Functional Analysis: Spectral Theory V.S. Sunder Institute of Mathematical Sciences Madras 600113 INDIA July 31, 2000 i ha ha ii ha ha iii ha ha iv ha ha v Preface This book grew out of a course of lectures on functional anal- ysis that the author gave during the winter semester of 1996 at the Institute of Mathematical Sciences, Madras. The one dif- ference between the course of lectures and these notes stems from the fact that while the audience of the course consisted of somewhat mature students in the Ph.D. programme, this book is intended for Master’s level students in India; in particular, an attempt has been made to fill in some more details, as well as to include a somewhat voluminous Appendix, whose purpose is to (at least temporarily) fill in the possible gaps in the background of the prospective Master’s level reader. The goal of this book is to begin with the basics of normed linear spaces, quickly specialise to Hilbert spaces and to get to the spectral theorem for (bounded as well as unbounded) oper- ators on separable Hilbert space. The first couple of chapters are devoted to basic proposi- tions concerning normed vector spaces (including the usual Ba- nach space results - such as the Hahn-Banach theorems, the Open Mapping theorem, Uniform boundedness principle, etc.) and Hilbert spaces (orthonormal bases, orthogonal complements, projections, etc.). The third chapter is probably what may not usually be seen in a first course on functional analysis; this is no doubt influenced by the author’s conviction that the only real way to understand the spectral theorem is as a statement concerning representations of commutative C∗-algebras. Thus, this chapter begins with the standard Gelfand theory of commutative Banach algebras, and proceeds to the Gelfand-Naimark theorem on commutative C∗-algebras; this is then followed by a discussion of represen- tations of (not necessarily commutative) C∗-algebras (including the GNS construction which establishes the correspondence be- tween cyclic representations and states on the C∗-algebra, as well as the so-called ‘non-commutative Gelfand Naimark theorem’ which asserts that C∗-algebras admit faithful representations on Hilbert space); the final section of this chapter is devoted to the Hahn-Hellinger classification of separable representations of a commutative C∗-algebra (or equivalently, the classification of vi separable spectral measures on the Borel σ-algebra of a compact Hausdorff space). The fourth chapter is devoted to more standard ‘operator the- ory’ in Hilbert space. To start with, the traditonal form of the spectral theorem for a normal operator on a separable Hilbert space is obtained as a special case of the theory discussed in Chapter 3; this is followed by a discussion of the polar decompo- sition of operators; we then discuss compact operators and the spectral decomposition of normal compact operators, as well as the singular value decomposition of general compact operators. The final section of this chapter is devoted to the classical facts concerning Fredholm operators and their ‘index theory’. The fifth and final chapter is a brief introduction to the the- ory of unbounded operators on Hilbert space; in particular, we establish the spectral and polar decomposition theorems. A fairly serious attempt has been made at making the treat- ment almost self-contained. There are seven sections in the Ap- pendix which are devoted to: (a) discussing and establishing some basic results in each of the following areas: linear algebra, transfinite considerations (including Zorn’s lemma and a ‘naive’ treatment of infinite cardinals), general topology, compact and locally compact Hausdorff spaces, and measure theory; and (b) a proof of the Stone-Weierstrass theorem, and finally, a statement of the Riesz Representation theorem (on measures and continu- ous functions). Most statements in the appendix are furnished with proofs, the exceptions to this being the sections on measure theory and the Riesz representation theorem. The intended objective of the numerous sections in the Ap- pendix is this: if a reader finds that (s)he does not know some ‘elementary’ fact from, say linear algebra, which seems to be used in the main body of the book, (s)he can go to the perti- nent section in the Appendix, to attempt a temporary stop-gap filler. The reader who finds the need to pick up a lot of ‘back- ground material’ from some section of the Appendix, should, at the earliest opportunity, try to fill in the area in which a lacuna in one’s training has been indicated. In particular, it should be mentioned that the treatment in Chapter 3 relies heavily on var- ious notions from measure theory, and the reader should master these prerequisites before hoping to master the material in this vii chapter. A few ’standard’ references have been listed in the brief bibliography; these should enable the reader to fill in the gaps in the appendix. Since the only way to learn mathematics is to do it, the book is punctuated with a lot of exercises; often, the proofs of some propositions in the text are relegated to the exercises; further, the results stated in the exercises are considered to be on the same footing as ‘properly proved’ propositions, in the sense that we have freely used the statements of exercises in the subsequent treatment in the text. Most exercises, whose solutions are not immediate, are furnished with fairly elaborate hints; thus, a stu- dent who is willing to sit down with pen and paper and ‘get her hands dirty’ should be able to go through the entire book without too much difficulty. Finally, all the material in this book is very ‘standard’ and no claims for originality are made; on the contrary, we have been heavily influenced by the treatment of [Sim] and [Yos] in various proofs; thus, for instance, the proofs given here for the Open Mapping Theorem, Urysohn’s lemma and Alexander’s sub- base theorem, are more or less the same as the ones found in [Sim], while the proofs of the Weierstrass as well as the Stone- Weierstrass theorems are almost identical to the ones in [Yos]; furthermore, the treatment in A.2 has been influenced a little by [Hal], but primarily, the proofs§ of this section on cardinal num- bers are a specialisation of the proofs in [MvN] for corresponding statements in a more general context. viii Acknowledgements : The author would like to thank the Institute of Mathematical Sciences for the excellent facilities it provided him during his preparation of these notes; he also wishes to thank the students of his class - in particular, M. Rajesh - for having made the process of giving the course an enjoyable experience; he would also like to thank R. Bhatia and M. Krishna for their constant encouragement in this venture. Finally, the author would like to thank Paul Halmos for hav- ing taught him most of this material in the first instance, for hav- ing been a great teacher, and later, colleague and friend. This book is fondly dedicated to him and to the author’s parents (for reasons too numerous to go into here). Contents 1 Normed spaces 1 1.1 Vectorspaces .................... 1 1.2 Normedspaces ................... 4 1.3 Linearoperators. 8 1.4 The Hahn-Banach theorem . 13 1.5 Completeness .................... 16 1.6 Some topological considerations . 28 2 Hilbert spaces 37 2.1 InnerProductspaces . 37 2.2 Somepreliminaries . 41 2.3 Orthonormalbases . 46 2.4 The Adjoint operator . 56 2.5 Strong and weak convergence . 67 3 C∗-algebras 79 3.1 Banach algebras . 79 3.2 Gelfand-Naimark theory . 93 3.3 Commutative C∗-algebras . 103 3.4 Representations of C∗-algebras . 117 3.5 The Hahn-Hellinger theorem . 128 4 Some Operator theory 147 4.1 Thespectraltheorem . 147 4.2 Polar decomposition . 151 4.3 Compact operators . 158 4.4 Fredholm operators and index . 171 ix x CONTENTS 5 Unbounded operators 189 5.1 Closed operators . 189 5.2 Symmetric and Self-adjoint operators . 197 5.3 Spectral theorem and polar decomposition . 204 A Appendix 217 A.1 Some linear algebra . 217 A.2 Transfinite considerations . 233 A.3 Topological spaces . 241 A.4 Compactness .................... 250 A.5 Measure and integration . 265 A.6 The Stone-Weierstrass theorem . 287 A.7 The Riesz Representation theorem . 299 Chapter 1 Normed spaces 1.1 Vector spaces We begin with the fundamental notion of a vector space. Definition 1.1.1 A vector space is a (non-empty) set V that comes equipped with the following structure: (a) There exists a mapping V V V , denoted (x, y) x+ y, referred to as vector addition× ,→ which satisfies the following7→ conditions, for arbitrary x, y, z V : ∈ (i) (commutative law) x + y = y + x; (ii) (associative law) (x + y)+ z = x +(y + z); (iii) (additive identity) there exists an element in V, always denoted by 0 (or 0V , if it is important to draw attention to V ), such that x +0= x; (iv) (additive inverses) for each x in V, there exists an ele- ment of V , denoted by x, such that x +( x) = 0. − − (b) There exists a map C V V , denoted by (α, x) αx, referred to as scalar multiplication× → , which satisfies the7→ follow- ing conditions, for arbitrary x, y V, α,β C : ∈ ∈ (i) (associative law) α(βx) = (αβ)x; (ii) (distributive law) α(x + y) = αx + αy, and (α + β)x = αx + βx; and (iii) (unital law) 1x = x. 1 2 CHAPTER 1. NORMED SPACES Remark 1.1.2 Strictly speaking, the preceding axioms define what is usually called a ‘vector space over the field of complex numbers’ (or simply, a complex vector space).