On contractions that are quasiaffine transforms of unilateral shifts

Abstract

It is known that if $T$ is a contraction of class $C_{10}$ and $I-T^{\ast }T$ is of trace class, then $T$ is a quasiaffine transform of a unilateral shift. Also it is known that if the multiplicity of a unilateral shift is infinite, the converse is not true. In this paper the converse for a finite multiplicity is proved: if $T$ is a contraction and $T$ is a quasiaffine transform of a unilateral shift of finite multiplicity, then $I-T^{\ast }T$ is of trace class. As a consequence we obtain that if a contraction $T$ has finite multiplicity and its characteristic function has an outer left scalar multiple, then $I-T^{\ast }T$ is of trace class. Also, it is known that if a contraction $T$ on a Hilbert space $\mathcal{H}$ is such that $\Vert b_{\lambda }(T)x\Vert \ge \delta \Vert x\Vert$ for every $\lambda \in \mathbb{D}$, $x\in \mathcal{H}$, with some $\delta >0$, and $$\underset{\lambda \in \mathbb{D}}{\sup }\Vert I-b_{\lambda }(T{)}^{\ast }{b}_{\lambda }(T){\Vert }_{{\mathfrak{S}}_1}<\infty$$ (here $b_{\lambda }$ is a Blaschke factor and ${\mathfrak{S}}_1$ is the trace class of operators), then $T$ is similar to an isometry. In this paper the converse for a finite multiplicity is proved: if $T$ is a contraction and $T$ is similar to an isometry of finite multiplicity, then $T$ satisfies the above conditions.

This is enlarged version of M. F. Gamal', On contractions that are quasiaffine transforms of unilateral shifts, Acta Sci. Math. (Szeged) 74(2008),757-767