Simon B. Operator Theory. A Comprehensive Course in Analysis. Part 4

American Mathematical Society, 2015. — 749 p. — (A Comprehensive Course in Analysis). — ISBN: 978-1-4704-2763-4, 978-1-4704-1103-9.A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level analysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and provide important historical background. Depth and breadth of exposition make this set a valuable reference source for almost all areas of classical analysis.
Part 4 focuses on operator theory, especially on a Hilbert space. Central topics are the spectral theorem, the theory of trace class and Fredholm determinants, and the study of unbounded self-adjoint operators. There is also an introduction to the theory of orthogonal polynomials and a long chapter on Banach algebras, including the commutative and non-commutative Gel'fand-Naimark theorems and Fourier analysis on general locally compact abelian groups.Preliminaries
Operator basics
Compact operators, mainly on a Hilbert space
Orthogonal polynomials
The spectral theorem
Banach algebras
Unbounded self-adjoint operators