Operator Theory by Example

529 Pages·2023·6.03 MB·English

by  Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

Operator Theory by Example
OxfordGraduateTextsinMathematics
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OXFORDGRADUATETEXTSINMATHEMATICS
1. Keith Hannabuss: An Introduction to Quantum Theory
2. Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis
3. James G. Oxley: Matroid Theory
4. N. J. Hitchin, G. B. Segal, and R. S. Ward: Integrable Systems: Twistors, Loop Groups, and Riemann
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5. Wulf Rossmann: Lie Groups: An Introduction Through Linear Groups
6. Qing. Liu: Algebraic Geometry and Arithmetic Curves
7. Martin R. Bridson and Simon M. Salamon (eds): Invitations to Geometry and Topology
8. Shmuel Kantorovitz: Introduction to Modern Analysis
9. Terry Lawson: Topology: A Geometric Approach
10. Meinolf Geck: An Introduction to Algebraic Geometry and Algebraic Groups
11. Alastair Fletcher and Vladimir Markovic: Quasiconformal Maps and Teichmu¨ller Theory
12. Dominic Joyce: Riemannian Holonomy Groups and Calibrated Geometry
13. Fernando Villegas: Experimental Number Theory
14. Péter Medvegyev: Stochastic Integration Theory
15. Martin A. Guest: From Quantum Cohomology to Integrable Systems
16. Alan D. Rendall: Partial Differential Equations in General Relativity
17. Yves Félix, John Oprea, and Daniel Tanré: Algebraic Models in Geometry
18. Jie Xiong: Introduction to Stochastic Filtering Theory
19. Maciej Dunajski: Solitons, Instantons, and Twistors
20. Graham R. Allan: Introduction to Banach Spaces and Algebras
21. James Oxley: Matroid Theory, Second Edition
22. Simon Donaldson: Riemann Surfaces
23. Clifford Henry Taubes: Differential Geometry: Bundles, Connections, Metrics and Curvature
24. Gopinath Kallianpur and P. Sundar: Stochastic Analysis and Diffusion Processes
25. Selman Akbulut: 4-Manifolds
26. Fon-Che Liu: Real Analysis
27. Dusa McDuff and Dietmar Salamon: Introduction to Symplectic Topology, Third Edition
28. Chris Heunen, Jamie Vicary: Categories for Quantum Theory: An Introduction
29. Shmuel Kantorovitz, Ami Viselter: Introduction to Modern Analysis, Second Edition
30. Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross: Operator Theory by Example

Operator Theory
by Example
STEPHAN RAMON GARCIA
W.M.KeckDistinguishedServiceProfessorandChair
oftheDepartmentofMathematicsandStatistics,PomonaCollege
JAVAD MASHREGHI
Professor,LavalUniversity
WILLIAM T. ROSS
RichardsonProfessorofMathematics,UniversityofRichmond
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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CONTENTS
Preface xv
Notation xvii
Overview xxi
1 HilbertSpaces... ... ... .. ... ... ... .. ... ... .. ... ... ... .. ... . 1
1.1 EuclideanSpace 1
1.2 TheSequenceSpaceℓ2 8
1.3 TheLebesgueSpace𝐿2[0,1] 10
1.4 AbstractHilbertSpaces 16
1.5 TheGram–SchmidtProcess 20
1.6 OrthonormalBasesandTotalOrthonormalSets 21
1.7 OrthogonalProjections 22
1.8 BanachSpaces 25
1.9 Notes 28
1.10 Exercises 29
1.11 HintsfortheExercises 39
2 DiagonalOperators ... ... ... .. ... ... .. ... ... ... .. ... ... .. .. 41
2.1 DiagonalOperators 41
2.2 Banach-SpaceInterlude 47
2.3 InverseofanOperator 48
2.4 SpectrumofanOperator 52
2.5 CompactDiagonalOperators 55
2.6 CompactSelfadjointOperators 57
2.7 Notes 60
2.8 Exercises 62
2.9 HintsfortheExercises 66
3 InfiniteMatrices ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. . 69
3.1 AdjointofanOperator 69
3.2 SpecialCaseofSchur’sTest 74
3.3 Schur’sTest 77
3.4 CompactnessandContractions 80
3.5 Notes 82
3.6 Exercises 83
3.7 HintsfortheExercises 91