Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 491-516.

$({\cal U}+{\cal K})$ -orbits, a block tridiagonal decomposition technique and a model with multiply connected spectrum

Authors: Michal Dostal
Author institution: Mexicka 4, Praha 10, CZ--101 00, Czech Republic

Summary: Two operators on a separable Hilbert space are

$\uk$ -{\it equivalent}

$A \uke B$ if

$A=R^{-1}BR$ , where

$R$ is invertible and

$R=U+K$ ,

$U$ unitary,

$K$ compact. The

$\uk$ -{\it orbit} of

$A$ is defined as

$\uk (A)=\{B\in \cB(\cH) : A \uke B \}$ . This orbit lies between the unitary and the similarity orbit. In addition, two

$\uk$ -equivalent operators are compalent. In this article we develop a block tridiagonal decomposition technique that allows us to show that an operator is in the

$\uk$ -orbit of another operator in some cases where the similarity of the two operators is apparent. We construct an essentially normal operator

$model$ with multiply connected

$non-essential$ spectrum and describe the closure of the

$\uk$ -orbit of this model.

Keywords:

${({\cal U}+{\cal K})}$ -orbit, essentially normal, model, multiply connected domain, block tridiagonal decomposition

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