An Introduction to Models and Decompositions in Operator Theory C. Kubrusly, PUC-RJ and LNCC, Rio de Janeiro, Brazil 0-8176-3992-6 * 1997-Spring * $34.50 * Hardcover * 148 pages

Decompositions and models for Hilbert-space operators have been very active research topics in operator theory over the past three decades. This book is intended as an introduction to this crucial part of operator theory, providing for the student a unified access, from an abstract point of view, to an active research field. It focuses on decompositions and models as if they were the main characters in a plot, chosen from a myriad of equally important characters and highlighted for their illustrative attributes. It has been written for an audience composed mainly of graduate students taking operator theory either as their major or as a support for applications in mathematics or in one of the sciences.

The approach is elementary in the sense that all proofs use only standard results of single operator theory. However a number of questions posed throughout the text provides the flavor of a research monograph in that it leads the reader to investigate some open problems, a number of them classical. This approach will lead the reader to visualize, even if only partially, the frontiers of a few directions in which operator theory has been developing. Although the material is mainly drawn from a variety of sources, there are some original contributions in the form of new intermediate results and simplified proofs.

Contents

Chapter 0. Preliminaries

0.1 Hilbert-Space Operators

0.2 Spectrum of an Operator

0.3 Convergence and Stability

0.4 Projections and Isometries

0.5 Invariant Subspaces

0.6 Spectral Theorem

Chapter 1. Equivalence

1.1 Parts

1.2 Norms

Chapter 2. Shifts

2.1 Unilateral Shifts

2.2 Bilateral Shifts

Chapter 3. Contractions

3.1 The Strong Limits of {T^*nTⁿ} and {TⁿT^*n}

3.2 The Isometry of V on R(A)^-

Chapter 4. Quasisimilarity

4.1 Invariant Subspaces

4.2 Hyperinvariant Subspaces

4.3 Contractions Quasisimilar to a Unitary Operator

Chapter 5. Decompositions

5.1 Nagy-Foias-Langer Decomposition

5.2 von Neumann-Wold Decomposition

5.3 A Decomposition for Contractions with A = A²

Chapter 6. Models

6.1 Rota's Model

6.2 de Branges-Rovnyak Refinement

6.3 Durszt Extension

Chapter 7. Applications

7.1 A Pattern for Contractions

7.2 Foguel Decomposition

Chapter 8. Similarity

8.1 Power Boundedness

8.2 Weak and Strong Stability

References

Index

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