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: $\def\uk{({\cal U}+{\cal K})}$ $\def\uke{\cong_{\cU+\cK}}$ $\def\cB{{\cal B}}$ $\def\cH{{\cal H}}$ $\def\cU{{\cal U}}$ $\def\cK{{\cal K}}$ Two operators on a separable Hilbert space are $\uk$-equivalent ($A \uke B$) if $A=R^{-1}BR$, where $R$ is invertible and $R=U+K$, $U$ unitary, $K$ compact. The $\uk$-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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