## An Introduction to Models and Decompositions in Operator TheoryBy 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 A² | 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 |

121 | |

129 | |

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An Introduction to Models and Decompositions in Operator Theory Carlos S. Kubrusly Limited preview - 2012 |

An Introduction to Models and Decompositions in Operator Theory Carlos S. Kubrusly No preview available - 2012 |

### Common terms and phrases

adjoint Af(A B+[H backward unilateral shift Banach-Steinhaus Theorem bilateral bounded operator boundedness canonical backward unilateral canonical unilateral shift Chapter coisometry commutes completely nonunitary contraction convergence countable denoted direct sum direct summand ensures exists finite-dimensional ft(A G+[H hence Hilbert space hyponormal implies inner product integer invariant subspace problem Lemma linear manifold linear transformation Moreover multiplicity ft Nagy-Foia§-Langer decomposition nonnegative operators nonscalar nontrivial hyperinvariant subspace nontrivial invariant subspace nonzero norm normal operator normaloid Note operator theory orthogonal subspaces partial isometry polar decomposition power bounded operator Proposition 3.1 quasiaffine transform quasisimilar readily verified recall reducing subspace scalar self-adjoint operators sequence shift of multiplicity space H spectral radius Spectral Theorem spectrum strict contraction strong stability strongly stable contraction subspace of H surjective T C B[H Take an arbitrary Theorem 5.1 trivially uniform stability unitarily equivalent unitary operator weak stability weakly stable x e H

### Popular passages

Page vi - Now one of them is in fashion, and most carriages go by that, now it's another and everything drives pell-mell there. And what governs this change of fashion has never yet been found out. At eight o'clock one morning they'll all be on another road, ten minutes later on a third, and half an hour after that on the first road again, and then they may stick to thaf road all day, but every minute there's the possibility of a change.

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.