内容简介

  算子理论在现代数学的许多重要领城诸如泛函分析、微分方程、指标论、表示论、数学物理中充当重要角色。
  《算子理论教程(影印版)》覆盖了算子理论的中心课题,叙述清晰简洁,读者很容易与Conway的写作产生互动。
  《算子理论教程(影印版)》前几章介绍和回顾了C*-代数、正规算子、紧算子和非正规算子,主题包含谱理论、泛函演算和Fredholm指标。此外,还论述了算子理论和解析函数之间某些深刻的联系。后续章节讲述了更高级的主题,包括C*-代数的表示、紧微扰和vonNeumann代数等。重要结果覆盖了诸如Sz.-Nagy伸缩定理、Weyl-vonNeumann-Berg定理和vonNeumann代数的分类,同样也讲述了对Fredholm理论的处理,这些高级论题均处于当今研究的中心。最后一章介绍了自返子空间,即由其不变子空间决定的算子子空间。这些连同超自返空间是现代非对称代数研究中成功的插曲之一。
  Conway教授的处理使《算子理论教程(影印版)》成为一本引人入胜但又相当缜密的教材,适合于已经上过泛函分析标准课程的研究生。

目录

Preface

Chapter1.IntroductiontoC*-Algebras
1.Definitionandexamples
2.AbelianC*-algebrasandtheFunctionalCalculus
3.ThepositiveelementsinaC*-algebra
4.Approximateidentities
5.IdealsinaC*-algebra
6.RepresentationsofaC*-algebra
7.PositivelinearfunctionalsandtheGNSconstruction

Chapter2.NormalOperators
8.SometopologiesonB(H)
9.Spectralmeasures
10.TheSpectralTheorem
11.Star-cyclicnormaloperators
12.Thecommutant
13.VonNeumannalgebras
14.AbelianvonNeumannalgebras
15.Thefunctionalcalculusfornormaloperators

Chapter3.CompactOperators
16.C*-algebrasofcompactoperators
17.Idealsofoperators
18.TraceclassandHilbert-Schmidtoperators
19.Thedualspacesofthecompactoperatorsandthetraceclass
20.Theweak-startopology
21.Inflationandthetopologies

Chapter4.SomeNon-NormalOperators
22.Algebrasandlattices
23.Isometries
24.Unilateralandbilateralshifts
25.SomeresultsonHardyspaces
26.Thefunctionalcalculusfortheunilateralshift
27.Weightedshifts
28.TheVolterraoperator
29.Bergmanoperators
30.Subnormaloperators
31.Essentiallynormaloperators

Chapter5.MoreonC*-Algebras
32.Irreduciblerepresentations
33.Positivemaps
34.Completelypositivemaps
35.Anapplication:SpectralsetsandtheSz.-NagyDilationTheorem
36.Quasicentralapproximateidentitites

Chapter6.CompactPerturbations
37.Behaviorofthespectrumunderacompactperturbation
38.Bpperturbationsofhermitianoperators
39.TheWeyl-vonNeumann-BergTheorem
40.Voiculescu'sTheorem
41.Approximatelyequivalentrepresentations
42.Someapplications

Chapter7.IntroductiontoVonNeumannAlgebras
43.Elementarypropertiesandexamples
44.TheKaplanskyDensityTheorem
45.ThePedersenUp-DownTheorem
46.Normalhomomorphismsandideals
47.Equivalenceofprojections
48.Classificationofprojections
49.Propertiesofprojections
50.ThestructureofTypeIalgebras
51.TheclassificationofTypeIalgebras
52.Operator-valuedmeasurablefunctions
53.Somestructuretheoryforcontinuousalgebras
54.Weak-starcontinuouslinearfunctionalsrevisited
55.Thecenter-valuedtrace

Chapter8.Reflexivity
56.Fundamentalsandexamples
57.Reflexiveoperatorsonfinitedimensionalspaces
58.Hyperreflexivesubspaces
59.Reflexivityandduality
60.HypereflexivevonNeumannalgebras
61.Someexamplesofoperators

Bibliography
Index
ListofSymbols

前言/序言

  ThegenesisofthisbookisapairofcoursesItaught,oneatIndianaUniversityandanotherattheUniversityofTennessee.Bothofthesefollowedastandardtwosemestercourseinfunctionalanalysis,thoughthisbookiswrittenwithonlyaonesemestercourseinfunctionalanalysisasitsprerequisite.Theaimistocoverwithvaryingdepthavarietyofsubjectsthatarecentraltooperatortheory.Manyofthesetopicshavetreatisesdevotedtotheirexplication,andsomecarehasbeentakentoalertthereadertoadditionalsourcesfordeeperstudy.Soyoucanthinkofthisbookasasequeltoabasicfunctionalanalysiscourseorasafoundationforthestudyofoperatortheory.
  Theprerequisitesforthisbookareabitfuzzy.Thereaderisassumedtoknowfundamentalfunctionalanalysis.Specificallyitwillbeassumedthatthereaderknowsthematerialinthefirstsevenchaptersof[ACFAl.(R,epeatedreferenceto[ACFA]ismadeinthistextanditisreferencedinthiswayratherthantheformofotherreferences.)Thedesireistomakethebookasaccessibleaspossible.Itisincumbentonmetoavoidthearroganceofassumingorrequiringfamiliaritywithoneofmypreviousbooks.ButIhavetosetthestandardsomewhere,soIwillassumethereaderknowswhatIbelievetobethematerialcommontoallbasiccoursesinfunctionalanalysis:thethreebasicprinciplesofBanachspaces,thedefinitionandelementarypropertiesoflocallyconvexspaces,thefoundationsofBanachalgebras,includingtheR.ieszfunctionalcalculus,andtherudimentsofoperatortheory,includingthespectraltheoryofcompactoperators.-Thisconstitutesthefirstsevenchaptersof[ACFA],
  ThisbookstartswithanintroductiontoC*-algebrasfollowedbyachapteronnormaloperatorsthatculminateswiththeSpectralTheoremandthefunctionalcalculus.ThereisconsiderableoverlapbetweenthesefirsttwochaptersandChaptersVIIIandIXin[ACFA].Ifthereaderreallyknowsthesefirstninechaptersof[ACFA],he/shewillbeabletoflythroughthefirsttwochaptershere.SuchareadercouldrealisticallybeginreadingChapter3ofthepresentbook.Thereis,however,materialinthefirsttwochaptersofthisbook,specifically4,thatdoesnotappearin[ACFA].

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