# Problems in Operator Theory

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

This is one of the few books available in the literature that contains problems
devoted entirely to the theory of operators on Banach spaces and Banach
lattices. The book contains complete solutions to the more than 600 exercises
in the companion volume, An Invitation to Operator Theory, Volume 50 in
the AMS series Graduate Studies in Mathematics, also by Abramovich and
Aliprantis.

The exercises and solutions contained in this volume serve many purposes.
First, they provide an opportunity to the readers to test their understanding
of the theory. Second, they are used to demonstrate explicitly technical
details in the proofs of many results in operator theory, providing the reader
with rigorous and complete accounts of such details. Third, the exercises
include many well-known results whose proofs are not readily available
elsewhere. Finally, the book contains 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 book can be very useful as a supplementary text to graduate courses in
operator theory, real analysis, function theory, integration theory, measure
theory, and functional analysis. It will also make a nice reference tool for
researchers in physics, engineering, economics, and finance.

#### Reviews & Endorsements

There are over 370 pages of mathematics here which provides a valuable supplement to that contained in the original textbook. The authors take the same care with attribution of results in these exercises … as they have done in the book … a worthy addition to any library (either individual or institutional) …

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Problems in Operator Theory

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Foreword xi12 free
- Chapter 1. Odds and Ends 114 free
- Chapter 2. Basic Operator Theory 6376
- Chapter 3. Operators on AL- and AM-spaees 87100
- Chapter 4. Special Classes of Operators 119132
- Chapter 5. Integral Operators 145158
- Chapter 6. Spectral Properties 189202
- Chapter 7. Some Special Spectra 215228
- Chapter 8. Positive Matrices 243256
- Chapter 9. Irreducible Operators 273286
- Chapter 10. Invariant Subspaces 299312
- §10.1. A Smorgasbord of Invariant Subspaces 299312
- §10.2. The Lomonosov Invariant Subspace Theorem 307320
- §10.3. Invariant Ideals for Positive Operators 310323
- §10.4. Invariant Subspaces of Families of Positive Operators 317330
- §10.5. Compact-friendly Operators 320333
- §10.6. Positive Operators on Banach Spaces with Bases 329342
- §10.7. Non-transitive Algebras 331344

- Chapter 11. The Daugavet Equation 335348
- §11.1. The Daugavet Equation and Uniform Convexity 335348
- §11.2. The Daugavet Property in AL- and AM-spaces 352365
- §11.3. The Daugavet Property in Banach Spaces 356369
- §11.4. The Daugavet Property in C(Ω)-spaces 359372
- §11.5. Slices and the Daugavet Property 365378
- §11.6. Narrow Operators 369382
- §11.7. Some Applications of the Daugavet Equation 372385

- Bibliography 375388
- Index 379392
- Back Cover Back Cover1402