**Graduate Studies in Mathematics**

Volume: 50;
2002;
530 pp;
Hardcover

MSC: Primary 46; 47; 28; 15;

Print ISBN: 978-0-8218-2146-6

Product Code: GSM/50

List Price: $84.00

Individual Member Price: $67.20

**Electronic ISBN: 978-1-4704-2099-4
Product Code: GSM/50.E**

List Price: $84.00

Individual Member Price: $67.20

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# An Invitation to Operator Theory

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*Y. A. Abramovich; C. D. 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.

#### Table of Contents

# Table of Contents

## An Invitation to Operator Theory

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents v6 free
- Foreword ix10 free
- Chapter 1. Odds and Ends 116 free
- Chapter 2. Basic Operator Theory 6984
- Chapter 3. Operators on AL-and AM-spaees 93108
- Chapter 4. Special Classes of Operators 123138
- Chapter 5. Integral Operators 179194
- Chapter 6. Spectral Properties 237252
- Chapter 7. Some Special Spectra 271286
- Chapter 8. Positive Matrices 315330
- Chapter 9. Irreducible Operators 347362
- Chapter 10. Invariant Subspaces 381396
- §10.1. A Smorgasbord of Invariant Subspaces 383398
- §10.2. The Lomonosov Invariant Subspace Theorem 393408
- §10.3. Invariant Ideals for Positive Operators 398413
- §10.4. Invariant Subspaces of Families of Positive Operators 409424
- §10.5. Compact-friendly Operators 425440
- §10.6. Positive Operators on Banach Spaces with Bases 436451
- §10.7. A Characterization of Non-transitive Algebras 440455
- §10.8. Comments on the Invariant Subspace Problem 449464

- Chapter 11. The Daugavet Equation 455470
- §11.1. The Daugavet Equation and Uniform Convexity 456471
- §11.2. The Daugavet Property in AL-and AM-spaces 467482
- §11.3. The Daugavet Property in Banach Spaces 471486
- §11.4. The Daugavet Property in C(Ω)-spaces 477492
- §11.5. Slices and the Daugavet Property 487502
- §11.6. Narrow Operators 493508
- §11.7. Some Applications of the Daugavet Equation 500515

- Bibliography 505520
- Index 521536
- Back Cover Back Cover1546

#### Readership

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