Compact operators that commute with a contraction

Abstract

Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that $I_H-T^{\ast }T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$ and continuous on $\bar\DD$, we show that $f(T)$ is compact if and only if $f$ vanishes on $σ(T)\cap \TT$, where $\sigma (T)$ is the spectrum of $T$ and $\TT$ the unit circle. If $f$ is just a bounded holomorphic function on $\DD$ we prove that $f(T)$ is compact if and only if ${\lim }_{n\to \infty }{T}^{n}f(T)=0$.

10p