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An Invitation to Operator Theory
Y. A. Abramovich, Indiana University-Purdue University, Indianapolis, IN, and C. D. Aliprantis, Purdue University, West Lafayette, IN
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2002; 530 pp; hardcover
Volume: 50
ISBN-10: 0-8218-2146-6
ISBN-13: 978-0-8218-2146-6
List Price: US$80 Member Price: US$64
Order Code: GSM/50

This item is also sold as part of the following set: GSMSET

Problems in Operator Theory - Y A Abramovich and C D Aliprantis

An Invitation to Operator Theory (Volume 50) and Problems in Operator Theory (Volume 51) - Y A Abramovich and C C Aliprantis

This book offers a comprehensive and reader-friendly exposition of the theory of linear operators on Banach spaces and Banach lattices. Abramovich and Aliprantis give a unique presentation that includes many new developments in operator theory and also draws together results that are spread over the vast literature. For instance, invariant subspaces of positive operators and the Daugavet equation are presented in monograph form for the first time.

The authors keep the discussion self-contained and use exercises to achieve this goal. The book contains over 600 exercises to help students master the material developed in the text. The exercises are of varying degrees of difficulty and play an important and useful role in the exposition. They help to free the proofs of the main results of some technical details but provide students with accurate and complete accounts of how such details ought to be worked out. The exercises also contain a considerable amount of additional material that includes many well-known results whose proofs are not readily available elsewhere.

The companion volume, Problems in Operator Theory, also by Abramovich and Aliprantis, is available from the AMS as Volume 51 in the Graduate Studies in Mathematics series, and it contains complete solutions to all exercises in An Invitation to Operator Theory.

The solutions demonstrate explicitly technical details in the proofs of many results in operator theory, providing the reader with rigorous and complete accounts of such details. Finally, the book offers a considerable amount of additional material and further developments. By adding extra material to many exercises, the authors have managed to keep the presentation as self-contained as possible. The best way of learning mathematics is by doing mathematics, and the book Problems in Operator Theory will help achieve this goal.

Prerequisites to each book are the standard introductory graduate courses in real analysis, general topology, measure theory, and functional analysis. An Invitation to Operator Theory is suitable for graduate or advanced courses in operator theory, real analysis, integration theory, measure theory, function theory, and functional analysis. Problems in Operator Theory is a very useful supplementary text in the above areas. Both books will be of great interest to researchers and students in mathematics, as well as in physics, economics, finance, engineering, and other related areas, and will make an indispensable reference tool.

Graduate students and researchers interested in mathematics, physics, economics, finance, engineering, and other related areas.

Reviews

"The book is a fine introduction to this particular part of operator theory ... In addition to the choice of material and the book being as well-written as one normally expects from these authors ... there are two features that set this book apart from others. The first is the great care that the authors give to correct attribution of original results ... and the second is the exercises that are included ... there are over 600 exercises ... The authors take the same care with the attribution of these exercises as they do with the results in the body of the text ... one could hardly wish for a better text than this one."

-- Mathematical Reviews

• Odds and ends
• Basic operator theory
• Operators on $$AL$$- and $$AM$$-spaces
• Special classes of operators
• Integral operators
• Spectral properties
• Some special spectra
• Positive matrices
• Irreducible operators
• Invariant subspaces
• The Daugavet equation
• Bibliography
• Index