On contractions with compact defects

Abstract

In 2005, the following question was posed by Duggal, Djordjevic, and Kubrusly: Assume that T is a contraction of the class C 10 such that I − T * T is compact and the spectrum of T is the unit disk. Can the isometric asymptote of T be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples. One of them are the operators of multiplication by independent variable in the closure of analytic polynomials in L 2(ν),where ν is an appropriate positive finite Borel measure on the closed unit disk. The second kind of examples is based on a theorem by Chevreau, Exner, and Pearcy. We obtain a contraction T satisfying all the needed conditions and such that I − T * T belongs to the Schatten–von Neumann classes $$ {\mathfrak{S}_p} $$ for all p > 1. We give an example of a contraction T such that I − T * T belongs to $$ {\mathfrak{S}_p} $$ for all p > 1, T is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following Bercovici and Kerchy, we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to a given absolutely continuous unitary operator, then this contraction T can be chosen so that I − T*T is compact. Bibliography: 29 titles.