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Problems in operator theory

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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.

Author(s): Y. A. Abramovich, Charalambos D. Aliprantis
Series: GSM051
Publisher: American Mathematical Society
Year: 2002

Language: English
Pages: 402

Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Foreword......Page 12
1.1. Banach Spaces. Operators. and Linear Functionals......Page 14
1.2. Banach Lattices and Positive Operators......Page 33
1.3. Bases in Banach Spaces......Page 44
1.4. Ultrapowers of Banach Spaces......Page 57
1.5. Vector-valued Functions......Page 61
1.6. Fundamentals of Measure Theory......Page 64
2.1. Bounded Below Operators ......Page 76
2.2. The Ascent and Descent of an Operator ......Page 81
2.3. Banach Lattices with Order Continuous Norms ......Page 84
2.4. Compact and Weakly Compact Positive Operators ......Page 91
3.1. AL- and A lf-spaces......Page 100
3.2. Complex Banach Lattices......Page 109
3.3. The Center of a Banach Lattice......Page 118
3.4. The Predual of a Principal Ideal......Page 124
4.1. Finite-rank Operators......Page 132
4.2. Multiplication Operators......Page 138
4.3. Lattice and Algebraic Homomorphisms......Page 142
4.4. Fredholm Operators ......Page 147
4.5. Strictly Singular Operators ......Page 152
5.1. The Basics of Integral Operators ......Page 158
5.2. Abstract Integral Operators ......Page 167
5.3. Conditional Expectations and Positive Projections ......Page 182
5.4. Positive Projections and Lattice-subspaces ......Page 193
6.1. The Spectrum of an Operator ......Page 202
6.2. Special Points of the Spectrum ......Page 210
6.3. The Resolvent of a Positive Operator ......Page 214
6.4. Functional Calculus ......Page 218
7.1. The Spectrum of a Compact Operator ......Page 228
7.2. Turning Approximate Eigenvalues into Eigenvalues ......Page 235
7.3. The Spectrum of a Lattice Homomorphism ......Page 243
7.4. The Order Spectrum of an Order Bounded Operator ......Page 245
7.5. The Essential Spectrum of a Bounded Operator ......Page 250
8.1. The Banach Lattices and ......Page 256
8.2. Operators on Finite Dimensional Spaces ......Page 264
8.3. Matrices with Non-negative Entries ......Page 275
8.4. Irreducible Matrices ......Page 278
8.5. The Perron-Frobenius Theorem ......Page 281
9.1. Irreducible and Expanding Operators ......Page 286
9.2. Ideal Irreducibility and the Spectral Radius ......Page 296
9.3. Band Irreducibility and the Spectral Radius ......Page 303
9.4. Krein Operators and C(fl)-spaces ......Page 306
10.1. A Smorgasbord of Invariant Subspaces ......Page 312
10.2. The Lomonosov Invariant Subspace Theorem ......Page 320
10.3. Invariant Ideals for Positive Operators ......Page 323
10.4. Invariant Subspaces of Families of Positive Operators ......Page 330
10.5. Compact-friendly Operators ......Page 333
10.6. Positive Operators on Banach Spaces with Bases ......Page 342
10.7. Non-transitive Algebras ......Page 344
11.1. The Daugavet Equation and Uniform Convexity ......Page 348
11.2. The Daugavet Property in AL- and AM-spaces ......Page 365
11.3. The Daugavet Property in Banach Spaces ......Page 369
11.4. The Daugavet Property in C(c2)-spaces ......Page 372
11.5. Slices and the Daugavet Property ......Page 378
11.6. Narrow Operators ......Page 382
11.7. Some Applications of the Daugavet Equation ......Page 385
Bibliography ......Page 388
Index......Page 392
Titles in This Series......Page 400
Back Cover......Page 402