On contractions with compact defects

In [8], the following question was posed: suppose that $T$ is a contraction of class $C_{10}$ such that $I-T^\ast T$ 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 is the operators of multiplication by the independent variable in the closure of analytic polynomials in $L^2(\nu)$, where $\nu$ is an appropiate positive finite Borel measure on the closed unit disk. The second kind of examples is based on Theorem 6.2 in [5]. We obtain a contraction $T$ satisfying all required conditions and such that $I-T^\ast T$ belongs to Schatten–von Neumann classes $\mathfrak S_p$ for all $p>1$. Also we give an example of a contraction $T$ such that $I-T^\ast T$ 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 [2], we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to an absolutely continuous unitary operator, then this contraction $T$ can be chosen such that $I-T^\ast T$ is compact. Bibl. – 29 titles.