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 ©SpringerNatureSingaporePteLtd.2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore 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