An Introduction to Models and Decompositions in Operator Theory

Carlos S. Kubrusly

ISBN 10: 1461273749 / ISBN 13: 9781461273745

Editore: Birkhäuser, Birkhäuser, 2012

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Druck auf Anfrage Neuware - Printed after ordering - 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.

Codice articolo 9781461273745

Dati bibliografici

• Titolo: An Introduction to Models and Decompositions in Operator Theory
• Editore: Birkhäuser, Birkhäuser
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 1461273749
• ISBN 13: 9781461273745
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 9 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 9 millimetri di profondità
• Peso: 236 grammi

Descrizione articolo

Riassunto

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.

Contenuti

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.