Operator Theory

A Comprehensive Course in Analysis #4

by Barry Simon

In the second half of 2015, the American Math Society will publish a five volume (total about 3000 pages) set of books that is a graduate analysis text with lots of additional bonus material. Included are hundreds of problems and copious notes which extend the text and provide historical background. Efforts have been made to find simple and elegant proofs and to keeping the writing style clear.


Eigenvalue Perturbation Theory, Operator Basics, Compact Operators, Orthogonal Polynomials, Spectral Theory, Banach Algebras, Unbounded Self-Adjoint Operators.

Selected topics include analytic functional calculus, polar decomposition, Hilbert-Schmidt and Riesz-Schauder theorems, Ringrose structure theorems, trace ideals, trace and determinant, Lidskii’s theorem, index theory for Fredholm operators, OPRL, OPUC, Bochner-Brenke theorem, Chebyshev polnomials, spectral measures, spectral multiplicity theory, trace class perturbations and Krein spectral shift, Gel’fand transform, Gel’fand-Naimark theorems, almost periodic functions, Gel’fand-Raikov and Peter-Weyl theorems, Fourier analysis on LCA groups, Wiener and Ingham tauberian theorems and the prime number theorem, Spectral and Stone’s theorem for unbounded self-adjoint operators, von Neumann theory of self-adjoint extensions, quadratic forms, Birman-Krein-Vershik theory of self adjoint extensions, Kato’s inequality, Beurling-Deny theorems, moment problems, Birman-Schwinger principle.

749 pages, Hardcover

First published December 21, 2015

About the author

Barry Simon is an eminent American mathematical physicist and the IBM Professor of Mathematics and Theoretical Physics (Emeritus) at Caltech, known for his prolific contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics (particularly Schrödinger operators), including the connections to atomic and molecular physics. He has authored more than 300 publications on mathematics and physics.

More particularly, his work has focused on broad areas of mathematical physics and analysis covering: quantum field theory, statistical mechanics, Brownian motion, random matrix theory, general nonrelativistic quantum mechanics (including N-body systems and resonances), nonrelativistic quantum mechanics in electric and magnetic fields, the semi-classical limit, the singular continuous spectrum, random and ergodic Schrödinger operators, orthogonal polynomials, and non-selfadjoint spectral theory.

Dr. Simon is a fellow of the American Mathematical Society (2012), a winner of the Henri Poincaré Prize (2012), a winner of the János Bolyai International Mathematical Prize (2015), a winner of the 2016 Steele Prize for Lifetime Achievement, and a winner of the Dannie Heineman Prize for Mathematical Physics (2018).