An Introduction to Models and Decompositions in Operator Theory

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Springer Science & Business Media, Aug 19, 1997 - Mathematics - 132 pages
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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.
 

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Contents

Equivalence
23
12 Norms
26
Shifts
36
22 Bilateral Shifts
44
Contractions
49
32 The Isometry V on ℛA
55
Quasisimilarity
61
42 Hyper invariant Subspaces
66
53 A Decomposition for Contractions with A A▓
83
Models
87
62 de BrangesRovnyak Refinement
92
63 Durszt Extension
95
Applications
101
72 Foguel Decomposition
103
Similarity
108
82 Weak and Strong Stability
114

43 Contractions Quasisimilar to a Unitary Operator
68
Decompositions
75
51 NagyFoiaşLanger Decomposition
76
52 von NeumannWold Decomposition
77

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Page 21 - It is said to be subnormal if it is the restriction of a normal operator to an invariant subspace.
Page ix - This work was supported in part by CNPq (Brazilian National Research Council) and FAPESP (Research Council of the State of Sao Paulo).
Page 123 - Funct. Anal. 67 (1986), 153-166. 15. , On the residual parts of completely non-unitary contractions, Acta Math. Hungar. 50 (1987), 127-145. 16. , Invariant subspaces of C, . -contractions with non-reductive unitary extensions, Bull. London Math. Soc. 19 (1987), 161-166. 17. , Isometric asymptotes of power bounded operators, Indiana Univ. Math. J. 38 (1989), 173-188. 18. HH Schaefer, Topological vector spaces. Springer, New York, 1971. 19. R. Schatten, Norm ideals of completely continuous operators,...
Page vii - 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 invariant subspace problem: does every Hilbert-space operator have a nontrivial invariant subspace?
Page 123 - IC Gohberg and MG Krein, Description of contraction operators which are similar to unitary operators, Functional Anal, and its Appl., 1, (1967), pp.
Page 21 - He also makes progress on the unsolved problem of showing that every contraction whose spectrum contains the unit circle has a nontrivial invariant subspace. (There is one ambiguity wich might confuse someone surveying the results of the paper. Every time the author assumes that a condition holds "for any 0<a< I" he intends to assume that it holds for every a such that 0<a < 1.) John P.
Page 121 - S. Brown, B. Chevreau, and C. Pearcy, Contractions with rich spectrum have invariant subspaces, J. Operator Theory 1 (1979) , 123-136.

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