El. knyga: 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. Atkinsons theorem.-
182. Weyls 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 Neumanns 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.