Overview

By a Hilbert-space operator we mean a bounded linear transformation be­ tween separable complex Hilbert spaces. Decompositions and models for Hilbert-space operators have been very active research topics in operator theory over the past three decades. The main motivation behind them is the in­ variant subspace problem: does every Hilbert-space operator have a nontrivial invariant subspace? This is perhaps the most celebrated open question in op­ erator theory. Its relevance is easy to explain: normal operators have invariant subspaces (witness: the Spectral Theorem), as well as operators on finite­ dimensional Hilbert spaces (witness: canonical Jordan form). If one agrees that each of these (i. e. the Spectral Theorem and canonical Jordan form) is important enough an achievement to dismiss any further justification, then the search for nontrivial invariant subspaces is a natural one; and a recalcitrant one at that. Subnormal operators have nontrivial invariant subspaces (extending the normal branch), as well as compact operators (extending the finite-dimensional branch), but the question remains unanswered even for equally simple (i. e. simple to define) particular classes of Hilbert-space operators (examples: hyponormal and quasinilpotent operators). Yet the invariant subspace quest has certainly not been a failure at all, even though far from being settled. The search for nontrivial invariant subspaces has undoubtly yielded a lot of nice results in operator theory, among them, those concerning decompositions and models for Hilbert-space operators. This book contains nine chapters.

Full Product Details

Author:   Carlos S. Kubrusly
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1997
Dimensions:   Width: 15.50cm , Height: 0.80cm , Length: 23.50cm
Weight:   0.237kg
ISBN:  

9781461273745

ISBN 10:   1461273749
Pages:   132
Publication Date:   13 October 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

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.- 1. Equivalence.- 1.1. Parts.- 1.2. Norms.- 2. Shifts.- 2.1. Unilateral Shifts.- 2.2. Bilateral Shifts.- 3. Contractions.- 3.1. The Strong Limits of {T*nTn} and {TnT*n}.- 3.2. The Isometry V on R(A)-.- 4. Quasisimilarity.- 4.1. Invariant Subspaces.- 4.2. Hyperinvariant Subspaces.- 4.3. Contractions Quasisimilar to a Unitary Operator.- 5. Decompositions.- 5.1. Nagy-Foia?—Langer Decomposition.- 5.2. von Neumann-Wold Decomposition.- 5.3. A Decomposition for Contractions with A = A2.- 6. Models.- 6.1. Rota’s Model.- 6.2. de Branges-Rovnyak Refinement.- 6.3. Durszt Extension.- 7. Applications.- 7.1. A Pattern for Contractions.- 7.2. Foguel Decomposition.- 8. Similarity.- 8.1. Power Boundedness.- 8.2. Weak and Strong Stability.- References.

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