A Hilbert Space Problem Book

The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem....

This is not an introduction to Hilbert space theory.

1. Vectors.- 1. Limits of quadratic forms.- 2. Schwarz inequality.- 3. Representation of linear functional.- 4. Strict convexity.- 5. Continuous curves.- 6. Uniqueness of crinkled arcs.- 7. Linear dimension.- 8. Total sets.- 9. Infinitely total sets.- 10. Infinite Vandermondes.- 11. T-totalsets.- 12. Approximate bases.- 2. Spaces.- 13. Vector sums.- 14. Lattice of subspaces.- 15. Vector sums and the modular law.- 16. Local compactness and dimension.- 17. Separability and dimension.- 18. Measure in Hilbert space.- 3. Weak Topology.- 19. Weak closure of subspaces.- 20. Weak continuity of norm and inner product.- 21. Semicontinuity of norm.- 22. Weak separability.- 23. Weak compactness of the unit ball.- 24. Weak metrizability of the unit ball.- 25. Weak closure of the unit sphere.- 26. Weak metrizability and separability.- 27. Uniform boundedness.- 28. Weak metrizability of Hilbert space.- 29. Linear functionals on l2.- 30. Weak completeness.- 4. Analytic Functions.- 31. Analytic Hilbert spaces.- 32. Basis for A2.- 33. Real functions in H2.- 34. Products in H2.- 35. Analytic characterization of H2.- 36. Functional Hilbert spaces.- 37. Kernel functions.- 38. Conjugation in functional Hilbert spaces.- 39. Continuity of extension.- 40. Radial limits.- 41. Bounded approximation.- 42. Multiplicativity of extension.- 43. Dirichlet problem.- 5. Infinite Matrices.- 44. Column-finite matrices.- 45. Schur test.- 46. Hilbert matrix.- 47. Exponential Hilbert matrix.- 48. Positivity of the Hilbert matrix.- 49. Series of vectors.- 6. Boundedness and Invertibility.- 50. Boundedness on bases.- 51. Uniform boundedness of linear transformations.- 52. Invertible transformations.- 53. Diminishablc complements.- 54. Dimension in inner-product spaces.- 55. Total orthonormal sets.- 56. Preservation of dimension.- 57. Projections of equal rank.- 58. Closed graph theorem.- 59. Range inclusion and factorization.- 60. Unbounded symmetric transformations.- 7. Multiplication Operators.- 61. Diagonal operators.- 62. Multiplications on l2.- 63. Spectrum of a diagonal operator.- 64. Norm of a multiplication.- 65. Boundedness of multipliers.- 66. Boundedness of multiplications.- 67. Spectrum of a multiplication.- 68. Multiplications on functional Hilbert spaces.- 69. Multipliers of functional Hilbert spaces.- 8. Operator Matrices.- 70. Commutative operator determinants.- 71. Operator determinants.- 72. Operator determinants with a finite entry.- 9. Properties of Spectra.- 73. Spectra and conjugation.- 74. Spectral mapping theorem.- 75. Similarity and spectrum.- 76. Spectrum of a product.- 77. Closure of approximate point spectrum.- 78. Boundary of spectrum.- 10. Examples of Spectra.- 79. Residual spectrum of a normal operator.- 80. Spectral parts of a diagonal operator.- 81. Spectral parts of a multiplication.- 82. Unilateral shift.- 83. Structure of the set of eigenvectors.- 84. Bilateral shift.- 85. Spectrum of a functional multiplication.- 11. Spectral Radius.- 86. Analyticity of resolvents.- 87. Non-emptiness of spectra.- 88. Spectral radius.- 89. Weighted shifts.- 90. Similarity of weighted shifts.- 91. Norm and spectral radius of a weighted shift.- 92. Power norms.- 93. Eigenvalues of weighted shifts.- 94. Approximate point spectrum of a weighted shift.- 95. Weighted sequence spaces.- 96. One-point spectrum.- 97. Analytic quasinilpotents.- 98. Spectrum of a direct sum.- 12. Norm Topology.- 99. Metric space of operators.- 100. Continuity of inversion.- 101. Interior of conjugate class.- 102. Continuity of spectrum.- 103. Semicontinuity of spectrum.- 104. Continuity of spectral radius.- 105. Normal continuity of spectrum.- 106. Quasinilpotent perturbations of spectra.- 13. Operator Topologies.- 107. Topologies for operators.- 108. Continuity of norm.- 109. Semicontinuity of operator norm.- 110. Continuity of adjoint.- 111. Continuity of multiplication.- 112. Separate continuity of multiplication.- 113. Sequential continuity of multiplication.- 114. Weak sequential continuity of squaring.- 115. Weak convergence of projections.- 14. Strong Operator Topology.- 116. Strong normal continuity of adjoint.- 117. Strong bounded continuity of multiplication.- 118. Strong operator versus weak vector convergence.- 119. Strong semicontinuity of spectrum.- 120. Increasing sequences of Hermitian operators.- 121. Square roots.- 122. Infimum of two projections.- 15. Partial Isometries.- 123. Spectral mapping theorem for normal operators.- 124. Decreasing squares.- 125. Polynomially diagonal operators.- 126. Continuity of the functional calculus.- 127. Partial isometries.- 128. Maximal partial isometries.- 129. Closure and connectedness of partial isometries.- 130. Rank, co-rank, and nullity.- 131. Components of the space of partial isometries.- 132. Unitary equivalence for partial isometries.- 133. Spectrum of a partial isometry.- 16. Polar Decomposition.- 134. Polar decomposition.- 135. Maximal polar representation.- 136. Extreme points.- 137. Quasinormal operators.- 138. Mixed Schwarz inequality.- 139. Quasinormal weighted shifts.- 140. Density of invertible operators.- 141. Connectedness of invertible operators.- 17. Unilateral Shift.- 142. Reducing subspaces of normal operators.- 143. Products of symmetries.- 144. Unilateral shift versus normal operators.- 145. Square root of shift.- 146. Commutant of the bilateral shift.- 147. Commutant of the unilateral shift.- 148. Commutant of the unilateral shift as limit.- 149. Characterization of isometries.- 150. Distance from shift to unitary operators.- 151. Square roots of shifts.- 152. Shifts as universal operators.- 153. Similarity to parts of shifts.- 154. Similarity to contractions.- 155. Wandering subspaces.- 156. Special invariant subspaces of the shift.- 157. Invariant subspaces of the shift.- 158. F. and M. Riesz theorem.- 159. Reducible weighted shifts.- 18. Cyclic Vectors.- 160. Cyclic vectors.- 161. Density of cyclic operators.- 162. Density of non-cyclic operators.- 163. Cyclicity of a direct sum.- 164. Cyclic vectors of adjoints.- 165. Cyclic vectors of a position operator.- 166. Totality of cyclic vectors.- 167. Cyclic operators and matrices.- 168. Dense orbits.- 19. Properties of Compactness.- 169. Mixed continuity.- 170. Compact operators.- 171. Diagonal compact operators.- 172. Normal compact operators.- 173. Hilbert-Schmidt operators.- 174. Compact versus Hilbert-Schmidt.- 175. Limits of operators of finite rank.- 176. Ideals of operators.- 177. Compactness on bases.- 178. Square root of a compact operator.- 179. Fredholm alternative.- 180. Range of a compact operator.- 181. Atkinson’s theorem.- 182. Weyl’s theorem.- 183. Perturbed spectrum.- 184. Shift modulo compact operators.- 185. Distance from shift to compact operators.- 20. Examples of Compactness.- 186. Bounded Volterra kernels.- 187. Unbounded Volterra kernels.- 188. Volterra integration operator.- 189. Skew-symmetric Volterra operator.- 190. Norm 1, spectrum {1}.- 191. Donoghue lattice.- 21. Subnormal Operators.- 192. Putnam-Fuglede theorem.- 193. Algebras of normal operators.- 194. Spectral measure of the unit disc.- 195. Subnormal operators.- 196. Quasinormal invariants.- 197. Minimal normal extensions.- 198. Polynomials in the shift.- 199. Similarity of subnormal operators.- 200. Spectral inclusion theorem.- 201. Filling in holes.- 202. Extensions of finite co-dimension.- 203. Hyponormal operators.- 204. Normal and subnormal partial isometries.- 205. Norm powers and power norms.- 206. Compact hyponormal operators.- 207. Hyponormal, compact imaginary part.- 208. Hyponormal idempotents.- 209. Powers of hyponormal operators.- 22. Numerical Range.- 210. Toeplitz-Hausdorff theorem.- 211. Higher-dimensional numerical range.- 212. Closure of numerical range.- 213. Numerical range of a compact operator.- 214. Spectrum and numerical range.- 215. Quasinilpotence and numerical range.- 216. Normality and numerical range.- 217. Subnormality and numerical range.- 218. Numerical radius.- 219. Normaloid, convexoid, and spectraloid operators.- 220. Continuity of numerical range.- 221. Power inequality.- 23. Unitary Dilations.- 222. Unitary dilations.- 223. Images of subspaces.- 224. Weak closures and dilations.- 225. Strong closures and extensions.- 226. Strong limits of hyponormal operators.- 227. Unitary power dilations.- 228. Ergodic theorem.- 229. von Neumann’s inequality.- 24. Commutators.- 230. Commutators.- 231. Limits of commutators.- 232. Kleinecke-Shirokov theorem.- 233. Distance from a commutator to the identity.- 234. Operators with large kernels.- 235. Direct sums as commutators.- 236. Positive self-commutators.- 237. Projections as self-commutators.- 238. Multiplicative commutators.- 239. Unitary multiplicative commutators.- 240. Commutator subgroup.- 25. Toeplitz Operators.- 241. Laurent operators and matrices.- 242. Toeplitz operators and matrices.- 243. Toeplitz products.- 244. Compact Toeplitz products.- 245. Spectral inclusion theorem for Toeplitz operators.- 246. Continuous Toeplitz products.- 247. Analytic Toeplitz operators.- 248. Eigenvalues of Hermitian Toeplitz operators.- 249. Zero-divisors.- 250. Spectrum of a Hermitian Toeplitz operator.- References.- List of Symbols.