theorem
Hint
Solution
P183. Perturbed spectrum
Hint
Solution
P184. Shift modulo compact operators
Hint
Solution
P185. Distance from shift to compact operators
Hint
Solution
20. Examples of Compactness (P186-191)
P186. Bounded Volterra kernels
Hint
Solution
P187. Unbounded Volterra kernels
Hint
Solution
P188. Volterra integration operator
Hint
Solution
P189. Skew-symmetric Volterra operator
Hint
Solution
P190. Norm 1, spectrum {1}
Hint
Solution
P191. Donoghue lattice
Hint
Solution
21. Subnormal Operators (P192-209)
P192. Putnam-Fuglede theorem
Hint
Solution
P193. Algebras of normal operators
Hint
Solution
P194. Spectral measure of the unit disc
Hint
Solution
P195. Subnormal operators
Hint
Solution
P196. Quasinormal invariants
Hint
Solution
P197. Minimal normal extensions
Hint
Solution
P198. Polynomials in the shift
Hint
Solution
P199. Similarity of subnormal operators
Hint
Solution
P200. Spectral inclusion theorem
Hint
Solution
P201. Filling in holes
Hint
Solution
P202. Extensions of finite co-dimension
Hint
Solution
P203. Hyponormal operators
Hint
Solution
P204. Normal and subnormal partial isometries
Hint
Solution
P205. Norm powers and power norms
Hint
Solution
P206. Compact hyponormal operators
Hint
Solution
P207. Hyponormal, compact imaginary part
Hint
Solution
P208. Hyponormal idempotents
Hint
Solution
P209. Powers of hyponormal operators
Hint
Solution
22. Numerical Range (P210-221)
P210. Toeplitz-Hausdorff theorem
Hint
Solution
P211. Higher-dimensional numerical range
Hint
Solution
P212. Closure of numerical range
Hint
Solution
P213. Numerical range of a compact operator
Hint
Solution
P214. Spectrum and numerical range
Hint
Solution
P215. Quasinilpotence and numerical range
Hint
Solution
P216. Normality and numerical range
Hint
Solution
P217. Subnormality and numerical range
Hint
Solution
P218. Numerical radius
Hint
Solution
P219. Normaloid, convexoid, and spectraloid operators
Hint
Solution
P220. Continuity of numerical range
Hint
Solution
P221. Power inequality
Hint
Solution
23. Unitary Dilations (P222-229)
P222. Unitary dilations
Hint
Solution
P223. Images of subspaces
Hint
Solution
P224. Weak closures and dilations
Hint
Solution
P225. Strong closures and extensions
Hint
Solution
P226. Strong limits of hyponormal operators
Hint
Solution
P227. Unitary power dilations
Hint
Solution
P228. Ergodic theorem
Hint
Solution
P229. von Neumann's inequality
Hint
Solution
24. Commutators (P230-240)
P230. Commutators
Hint
Solution
P231. Limits of commutators
Hint
Solution
P232. Kleinecke-Shirokov theorem
Hint
Solution
P233. Distance from a commutator to the identity
Hint
Solution
P234. Operators with large kernels
Hint
Solution
P235. Direct sums as commutators
Hint
Solution
P236. Positive self-commutators
Hint
Solution
P237. Projections as self-commutators
Hint
Solution
P238. Multiplicative commutators
Hint
Solution
P239. Unitary multiplicative commutators
Hint
Solution
P240. Commutator subgroup
Hint
Solution
25. Toeplitz Operators (P241-250)
P241. Laurent operators and matrices
Hint
Solution
P242. Toeplitz operators and matrices
Hint
Solution
P243. Toeplitz products
Hint
Solution
P244. Compact Toeplitz products
Hint
Solution
P245. Spectral inclusion theorem for Toeplitz operators
Hint
Solution
P246. Continuous Toeplitz products
Hint
Solution
P247. Analytic Toeplitz operators
Hint
Solution
P248. Eigenvalues of Hermitian Toeplitz operators
Hint
Solution
P249. Zero-divisors
Hint
Solution
P250. Spectrum of a Hermitian Toeplitz operator
Hint
Solution
HINTS (see the bookmarks under each problem)
SOLUTIONS (see the bookmarks under each problem)
References
List of Symbols
Index