[pt] O problema do subespaço invariante consiste na seguinte pergunta: será que todo operador (i.e., transformação linear limitada) atuando em um espaço de Hilbert separável (complexo de dimensão infinita) tem subespaço invariante nãotrivial? Este é, possivelmente, o mais importante problema em aberto na teoria de operadores. Em particular, o problema do subespaço invariante permanece em aberto (pelo menos até a presente data) para operadores hiponormais, ou seja, ainda não se sabe se todo operador hiponormal (atuando em um espaço de Hilbert complexo separável) tem subespaço invariante não-trivial. O objetivo desta dissertação é apresentar, de maneira unificada, um levantamento sobre subespaços invariantes para operadores hiponormais. Inicialmente, o problema do subespaço invariante é abordado em sua forma geral (sem restrição a classes de operadores) onde diversos resultados clássicos são expostos. Em seguida, o problema específico de se encontrar subespaços invariantes para operadores hiponormais é apresentado de maneira sistemática. Em particular, investigamos propriedades do espectro de um operador hiponormal que não tenha subespaço invariante não trivial. / [en] The invariant subspace problem is: does every operator acting on an infinite-dimensional complex separable Hilbert space have a nontrivial invariant subspace? This is, probably, the most important open question in the operator theory. In particular, the problem of the invariant subspace remains open (at least until now) for hyponormal operators, that is, it is still unknown whether every hyponormal operator (on a complex separable Hilbert space) has a nontrivial invariant subspace. The purpose of these dissertation is to present, in an unified way, a survey on invariant subspaces for hyponormal operators. At first, the invariant subspace problem is posed in a general form (without any restriction on the operator classes), where some of classical results are discussed. Secondly, the specific problem of finding invariant subspaces for hyponormal operators is presented in a systematic way and, in particular, we show some characteristics of the spectrum of a hyponormal operator with no nontrivial invariant subspace.

This paper demonstrates the properties of a Hilbert structure. In order to have a Hilbert structure it is necessary to satisfy certain properties or axioms. The main body of the paper is centered on six questions that develop these ideas.

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In this thesis we have investigated two different types of problems in multivariable operator theory. The first one deals with the defect sequence for contractive tuples and maximal con-tractive tuples. These condone deals with the wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in thefollowing two sections. (I) The Defect Sequence for ContractiveTuples LetT=(T1,...,Td)bead-tuple of bounded linear operators on some Hilbert space H. We say that T is a row contraction, or, acontractive tuplei f the row operator (Pl refer the abstract pdf file)

A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results.

Indiana University-Purdue University Indianapolis (IUPUI) / Invariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.