An invitation to operator theory / Y.A. Abramovich, C.D. Aliprantis.

Format:

Book

Author/Creator:

Abramovich, Y. A. (Yuri A.), author.

Aliprantis, Charalambos D., author.

Series:

Graduate studies in mathematics ; v. 50.

Graduate studies in mathematics, 1065-7339 ; volume 50

Language:

English

Subjects (All):

Operator theory.

Physical Description:

1 online resource (530 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

This book offers a comprehensive and reader-friendly exposition of the theory of linear operators on Banach spaces and Banach lattices using their topological and order structures and properties. Abramovich and Aliprantis give a unique presentation that includes many new and very recent developments in operator theory and also draws together results which are spread over the vast literature. For instance, invariant subspaces of positive operators and the Daugavet equation arepresented 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 importantand 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 inthe 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 ofsuch 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, andfunctional 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. Bothbooks 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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Foreword""; ""Chapter 1. Odds and Ends""; ""Â1.1. Banach Spaces, Operators, and Linear Functional""; ""Â1.2. Banach Lattices and Positive Operators""; ""Â1.3. Bases in Banach Spaces""; ""Â1.4. Ultrapowers of Banach Spaces""; ""Â1.5. Vector-valued Functions""; ""Â1.6. Fundamentals of Measure Theory""; ""Chapter 2. Basic Operator Theory""; ""Â2.1. Bounded Below Operators""; ""Â2.2. The Ascent and Descent of an Operator""; ""Â2.3. Banach Lattices with Order Continuous Norms""; ""Â2.4. Compact and Weakly Compact Positive Operators""

""Chapter 3. Operators on AL-and AM-spaees""""Â3.1. AL-and AM-spaces""; ""Â3.2. Complex Banach Lattices""; ""Â3.3. The Center of a Banach Lattice""; ""Â3.4. The Predual of a Principal Ideal""; ""Chapter 4. Special Classes of Operators""; ""Â4.1. Finite-rank Operators""; ""Â4.2. Multiplication Operators""; ""Â4.3. Lattice and Algebraic Homomorphisms""; ""Â4.4. Fredholm Operators""; ""Â4.5. Strictly Singular Operators""; ""Chapter 5. Integral Operators""; ""Â5.1. The Basics of Integral Operators""; ""Â5.2. Abstract Integral Operators""

""Â5.3. Conditional Expectations and Positive Projections""""Â5.4. Positive Projections and Lattice-subspaces""; ""Chapter 6. Spectral Properties""; ""Â6.1. The Spectrum of an Operator""; ""Â6.2. Special Points of the Spectrum""; ""Â6.3. The Resolvent of a Positive Operator""; ""Â6.4. Functional Calculus""; ""Chapter 7. Some Special Spectra""; ""Â7.1. The Spectrum of a Compact Operator""; ""Â7.2. Turning Approximate Eigenvalues into Eigenvalues""; ""Â7.3. The Spectrum of a Lattice Homomorphism""; ""Â7.4. The Order Spectrum of an Order Bounded Operator""

""Â7.5. The Essential Spectrum of a Bounded Operator""""Chapter 8. Positive Matrices""; ""Â8.1. The Banach Lattices M[sub(n)](R) and M[sub(n)](C)""; ""Â8.2. Operators on Finite Dimensional Spaces""; ""Â8.3. Matrices with Non-negative Entries""; ""Â8.4. Irreducible Matrices""; ""Â8.5. The Perron-Frobenius Theorem""; ""Chapter 9. Irreducible Operators""; ""Â9.1. Irreducible and Expanding Operators""; ""Â9.2. Ideal Irreducibility and the Spectral Radius""; ""Â9.3. Band Irreducibility and the Spectral Radius""; ""Â9.4. Krein Operators on C(Ω)-spaces""

""Chapter 10. Invariant Subspaces""""Â10.1. A Smorgasbord of Invariant Subspaces""; ""Â10.2. The Lomonosov Invariant Subspace Theorem""; ""Â10.3. Invariant Ideals for Positive Operators""; ""Â10.4. Invariant Subspaces of Families of Positive Operators""; ""Â10.5. Compact-friendly Operators""; ""Â10.6. Positive Operators on Banach Spaces with Bases""; ""Â10.7. A Characterization of Non-transitive Algebras""; ""Â10.8. Comments on the Invariant Subspace Problem""; ""Chapter 11. The Daugavet Equation""; ""Â11.1. The Daugavet Equation and Uniform Convexity""

""Â11.2. The Daugavet Property in AL-and AM-spaces""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 505-520) and index.

Description based on print version record.

ISBN:

1-4704-2099-6