Operators on Hilbert Space

107 Pages·2016·1.331 MB·English

by  V. S. Sunder

Texts and Readings in Mathematics 71
V.S. Sunder
Operators on
Hilbert Space
Texts and Readings in Mathematics
Volume 71
Advisory Editor
C.S. Seshadri, Chennai Mathematical Institute, Chennai
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi
Editor
Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur
V. Balaji, Chennai Mathematical Institute, Chennai
R.B. Bapat, Indian Statistical Institute, New Delhi
V.S. Borkar, Indian Institute of Technology Bombay, Mumbai
T.R. Ramadas, Chennai Mathematical Institute, Chennai
V. Srinivas, Tata Institute of Fundamental Research, Mumbai
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V.S. Sunder
Operators on Hilbert Space
123
V.S. Sunder
Department ofMathematics
Institute of Mathematical Sciences
Chennai, Tamil Nadu
India
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Preface
This book was born out of a desire to have a brief introduction to op-
erator theory – the spectral theorem (arguably the most important theorem
in Hilbert space theory), polar decomposition, compact operators, trace-class
operators, etc., which would involve a minimum of initial spadework (avoiding
such digressions as, for example, the Gelfand theory of commutative Banach
algebras), and which only needed simple facts from a first semester graduate
course on Functional Analysis. I believe the cleanest formulation of the spec-
traltheorem is asastatementoftheexistenceanduniquenessof appropriately
homeomorphic (continuous and measurable) functional calculi of one or more
pairwise commuting self-adjoint, and more generally normal, operators on a
separable Hilbert space, rather than one about spectral measures.
This book may be thought of as a re-take of my earlier book ([Sun]) on
Functional Analysis, but with so many variations as to not really look like a
‘second edition’: the operator algebraic point of view is minimised drastically,
resulting in an essentially operator-theoretic proof of the spectral theorem –
first for self-adjoint, and later for normal, operators. What is probably new
here is what I call the joint spectrum of a family of commuting self-adjoint
operators, a new proof of the Fuglede theorem on the commutant of a nor-
mal operator being *-closed, and the extension of the spectral theorem to a
familyofcommutingnormaloperators.Thethirdchaptercontains,inaddition
to everything in the fourth chapter of [Sun], a section about Hilbert-Schmidt
and trace-class operators, and the duality results involving compact operators,
trace-class operators and all bounded operators.
This book is fondly dedicated to the memory of Paul Halmos.
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Acknowledgements
I wish to record my appreciation of the positive encouragement of Rajendra
Bhatia(theManagingEditoroftheseriesinwhich[Sun]appeared)toconsider
coming up with a second edition but with enough work put in rather than a
sloppy cut-and-paste mish-mash. Even though I have lifted fairly large chunks
from[Sun],Ibelievethereisenoughnewmaterialheretomeritthisbookhaving
a different name rather than be thought of as the second edition of the older
book. I will be remiss if I did not record my gratitude to (i) one of the referees
who displayed tremendous patience in wading through the manuscript in spite
of instances of my sloppiness being everywhere dense in it, and painstakingly
preparedareportthatpointedoutmanyhowlerswhichwerefortunatelycaught
before this appeared in print, and (ii) my long lasting colleagues and friends,
Kesavan and Vijay, for having kindly allowed me to subject them to proof-
reading stints with my typo-prone manuscript.
Finally, it is a pleasure to thank the Institute of Mathematical Sciences for
the wonderful infrastructural facilities and for the congenial atmosphere it has
providedmealltheseyears.ImustalsothanktheDST(anditsformerSecretary
Dr. Ramasami, in particular) for interpreting their rules to permit my use of
theJCBoseFellowshipinawonderfulexampleof‘reasonableaccommodation’
which allowed me to continue being a ‘working mathematician’ in spite of my
not-so-great health.
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Contents
Preface v
Acknowledgements vii
AbouttheAuthor xi
1 Hilbert space 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Inner Product spaces . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Hilbert spaces : examples . . . . . . . . . . . . . . . . . . . . 5
1.4 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Approximate eigenvalues . . . . . . . . . . . . . . . . . . . . 20
1.7 Important classes of operators . . . . . . . . . . . . . . . . . 21
1.7.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.2 Isometric versus Unitary . . . . . . . . . . . . . . . . . . 24
2 The Spectral Theorem 31
2.1 C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Cyclic representations and measures . . . . . . . . . . . . . . 33
2.3 Spectral Theorem for self-adjoint operators . . . . . . . . . . 38
2.4 The spectral subspace for an interval. . . . . . . . . . . . . . 41
2.5 Finitely many commuting self-adjoint operators . . . . . . . 43
2.6 The Spectral Theorem for a normal operator . . . . . . . . . 45
2.7 Several commuting normal operators . . . . . . . . . . . . . . 48
2.7.1 The Fuglede Theorem . . . . . . . . . . . . . . . . . . . 48
2.7.2 Functional calculus for several commuting
normal operators . . . . . . . . . . . . . . . . . . . . . . 51
2.8 Typical uses of the spectral theorem . . . . . . . . . . . . . . 52
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x Contents
3 Beyond normal operators 55
3.1 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Compact operators. . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 von Neumann-Schatten ideals . . . . . . . . . . . . . . . . . . 68
3.3.1 Hilbert-Schmidt operators . . . . . . . . . . . . . . . . . 69
3.3.2 Trace-class operators . . . . . . . . . . . . . . . . . . . . 74
3.3.3 Duality results . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . 79
Appendix 91
Bibliography 95
Index 97
TextsandReadingsinMathematics 99
About the Author
V.S.Sunder isprofessorofmathematicsattheInstituteofMathematicalSciences
(commonly known as MATSCIENCE), Chennai, India. He specialises in subfac-
tors,operatoralgebrasandfunctionalanalysisingeneral.In1996,hewasawarded
theShantiSwarupBhatnagarPrizeforScienceandTechnology,thehighestscience
award inIndia,in themathematical sciences category.He isone ofthe first Indian
operator algebraists. In addition to publishing over 60 papers, he has written six
books including at least three monographs at the graduate level or higher on von
Neumann algebras. One of the books was co-authored with Vaughan Jones, an
operator algebraist, who has received the Fields Medal.
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