Linear Operators in Hilbert Spaces

The purpose of this book is to give an introduction to the theory of linear operators on Hilbert spaces and then to proceed to the interesting applications of differential operators to mathematical physics. Besides the usual introductory courses common to both mathematicians and physicists, only a fundamental knowledge of complex analysis and of ordinary differential equations is assumed. The most important results of Lebesgue integration theory, to the extent that they are used in this book, are compiled with complete proofs in Appendix A. I hope therefore that students from the fourth semester on will be able to read this book without major difficulty. However, it might also be of some interest and use to the teaching and research mathematician or physicist, since among other things it makes easily accessible several new results of the spectral theory of differential operators. In order to limit the length of the text, I present the results of abstract functional analysis only insofar as they are significant for this book. I prove those theorems (for example, the closed graph theorem) that also hold in more general Banach spaces by Hilbert space methods whenever this leads to simplification. The typical concepts of Hilbert space theory, "orthogonal" and "self-adjoint," stand clearly at the center. The spectral theorem for self-adjoint operators and its applications are the central topics of this book. A detailed exposition of the theory of expansions in terms of generalized eigenfunctions and of the spectral theory of ordinary differential operators (Weyl-Titchmarsh-Kodaira) was not possible within the framework of this book. In the first three chapters pre-Hilbert spaces and Hilbert spaces are introduced, and their basic geometric and topologic properties are proved. Chapters 4 and 5 contain the fundamentals of the theory of (not necessarily bounded) linear operators on Hilbert spaces, including general spectral theory. Besides the numerous examples scattered throughout the text, in Chapter 6 certain important classes of linear operators are studied in detail. Chapter 7 contains the spectral theory of self-adjoint operators (first for compact operators, and then for the general case), as well as some important consequences and a detailed characterization of the spectral points. In Chapter 8 von Neumann's extension theory for symmetric operators is developed and is applied to, among other things, the Sturm-Liouville operators. Chapter 9 provides some important results of perturbation theory for self-adjoint operators. Chapter 10 begins with proofs of the most significant facts about Fourier transforms in L2(Mm)> applications to partial differential operators, in particular to Schrodinger and Dirac operators, follow. Finally, Chapter 11 gives a short introduction to (time dependent) scattering theory with some typical results; to my regret, I could only touch upon the far reaching results of recent years. Exercises are not used later in the text, with a few exceptions. They mainly serve to deepen understanding of the material and give opportunity for practice; however, I often use them to formulate further results which I cannot treat in the text. The level of difficulty of the exercises varies widely. Because I give many exercises with detailed hints, they can be solved in general without much difficulty. Now I want to very heartily thank all those who helped me with the production of this book. Mrs. Hose turned my notes into an excellent typed manuscript with infinite diligence. Messrs. R. Hollstein, D. Keim and H. Kuch spent much time reading the whole manuscript and discussing with me their suggestions for improvement. Messrs. R. Colgen and W. Stork helped me with the proofreading. I thank the publisher and the editors for their pleasant cooperation. My teacher Konrad Jorgens inspired me to study this subject; he influenced the present exposition in several ways. I dedicate this volume to his memory.