Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators

Abstract

This book has its roots in a course I taught for many years at the University of Paris. It is intended for students who have a good background in real analysis (as expounded, for instance, in the textbooks of G. B. Folland [2], A. W. Knapp [1], and H. L. Royden [1]). I conceived a program mixing elements from two distinct “worlds”: functional analysis (FA) and partial differential equations (PDEs).The first part deals with abstract results in FA and operator theory. The second part concerns the study of spaces of functions (of one or more real variables) having specific differentiability properties: the celebrated Sobolev spaces, which lie at the heart of the modern theory of PDEs. I show how the abstract results from FA can be applied to solve PDEs. The Sobolev spaces occur in a wide range of questions, in both pure and applied mathematics. They appear in linear and nonlinear PDEs that arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, and physics. They belong to the toolbox of any graduate student in analysis. Unfortunately, FA and PDEs are often taught in separate courses, even though they are intimately connected. Many questions tackled in FAoriginated in PDEs (for a historical perspective, see, e.g., J. Dieudonné [1] and H. Brezis–F. Browder [1]). There is an abundance of books (even voluminous treatises) devoted to FA. There are also numerous textbooks dealing with PDEs. However, a synthetic presentation intended for graduate students is rare. and I have tried to fill this gap. Students who are often fascinated by the most abstract constructions in mathematics are usually attracted by the elegance of FA. On the other hand, they are repelled by the never- ending PDE formulas with their countless subscripts. I have attempted to present a “smooth” transition from FA to PDEs by analyzing first the simple case of one- dimensional PDEs (i.e., ODEs—ordinary differential equations), which looks much more manageable to the beginner. In this approach, I expound techniques that are possibly too sophisticated for ODEs, but which later become the cornerstones of the PDE theory. This layout makes it much easier for students to tackle elaborate higher-dimensional PDEs afterward. A previous version of this book, originally published in 1983 in French and fol- lowed by numerous translations, became very popular worldwide, and was adopted as a textbook in many European universities.A deficiency of the French text was the lack of exercises. The present book contains a wealth of problems. I plan to add even more in future editions. I have also outlined some recent developments, especially in the direction of nonlinear PDEs.