Elements of Hilbert Spaces and Operator Theory

528 Pages·2017·2.99 MB·English

by  Harkrishan Lal Vasudeva

Harkrishan Lal Vasudeva
Elements of
Hilbert Spaces
and Operator
Theory
Elements of Hilbert Spaces and Operator Theory
Harkrishan Lal Vasudeva
Elements of Hilbert Spaces
and Operator Theory
With contributions from Satish Shirali
123
Harkrishan LalVasudeva
Indian Institute of Science Education
andResearch
Mohali,Punjab
India
ISBN978-981-10-3019-2 ISBN978-981-10-3020-8 (eBook)
DOI 10.1007/978-981-10-3020-8
LibraryofCongressControlNumber:2016957499
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To
Siddhant, Ashira and Shrayus
Preface
Algebraicandtopologicalstructurescompatiblyplaced onthesameunderlyingset
lead to the notions of topological semigroups, groups and vector spaces, among
others. It is then natural to consider concepts such as continuous homomorphisms
andcontinuouslineartransformationsbetweenabove-saidobjects.Byan‘operator’,
we mean a continuous linear transformation of anormed linear space into itself.
Functional analysis was developed around the turn of the last century by the
pioneeringworkofBanach,Hilbert,vonNeumann,Rieszandothers.Withinafew
years,afteranamazingburstofactivity,itwaswelldevelopedasamajorbranchof
mathematics. It is a unifying framework for many diverse areas such as Fourier
series, differential and integral equations, analytic function theory and analytic
number theory. The subject continues to grow and attracts the attention of some
of the finest mathematicians of the era.
Ageneralisationofthemethodsofvectoralgebraandcalculusmanifestsitselfin
the mathematical concept of a Hilbert space, named after the celebrated mathe-
matician Hilbert. It extends these methods from two-dimensional and
three-dimensionalEuclideanspaces tospaces with anyfinite orinfinite dimension.
These are inner product spaces, which allow the measurement of angles and
lengths; once completed, they possess enough limits in the space so that the
techniquesofanalysiscanbeused.Theirdiverseapplicationsattracttheattentionof
physicists, chemists and engineers alike in good measure.
Chapter 1 establishes notations used in the text and collects results from vector
spaces,metric spaces,Lebesgue integrationandrealanalysis. Noattempthasbeen
made to prove the results included under the above topics. It is assumed that the
readerisfamiliarwiththem.Appropriatereferenceshave,however,beenprovided.
Chapter 2 includes in some details the study of inner product spaces and their
completions. The space L2(X, M, l), where X, M and l denote, respectively, a
nonemptyset,ar-algebraofsubsetsofitandanextendednonnegativereal-valued
measure,hasbeenstudied.Thetheoremofcentralimportanceintheanalysisdueto
Riesz and Fischer, namely that L2(X, M, l) is a complete metric space, has been
proved. So has been the result, namely, the space A(X) of holomorphic functions
defined on a bounded domain X is complete. To make the book useful to
vii
viii Preface
probabilists, statisticians, physicists, chemists and engineers, we have included
many applied topics: Legendre, Hermite, Laguerre polynomials, Rademacher
functions,FourierseriesandPlancherel’stheorem.Suchapplicationsoftheabstract
theory are also of significance for the pure mathematician who wants to know the
origin of the subject. This chapter also contains the study of linear functionals on
Hilbert spaces; more specifically, Riesz Representation Theorem, the dual of a
Hilbert space is itself a Hilbert space and the fact that these spaces constitute
importantexamplesofreflexivenormedlinearspaces.ApplicationsofHilbertspace
theorytodifferentbranchesofmathematics,suchasapproximationtheory(Müntz’
Theorem), measure theory (Radon–Nikodým Theorem), Bergman kernel and
conformal mapping (analytic function theory), are included in Chap. 2.
A major portion of this book is devoted to the study of operators in Hilbert
spaces.ItiscarriedoutinChaps.3and4.ThesetofoperatorsinaHilbertspaceH,
equippedwiththeuniformnorm,isdenotedbyBðHÞ.Somewell-knownclassesof
operators have been defined. Under compact operators, Fredholm theory has been
discussed.TheMeanErgodicTheoremhasbeenprovedasanapplicationattheend
ofChap.3.Spectrumofanoperatoristhekeytotheunderstandingoftheoperator.
Properties of the spectrum of different classes of operators, such as normal oper-
ators,self-adjointoperators,unitaries,isometriesandcompactoperators,havebeen
discussedunderappropriateheadings.Here,thepropertiesofthespectrumspecific
to the class of operators under consideration are studied. A large number of
examples of operators together with their spectrum and its splitting into point
spectrum,continuousspectrum,residualspectrum,approximatepointspectrumand
compression spectrum have been painstakingly worked out. It is expected that the
treatment will aid the understanding of the reader. The treatment of polar decom-
position of an operator is different from the ones available in books. Numerical
range and numerical radius of an operator have been defined. The spectral radius
and the numerical radius of an operator have been compared. Professor Ajit Iqbal
Singh deserves special thanks for the help she rendered while this part was being
written. Spectral theorems, which reveal almost everything about the operators,
havebeenaccordedspecialtreatmentinthetext.Afterprovingthespectraltheorem
for compact normal operators, spectral theorems for self-adjoint operators and
normaloperatorshavebeenproved.Here,wehavebeenguidedbythefundamental
principle of pedagogy that repetition helps in imbibing rather subtle techniques
neededforprovingthespectraltheorems.Abird’seyeviewofinvariantsubspaces
withspecialattentiontotheVolterraoperatorisincluded.Weclosethechapterwith
a brief introduction to unbounded operators.
Chapter 5 contains important theorems followed by applications from Banach
spaces.
The final chapter contains hints and solutions to the 166 problems listed under
various sections. These are over and above the numerous detailed examples scat-
tered all over the text.
Chandigarh, India Harkrishan Lal Vasudeva
Contents
1 Preliminaries ... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Vector Spaces.. .... ..... .... .... .... .... .... ..... .... 1
1.2 Metric Spaces.. .... ..... .... .... .... .... .... ..... .... 5
1.3 Lebesgue Integration. ..... .... .... .... .... .... ..... .... 11
1.4 Zorn’s Lemma . .... ..... .... .... .... .... .... ..... .... 18
1.5 Absolute Continuity . ..... .... .... .... .... .... ..... .... 18
2 Inner Product Spaces .... ..... .... .... .... .... .... ..... .... 21
2.1 Definition and Examples... .... .... .... .... .... ..... .... 21
2.2 Norm of a Vector... ..... .... .... .... .... .... ..... .... 26
2.3 Inner Product Spaces as Metric Spaces.... .... .... ..... .... 34
2.4 The Space L2 (X, M, µ) .. .... .... .... .... .... ..... .... 40
2.5 A Subspace of L2(X, M, µ) .... .... .... .... .... ..... .... 46
2.6 The Hilbert Space A(X) ... .... .... .... .... .... ..... .... 48
2.7 Direct Sum of Hilbert Spaces... .... .... .... .... ..... .... 53
2.8 Orthogonal Complements.. .... .... .... .... .... ..... .... 59
2.9 Complete Orthonormal Sets .... .... .... .... .... ..... .... 78
2.10 Orthogonal Decomposition and Riesz Representation. ..... .... 102
2.11 Approximation in Hilbert Spaces .... .... .... .... ..... .... 123
2.12 Weak Convergence.. ..... .... .... .... .... .... ..... .... 127
2.13 Applications... .... ..... .... .... .... .... .... ..... .... 137
3 Linear Operators.... .... ..... .... .... .... .... .... ..... .... 153
3.1 Basic Definitions ... ..... .... .... .... .... .... ..... .... 153
3.2 Bounded and Continuous Linear Operators .... .... ..... .... 156
3.3 The Algebra of Operators.. .... .... .... .... .... ..... .... 167
3.4 Sesquilinear Forms.. ..... .... .... .... .... .... ..... .... 175
3.5 The Adjoint Operator..... .... .... .... .... .... ..... .... 182
3.6 Some Special Classes of Operators... .... .... .... ..... .... 192
3.7 Normal, Unitary and Isometric Operators.. .... .... ..... .... 205
ix
x Contents
3.8 Orthogonal Projections.... .... .... .... .... .... ..... .... 216
3.9 Polar Decomposition. ..... .... .... .... .... .... ..... .... 222
3.10 An Application. .... ..... .... .... .... .... .... ..... .... 229
4 Spectral Theory and Special Classes of Operators.. .... ..... .... 233
4.1 Spectral Notions.... ..... .... .... .... .... .... ..... .... 233
4.2 Resolvent Equation and Spectral Radius... .... .... ..... .... 238
4.3 Spectral Mapping Theorem for Polynomials.... .... ..... .... 242
4.4 Spectrum of Various Classes of Operators . .... .... ..... .... 248
4.5 Compact Linear Operators . .... .... .... .... .... ..... .... 263
4.6 Hilbert–Schmidt Operators. .... .... .... .... .... ..... .... 279
4.7 The Trace Class .... ..... .... .... .... .... .... ..... .... 285
4.8 Spectral Decomposition for Compact Normal Operators.... .... 294
4.9 Spectral Measure and Integral... .... .... .... .... ..... .... 305
4.10 Spectral Theorem for Self-adjoint Operators.... .... ..... .... 317
4.11 Spectral Mapping Theorem For Bounded Normal Operators .... 331
4.12 Spectral Theorem for Bounded Normal Operators ... ..... .... 337
4.13 Invariant Subspaces . ..... .... .... .... .... .... ..... .... 343
4.14 Unbounded Operators..... .... .... .... .... .... ..... .... 351
5 Banach Spaces.. .... .... ..... .... .... .... .... .... ..... .... 373
5.1 Definition and Examples... .... .... .... .... .... ..... .... 373
5.2 Finite-Dimensional Spaces and Riesz Lemma... .... ..... .... 384
5.3 Linear Functionals and Hahn–Banach Theorem . .... ..... .... 393
5.4 Baire Category Theorem and Uniform Boundedness
Principle.. .... .... ..... .... .... .... .... .... ..... .... 401
5.5 Open Mapping and Closed Graph Theorems ... .... ..... .... 409
6 Hints and Solutions.. .... ..... .... .... .... .... .... ..... .... 417
6.1 Problem Set 2.1 .... ..... .... .... .... .... .... ..... .... 417
6.2 Problem Set 2.2 .... ..... .... .... .... .... .... ..... .... 418
6.3 Problem Set 2.3 .... ..... .... .... .... .... .... ..... .... 422
6.4 Problem Set 2.4 .... ..... .... .... .... .... .... ..... .... 427
6.5 Problem Set 2.5 .... ..... .... .... .... .... .... ..... .... 428
6.6 Problem Set 2.6 .... ..... .... .... .... .... .... ..... .... 428
6.7 Problem Set 2.8 .... ..... .... .... .... .... .... ..... .... 429
6.8 Problem Set 2.9 .... ..... .... .... .... .... .... ..... .... 437
6.9 Problem Set 2.10 ... ..... .... .... .... .... .... ..... .... 442
6.10 Problem Set 2.11 ... ..... .... .... .... .... .... ..... .... 448
6.11 Problem Set 2.12 ... ..... .... .... .... .... .... ..... .... 450
6.12 Problem Set 3.2 .... ..... .... .... .... .... .... ..... .... 454
6.13 Problem Set 3.3 .... ..... .... .... .... .... .... ..... .... 462
6.14 Problem Set 3.4 .... ..... .... .... .... .... .... ..... .... 465
6.15 Problem Set 3.5 .... ..... .... .... .... .... .... ..... .... 466
6.16 Problem Set 3.6 .... ..... .... .... .... .... .... ..... .... 466
6.17 Problem Set 3.7 .... ..... .... .... .... .... .... ..... .... 470
Contents xi
6.18 Problem Set 3.8 .... ..... .... .... .... .... .... ..... .... 475
6.19 Problem Set 3.9 .... ..... .... .... .... .... .... ..... .... 478
6.20 Problem Set 4.1 .... ..... .... .... .... .... .... ..... .... 479
6.21 Problem Set 4.2 .... ..... .... .... .... .... .... ..... .... 485
6.22 Problem Set 4.4 .... ..... .... .... .... .... .... ..... .... 487
6.23 Problem Set 4.5 .... ..... .... .... .... .... .... ..... .... 487
6.24 Problem Set 4.6 .... ..... .... .... .... .... .... ..... .... 504
6.25 Problem Set 4.7 .... ..... .... .... .... .... .... ..... .... 507
6.26 Problem Set 4.8 .... ..... .... .... .... .... .... ..... .... 511
6.27 Problem Set 4.9 .... ..... .... .... .... .... .... ..... .... 512
References.... .... .... .... ..... .... .... .... .... .... ..... .... 515
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 517