From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... 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. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book."

Inhalt

Problem.- 1. Vectors and Spaces.- 1. Limits of quadratic forms.- 2. Representation of linear functional.- 3. Strict convexity.- 4. Continuous curves.- 5. Linear dimension.- 6. Infinite Vandermondes.- 7. Approximate bases.- 8. Vector sums.- 9. Lattice of subspaces.- 10. Local compactness and dimension.- 11. Separability and dimension.- 12. Measure in Hilbert space.- 2. Weak Topology.- 13. Weak closure of subspaces.- 14. Weak continuity of norm and inner product.- 15. Weak separability.- 16. Uniform weak convergence.- 17. Weak compactness of the unit ball.- 18. Weak metrizability of the unit ball.- 19. Weak metrizability and separability.- 20. Uniform boundedness.- 21. Weak metrizability of Hubert space.- 22. Linear functionate on l2.- 23. Weak completeness.- 3. Analytic Functions.- 24. Analytic Hilbert spaces.- 25. Basis for A2.- 26. Real functions in H2.- 27. Products in H2.- 28. Analytic characterization of H2.- 29. Functional Hilbert spaces.- 30. Kernel functions.- 31. Continuity of extension.- 32. Radial limits.- 33. Bounded approximation.- 34. Multiplicativity of extension.- 35. Dirichlet problem.- 4. Infinite Matrices.- 36. Column-finite matrices.- 37. Schur test.- 38. Hilbert matrix.- 5. Boundedness and Invertibility.- 39. Boundedness on bases.- 40. Uniform boundedness of linear transformations.- 41. Invertible transformations.- 42. Preservation of dimension.- 43. Projections of equal rank.- 44. Closed graph theorem.- 45. Unbounded symmetric transformations.- 6. Multiplication Operators.- 46. Diagonal operators.- 47. Multiplications on l2.- 48. Spectrum of a diagonal operator.- 49. Norm of a multiplication.- 50. Boundedness of multipliers.- 51. Boundedness of multiplications.- 52. Spectrum of a multiplication.- 53. Multiplications on functional Hilbert spaces.- 54. Multipliers of functional Hilbert spaces.- 7. Operator Matrices.- 55. Commutative operator determinants.- 56. Operator determinants.- 57. Operator determinants with a finite entry.- 8. Properties of Spectra.- 58. Spectra and conjugation.- 59. Spectral mapping theorem.- 60. Similarity and spectrum.- 61. Spectrum of a product.- 62. Closure of approximate point spectrum.- 63. Boundary of spectrum.- 9. Examples of Spectra.- 64. Residual spectrum of a normal operator.- 65. Spectral parts of a diagonal operator.- 66. Spectral parts of a multiplication.- 67. Unilateral shift.- 68. Bilateral shift.- 69. Spectrum of a functional multiplication.- 70. Relative spectrum of shift.- 71. Closure of relative spectrum.- 10. Spectral Radius.- 72. Analyticity of resolvents.- 73. Non-emptiness of spectra.- 74. Spectral radius.- 75. Weighted shifts.- 76. Similarity of weighted shifts.- 77. Norm and spectral radius of a weighted shift.- 78. Eigenvalues of weighted shifts.- 79. Weighted sequence spaces.- 80. One-point spectrum.- 81. Spectrum of a direct sum.- 82. Reid's inequality.- 11. Norm Topology.- 83. Metric space of operators.- 84. Continuity of inversion.- 85. Continuity of spectrum.- 86. Semicontinuity of spectrum.- 87. Continuity of spectral radius.- 12. Strong and Weak Topologies.- 88. Topologies for operators.- 89. Continuity of norm.- 90. Continuity of adjoint.- 91. Continuity of multiplication.- 92. Separate continuity of multiplication.- 93. Sequential continuity of multiplication.- 94. Increasing sequences of Hermitian operators.- 95. Square roots.- 96. Infimum of two projections.- 13. Partial Isometries.- 97. Spectral mapping theorem for normal operators.- 98. Partial isometries.- 99. Maximal partial isometries.- 100. Closure and connectedness of partial isometries.- 101. Rank, co-rank, and nullity.- 102. Components of the space of partial isometries.- 103. Unitary equivalence for partial isometries.- 104. Spectrum of a partial isometry.- 105. Polar decomposition.- 106. Maximal polar representation.- 107. Extreme points.- 108. Quasinormal operators.- 109. Density of invertible operators.- 110. Connectedness of invertible operators.- 14. Unilateral Shift.- 111. Reducing subspaces of normal operators.- 112. Products of symmetries.- 113. Unilateral shift versus normal operators.- 114. Square root of shift.- 115. Commutant of the bilateral shift.- 116. Commutant of the unilateral shift.- 117. Commutant of the unilateral shift as limit.- 118. Characterization of isometries.- 119. Distance from shift to unitary operators.- 120. Reduction by the unitary part.- 121. Shifts as universal operators.- 122. Similarity to parts of shifts.- 123. Wandering subspaces.- 124. Special invariant subspaces of the shift.- 125. Invariant subspaces of the shift.- 126. Cyclic vectors.- 127. The F. and M. Riesz theorem.- 128. The F. and M. Riesz theorem generalized.- 129. Reducible weighted shifts.- 15. Compact Operators.- 130. Mixed continuity.- 131. Compact operators.- 132. Diagonal compact operators.- 133. Normal compact operators.- 134. Kernel of the identity.- 135. Hilbert-Schmidt operators.- 136. Compact versus Hilbert-Schmidt.- 137. Limits of operators of finite rank.- 138. Ideals of operators.- 139. Square root of a compact operator.- 140. Fredholm alternative.- 141. Range of a compact operator.- 142. Atkinson's theorem.- 143. Weyl's theorem.- 144. Perturbed spectrum.- 145. Shift modulo compact operators.- 146. Bounded Volterra kernels.- 147. Unbounded Volterra kernels.- 148. The Volterra integration operator.- 149. Skew-symmetric Volterra operator.- 150. Norm 1, spectrum {1}.- 151. Donoghue lattice.- 16. Subnormal Operators.- 152. The Putnam-Fuglede theorem.- 153. Spectral measure of the unit disc.- 154. Subnormal operators.- 155. Minimal normal extensions.- 156. Similarity of subnormal operators.- 157. Spectral inclusion theorem.- 158. Filling in holes.- 159. Extensions of finite co-dimension.- 160. Hyponormal operators.- 161. Normal and subnormal partial isometries.- 162. Norm powers and power norms.- 163. Compact hyponormal operators.- 164. Powers of hyponormal operators.- 165. Contractions similar to unitary operators.- 17. Numerical Range.- 166. The Toeplitz-Hausdorff theorem.- 167. Higher-dimensional numerical range.- 168. Closure of numerical range.- 169. Spectrum and numerical range.- 170. Quasinilpotence and numerical range.- 171. Normality and numerical range.- 172. Subnormality and numerical range.- 173. Numerical radius.- 174. Normaloid, convexoid, and spectraloid operators.- 175. Continuity of numerical range.- 176. Power inequality.- 18. Unitary Dilations.- 177. Unitary dilations.- 178. Unitary power dilations.- 179. Ergodic theorem.- 180. Spectral sets.- 181. Dilations of positive definite sequences.- 19. Commutators of Operators.- 182. Commutators.- 183. Limits of commutators.- 184. The Kleinecke-Shirokov theorem.- 185. Distance from a commutator to the identity.- 186. Operators with large kernels.- 187. Direct sums as commutators.- 188. Positive self-commutators.- 189. Projections as self-commutators.- 190. Multiplicative commutators.- 191. Unitary multiplicative commutators.- 192. Commutator subgroup.- 20. Toeplitz Operators.- 193. Laurent operators and matrices.- 194. Toeplitz operators and matrices.- 195. Toeplitz products.- 196. Spectral inclusion theorem for Toeplitz operators.- 197. Analytic Toeplitz operators.- 198. Eigenvalues of Hermitian Toeplitz operators.- 199. Spectrum of a Hermitian Toeplitz operator.- Hint.- 1. Vectors and Spaces.-…