A Primer on Spectral Theory

by Bernard Aupetit

Overview

This textbook provides an introduction to the new techniques of subharmonic functions and analytic multifunctions in spectral theory. Topics include the basic results of functional analysis, bounded operations on Banach and Hilbert spaces, Banach algebras, and applications of spectral subharmonicity. Each chapter is followed by exercises of varying difficulty. Much of the subject matter, particularly in spectral theory, operator theory and Banach algebras, contains new results.

Product Details

ISBN-13:	9780387973906
Publisher:	Springer New York
Publication date:	12/19/1990
Series:	Universitext
Edition description:	Softcover reprint of the original 1st ed. 1991
Pages:	193
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

I. Some Reminders of Functional Analysis.- II. Some Classes of Operators.- §1. Finite-Dimensional Operators.- §2. Bounded Linear Operators on a Banach Space.- §3. Bounded Linear Operator on a Hilbert Space.- III. Banach Algebras.- §1. Definition and Examples.- §2. Invertible Elements and Spectrum.- §3. Holomorphic Functional Calculus.- §4. Analytic Properties of the Spectrum.- IV. Representation Theory.- §1. Gelfand Theory for Commutative Banach Algebras.- §2. Representation Theory for Non-Commutative Banach Algebras.- V. Some Applications of Subharmonicity.- §1. Some Elementary Applications.- §2. Spectral Characterizations of Commutative Banach Algebras.- §3. Spectral Characterizations of the Radical.- §4. Spectral Characterizations of Finite-Dimensional Banach Algebras.- §5. Automatic Continuity for Banach Algebra Morphisms.- §6. Elements with Finite Spectrum.- §7. Inessential Elements.- VI. Representation of Cσ-algebras and the Spectral Theorem.- §1. Banach Algebras with Involution.- §2. Cσ-algebras.- §3. The Spectral Theorem.- §4. Applications.- VII. An Introduction to Analytic Multifunctions.- §1. Definitions and General Properties.- §2. The Oka-Nishino Theorem and Its Applications.- §3. Distribution of Values of Analytic Multifunctions.- §4. Conclusion.- §1. Subharmonic Functions and Capacity.- §2. Functions of Several Complex Variables.- References.- Author and Subject Index.

Editorial Reviews

An introduction to the new techniques of subharmonic functions and analytic multifunctions in spectral theory. Topics include the basic results of functional analysis, bounded operators on Banach and Hilbert spaces, Banach algebras, and applications of spectral subharmonicity. Each chapter is followed by exercises of varying difficulty. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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