Operator theory is a significant part of many important areas of modern mathematics: functional analysis, differential equations, index theory, representation theory, mathematical physics, and more. This text covers the central themes of operator theory, presented with the excellent clarity and style that readers have come to associate with Conway's writing. Early chapters introduce and review material on C*algebras, normal operators, compact operators and nonnormal operators. The topics include the spectral theorem, the functional calculus and the Fredholm index. Also, some deep connections between operator theory and analytic functions are presented. Later chapters cover more advanced topics, such as representations of C*algebras, compact perturbations and von Neumann algebras. Major results, such as the Sz.Nagy Dilation Theorem, the Weylvon NeumannBerg Theorem and the classification of von Neumann algebras, are covered, as is a treatment of Fredholm theory. These advanced topics are at the heart of current research. The last chapter gives an introduction to reflexive subspaces, i.e., subspaces of operators that are determined by their invariant subspaces. These, along with hyperreflexive spaces, are one of the more successful episodes in the modern study of asymmetric algebras. Professor Conway's authoritative treatment makes this a compelling and rigorous course text, suitable for graduate students who have had a standard course in functional analysis. Readership Graduate students and research mathematicians interested in operator theory. Reviews "John B. Conway belongs to the best authors of basic textbooks ... The present book continues this tradition of clear and elegant way of presentation. ... this book can be highly recommended for students of operator theory as well as to experts in the field who will find many interesting ideas there."  Mathematica Bohemica "Conway's book adds a complementary volume of study for those just becoming acquainted with the field ... shares ... a style which is relaxed, yet concise ... recommend it to anyone wishing to gain a better understanding of operator theory."  Bulletin of the London Mathematical Society "This is an excellent course in operator theory and operator algebras ... leads the reader to deep new results and modern research topics ... the author has done more than just write a good bookhe has managed to reveal the unspeakable charm of the subject, which is indeed the `source of happiness' for operator theorists."  Mathematical Reviews Table of Contents  Introduction to C*algebras
 Normal operators
 Compact operators
 Some nonnormal operators
 More on C*algebras
 Compact perturbations
 Introduction to von Neumann algebras
 Reflexivity
 Bibliography
 Index
 List of symbols
