Functional analysis: spectral theory (Q1977448)

The book was written on the base of the course delivered by the author to the Ph.D. programme students at the Institute of Mathematical Sciences, Madras. The idea of the course as well as its realization in the book is to constitute and present the background of the basic part of modern functional analysis. The style of presentation is very attractive saving the advantages of the lectures behind the interested auditorium and having mainly the form of dialog with the reader. In spite of the subtitle of the book it covers the main results of functional analysis giving the perspective for further developing in its certain directions. It is compactly written and basically self-contained. The book consists of five chapters, an Appendix, a short list of classical books which can help the reader to be more familiar with the results presented, and an Index. Standard bases for any modern book on analysis are constituted by normed spaces and Hilbert spaces. Both theories are discussed here (Ch. 1 and Ch. 2 respectively). The presentation is short and straightforward having in mind the level of the auditorium. Anyway all basic facts are given and all really important questions are discussed. The further discussion leads the author to pay attention to certain properties of operators, in particular to the notion of adjoint operator, to the weak and strong topology of the vector space of linear bounded operators acting from one Hilbert space to another. The central part of the book is Ch. 3 (``\(C^{\ast}\)-algebras''). Here the modern language of the spectral theory of operators is described. Beginning from the very basic definitions in the Banach algebra theory the author then presents the bases for the remarkable theory of ideals on normed algebras (Gelfand-Naimark theory). The real description of \(C^{\ast}\)-algebras, i.e. Banach algebras with an involution, is started with the study of the commutative case allowing to give many attractive examples and to clarify the main facts of this theory. Central among them is in no doubt the representation of an abstract \(C^{\ast}\)-algebra as operators on Hilbert spaces. It is described up to the classification of a separable commutative \(C^{\ast}\)-algebra (Hahn-Hellinger theorem). The next two chapters are devoted to the presentation of the spectral theory of operators (Ch. 4 ``Some Operator Theory'', and Ch. 5 ``Unbounded operators''). They use to apply the material of the preceding chapters in order to study operators on Hilbert spaces. The spectral theorem for bounded operators is the starting point of this study. It is accompanied by the polar decomposition theorem. These results show the way to answer certain questions concerning general bounded operators on Hilbert spaces -- first to decompose the operator and to reduce the question to the case of positive operators, second, to study the later by using the spectral theorem. Compact operators and Hilbert-Schmidt theory are developed later. The results presented allow the author to end with the essential description of the Fredholm operator theory. The main question at the discussion of unbounded operators on Hilbert spaces is how to discover the classes of such operators for which it is possible (and natural!) to develop the spectral theory analogous to that for bounded operators. From this point of view the last chapter is not as complete as the previous ones. But the aim of it is to initiate people to study some problems in this branch rather than to give them a final receipt how to do it. In order to make the text basically self-contained the author presents in the Appendix certain needed facts from linear algebra (up to Zorn's lemma and its consequences), and from the theory of topological vector spaces (including the notion of compactness). Basic constructions and results from Lebesgue integration theory are given, too. Finally, two theorems which are necessarily to be known to any analyst, namely the Stone-Weierstrass and the Riesz representation theorems are described. This book combines the advantages of lecture notes, of a textbook, and of a reference book. It includes the basic facts of modern analysis as well as instructive and far reaching exercises.