希尔伯特空间问题集

　　This book was written for the active reader. The first part consists of problems， frequently preceded by definitions and motivation， and some-times followed by corollaries and historical remarks. Most of the problems are statements to be proved， but some are questions （is it？， what is？）， and some are challenges （construct， determine）. The second part， a very short one， consists of hints. A hint is a word， or a paragraph， usually intended to help the reader find a solution. The hint itself is not necessarily a con-densed solution of the problem; it may just point to what I regard as the heart of the matter. Sometimes a problem contains a trap， and the hint may serve to chide the reader for rushing in too recklessly. The third part， the longest， consists of solutions： proofs， answers， or constructions， depending on the nature of the problem

暂缺《希尔伯特空间问题集》作者简介

1.VECTORS
1.Limitsofquadraticforms
2.Schwarzinequality
3.Representationoflinearfunctionals
4.Strictconvexity
5.Continuouscurves
6.Uniquenessofcrinkledarcs
7.Lineardimension
8.Totalsets
9.Infinitelytotalsets
10.InfiniteVandermondes
11.T-totalsets
12.Approximatebases
2.SPACES
13.Vectorsums
14.Latticeofsubspaces
15.Vectorsumsandthemodularlaw
16.Localcompactnessanddimension
17.Separabilityanddimension
18.MeasureinHiibertspace
3.WEAKTOPOLOGY
19.Weakclosureofsubspaces
20.Weakcontinuityofnormandinnerproduct
21.Semicontinuityofnorm
22.Weakseparability
23.Weakcompactnessoftheunitbali
24.Weakmetrizabilitvoftheunitball
25.Weakclosureoftheunitsphere
26.Weakmetrizabilityandseparability
27.Uniformboundedness
28.WeakmetrizabilityofHilbertspace
29.Linearfunctionalson/2
30.Weakcompleteness
4.ANALYTICFUNCTIONS
31.AnalyticHilbertspaces
32.BasisforAe
33.RealfunctionsinHe
34.ProductsinH2
35.AnalyticcharacterizationofH2
36.FunctionalHilbcrtspaces
37.Kernelfunctions
38.ConjugationinfunctionalHilbertspaces
39.Continuityofextension
40.Radiallimits
41.Boundedapproximation
42.Multiplicativityofextension
43.Dirichletproblem
5.INFINITEMATRICES
44.Column-finitematrices
45.Schurtest
46.Hilbertmatrix
47.ExponentialHilbertmatrix
48.PositivityoftheHilbertmatrix
49.Seriesofvectors
6.BOUNDEDNESSANDINVERTIBILITY
50.Boundednessonbases
51.Uniformboundednessoflineartransformations
52.lnvertibletransformations
53.Diminishablecomplements
54.Dimensionininner-productspaces
55.Totalorthonormalsets
56.Preservationofdimension
57.Projectionsofequalrank
58.Closedgraphtheorem
59.Rangeinclusionandfactorization
60.Unboundedsymmetrictransformations
7.MULTIPLICATIONOPERATORS
61.Diagonaloperators
62.Multiplicationson12
63.Spectrumofadiagonaloperator
64.Normofamultiplication
65.Boundednessofmultipliers
66.Boundednessofmultiplications
67.Spectrumofamultiplication
68.MultiplicationsonfunctionalHilbertspaces
69.MultipliersoffunctionalHiibertspaces
8.OPERATORMATRICES
70.Commutativeoperatordeterminants
71.Operatordeterminants
72.Operatordeterminantswithafiniteentry
9.PROPERTIESOFSPECTRA
73.Spectraandconjugation
74.Spectralmappingtheorem
75.Similarityandspectrum
76.Spectrumofaproduct
77.Closureofapproximatepointspectrum
78.Boundaryofspectrum
10.EXAMPLESOFSPECTRA
79.Residualspectrumofanormaloperator
80.Spectralpartsofadiagonaloperator
81.Spectralpartsofamultiplication
82.Unilateralshift
83.Structureofthesetofeigenvectors
84.Bilateralshift
85.Spectrumofafunctionalmultiplication
11.SPECTRALRADIUS
86.Analyticityofresolvents
87.Non-emptinessofspectra
88.Spectralradius
89.Weightedshifts
90.Similarityofweightedshifts
91.Normandspectralradiusofaweightedshift
92.Powernorms
93.Eigenvaluesofweightedshifts
94.Approximatepointspectrumofaweightedshift
95.Weightedsequencespaces
96.One-pointspectrum
97.Analyticquasinilpotents
98.Spectrumofadirectsum
12.NORMTOPOLOGY
99.Metricspaceofoperators
100.Continuityofinversion
101.Interiorofconjugateclass
102.Continuityofspectrum
103.Semicontinuityofspectrum
104.Continuityofspectralradius
105.Normalcontinuityofspectrum
106.Quasinilpotentperturbationsofspectra
13.OPERATORTOPOLOGIES
107.Topologiesforoperators
108.Continuityofnorm
109.Semicontinuityofoperatornorm
110.Continuityofadjoint
111.Continuityofmultiplication
112.Separatecontinuityofmultiplication
113.Sequentialcontinuityofmultiplication
114.Weaksequentialcontinuityofsquaring
115.Weakconvergenceofprojections
14.STRONGOPERATORTOPOLOGY
116.Strongnormalcontinuityofadjoint
117.Strongboundedcontinuityofmultiplication
118.Strongoperatorversusweakvectorconvergence
119.Strongsemicontinuityofspectrum
120.IncreasingsequencesofHermitianoperators
121.Squareroots
122.Infimumoftwoprojections
15.PARTIAL]SOMETRIES
123.Spectralmappingtheoremfornormaloperators
124.Decreasingsquares
125.Polynomiallydiagonaloperators
126.Continuityofthefunctionalcalculus
127.Partialisometrics
128.Maximalpartialisomctries
129.Closureandconnectednessofpartialisometrics
130.Rank，co-rank，andnullity
131.Componentsofthespaceofpartialisometries
132.Unitaryequivalenceforpartialisometries
133.Spectrumofapartialisometry
16.POLARDECOMPOSITION
134.Polardecomposition
135.Maximalpolarrepresentation
136.Extremepoints
137.Quasinormaioperators
138.MixedSehwarzinequality
139.Quasinormalweightedshifts
140.Densityofinvertibleoperators
141.Connectednessofinvertibleoperators
17.UNILATERALSHIFT
142.Reducingsubspacesofnormaloperators
143.Productsofsymmetries
144.Unilateralshiftversusnormaloperators
145.Squarerootofshift
146.Commutantofthebilateralshift
147.Commutantoftheunilateralshift
148.Commutantoftheunilateralshiftaslimit
149.Characterizationofisometrics
150.Distancefromshifttounitaryoperators
151.Squarerootsofshifts
152.Shiftsasuniversaloperators
153.Similaritytopartsofshifts
154.Similaritytocontractions
155.Wanderingsubspaces
156.Specialinvariantsubspacesoftheshift
157.lnvariantsubspacesoftheshift
158.F.andM.Riesztheorem
159.Reducibleweightedshifts
18.CYCLICVECTORS
160.Cyclicvectors
161.Densityofcyclicoperators
162.Densityofnon-cyclicoperators
163.Cyclicityofadirectsum
164.Cyclicvectorsofadjoints
165.Cyclicvectorsofapositionoperator
166.Totalityofcyclicvectors
167.Cyclicoperatorsandmatrices
168.Denseorbits
19.PROPERTIESOFCOMPACTNESS
169.Mixedcontinuity
170.Compactoperators
171.Diagonalcompactoperators
172.Normalcompactoperators
173.Hilbert-Schmidtoperators
174.CompactversusHilbert-Schmidt
175.Limitsofoperatorsoffiniterank
176.Idealsofoperators
177.Compactnessonbases
178.Squarerootofacompactoperator
179.Fredholmalternative
180.Rangeofacompactoperator
181.Atkinsonstheorem
182.Weylstheorem
183.Perturbedspectrum
184.Shiftmodulocompactoperators
185.Distancefromshifttocompactoperators
EXAMPLESOFCOMPACTNESS
186.BoundedVolterrakernels
187.UnboundedVoltcrrakernels
188.Volterraintegrationoperator
189.Skew-symmetricVolterraoperator
190.Norm1，spectrum{1}
191.Donoghuelattice
SUBNORMALOPERATORS
192.Putnam-Fugledetheorem
193.Algebrasofnormaloperators
194.Spectralmeasureoftheunitdisc
195.Subnormaloperators
196.Quasinormalinvariants
197.Minimalnormalextensions
198.Polynomialsintheshift
199.Similarityofsubnormaloperators
200.Spectralinclusiontheorem
201.Fillinginholes
202.Extensionsoffiniteco-dimension
203.Hyponormaloperators
204.Normalandsubnormalpartialisometrics
205.Normpowersandpowernorms
206.Compacthyponormaloperators
207.Hyponormal，compactimaginarypart
208.Hyponormalidempotents
209.Powersofhyponormaloperators
NUMERICALRANGE
210.Toeplitz-Hausdorfftheorem
211.Higher-dimensionalnumericalrange
212.Closureofnumericalrange
213.Numericalrangeoracompactoperator
214.Spectrumandnumericalrange
215.Quasinilpotenceandnumericalrange
216.Normalityandnumericalrange
217.Subnormalityandnumericalrange
218.Numericalradius
219.Normaloid，convexoid，andspectraloidoperators
220.Continuityofnumericalrange
221.Powerinequality
UNITARYDILATIONS
222.Unitarydilations
223.Imagesofsubspaces
224.Weakclosuresanddilations
225.Strongclosuresandextensions
226.Stronglimitsofhyponormaloperators
227.Unitarypowerdilations
228.Ergodictheorem
229.yonNcumannsinequality
24.COMMUTATORS
230.Commutators
231.Limitsofcommutators
232.Kleinecke-Shirokovtheorem
233.Distancefromacommutatortotheidentity
234.Operatorswithlargekernels
235.Directsumsascommutators
236.Positiveserf-commutators
237.Projectionsasself-commutators
238.Multiplicativecommutators
239.Unitarymultiplicativecommutators
240.Commutatorsubgroup
25.TOEPLITZOPERATORS
241.Laurentoperatorsandmatrices
242.Toeplitzoperatorsandmatrices
243.Toeplitzproducts
244.CompactToeplitzproducts
245.SpectralinclusiontheoremforToeplitzoperators
246.ContinuousToeplitzproducts
247.AnalyticToeplitzoperators
248.EigenvaluesofHermitianToeplitzoperators
249.Zero-divisors
250.SpectrumofaHermitianToeplitzoperator
REFERENCES
LISTOFSYMBOLS
INDEX