A Hilbert Space Problem Book

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.