Problems in Operator Theory (Graduate Studies in Mathematics, V. 51)

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 Chapter 1. Odds And Ends Chapter 2. Basic Operator Theory Chapter 3. Operators On $al$- And $am$-spaces Chapter 4. Special Classes Of Operators Chapter 5. Integral Operators Chapter 6. Spectral Properties Chapter 7. Some Special Spectra Chapter 8. Positive Matrices Chapter 9. Irreducible Operators Chapter 10. Invariant Subspaces Chapter 11. The Daugavet Equation Y.a. Abramovich, C.d. Aliprantis. Includes Bibliographical References (p. 375-378) And Index. This book contains complete solutions to the more than six hundred exercises in the authors' book: Invitation to operator theory--foreword Problem 1.1.1. If x+y = x + y for two vectors x and y in a normed space, then show that ax+By = x + B y for all scalars a,B 0.