Table Of ContentA Short Course on
Spectral Theory
William Arveson
Springer
209
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
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William Arveson
A Short Course on
Spectral Theory
WilliamArveson
DepartmentofMathematics
UniversityofCalifornia,Berkeley
Berkeley,CA94720-0001
USA
[email protected]
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment MathematicsDepartment
SanFranciscoState EastHall UniversityofCalifornia,
University UniversityofMichigan Berkeley
SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840
USA USA USA
MathematicsSubjectClassification(2000):46-01,46Hxx,46Lxx,47Axx,58C40
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Arveson,William.
Ashortcourseonspectraltheory/WilliamArveson.
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ISBN0-387-95300-0(alk.paper)
1.Spectraltheory(Mathematics) I.Title. II.Series.
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To Lee
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Preface
This book presents the basic tools of modern analysis within the context of
what might be called the fundamental problem of operator theory: to cal-
culatespectraofspecificoperatorsoninfinite-dimensionalspaces,especially
operators on Hilbert spaces. The tools are diverse, and they provide the
basisformorerefinedmethodsthatallowonetoapproachproblemsthatgo
well beyond the computation of spectra; the mathematical foundations of
quantum physics, noncommutative K-theory, and the classification of sim-
ple C∗-algebras being three areas of current research activity that require
mastery of the material presented here.
The notion of spectrum of an operator is based on the more abstract
notion of the spectrum of an element of a complex Banach algebra. Af-
ter working out these fundamentals we turn to more concrete problems of
computing spectra of operators of various types. For normal operators, this
amounts to a treatment of the spectral theorem. Integral operators require
the development of the Riesz theory of compact operators and the ideal L2
of Hilbert–Schmidt operators. Toeplitz operators require several important
tools;inordertocalculatethespectraofToeplitzoperatorswithcontinuous
symbol one needs to know the theory of Fredholm operators and index, the
structureoftheToeplitzC∗-algebraanditsconnectionwiththetopologyof
curves, and the index theorem for continuous symbols.
I have given these lectures several times in a fifteen-week course at
Berkeley (Mathematics 206), which is normally taken by first- or second-
yeargraduatestudentswithafoundationinmeasuretheoryandelementary
functional analysis. It is a pleasure to teach that course because many deep
and important ideas emerge in natural ways. My lectures have evolved sig-
nificantlyovertheyears,buthavealwaysfocusedonthenotionofspectrum
and the role of Banach algebras as the appropriate modern foundation for
such considerations. For a serious student of modern analysis, this material
is the essential beginning.
Berkeley, California William Arveson
July 2001
vii
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Contents
Preface vii
Chapter 1. Spectral Theory and Banach Algebras 1
1.1. Origins of Spectral Theory 1
1.2. The Spectrum of an Operator 5
1.3. Banach Algebras: Examples 7
1.4. The Regular Representation 11
1.5. The General Linear Group of A 14
1.6. Spectrum of an Element of a Banach Algebra 16
1.7. Spectral Radius 18
1.8. Ideals and Quotients 21
1.9. Commutative Banach Algebras 25
1.10. Examples: C(X) and the Wiener Algebra 27
1.11. Spectral Permanence Theorem 31
1.12. Brief on the Analytic Functional Calculus 33
Chapter 2. Operators on Hilbert Space 39
2.1. Operators and Their C∗-Algebras 39
2.2. Commutative C∗-Algebras 46
2.3. Continuous Functions of Normal Operators 50
2.4. The Spectral Theorem and Diagonalization 52
2.5. Representations of Banach ∗-Algebras 57
2.6. Borel Functions of Normal Operators 59
2.7. Spectral Measures 64
2.8. Compact Operators 68
2.9. Adjoining a Unit to a C∗-Algebra 75
2.10. Quotients of C∗-Algebras 78
Chapter 3. Asymptotics: Compact Perturbations and Fredholm
Theory 83
3.1. The Calkin Algebra 83
3.2. Riesz Theory of Compact Operators 86
3.3. Fredholm Operators 92
3.4. The Fredholm Index 95
Chapter 4. Methods and Applications 101
4.1. Maximal Abelian von Neumann Algebras 102
ix
