An Introduction to Models and Decompositions in Operator Theory

Carlos S. Kubrusly

ISBN 10: 0817639926 / ISBN 13: 9780817639921

Published by Birkhäuser Boston, 1997

About this Item

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.

Seller Inventory # 9780817639921

Bibliographic Details

• Title: An Introduction to Models and Decompositions in Operator Theory
• Publisher: Birkhäuser Boston
• Publication Date: 1997
• Language: English
• ISBN 10: 0817639926
• ISBN 13: 9780817639921
• Binding: Buch
• Condition: Neu
• Dimensions: 241W x 160H x 13D millimeters 241 millimeters width by 160 millimeters height by 13 millimeters depth
• Weight: 395 grams

About this title

Synopsis

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.

Synopsis

Decompositions and models for Hilbert-space operators have been active research topics in recent years, and this book is intended as an introduction to this area of operator theory, working from an abstract point of view. The approach is elementary in that all the proofs only use standard results of single operator thoery, although many of the questions posed in the text lead on to open problems.