# Compact operators that commute with a contraction - Mathematics > Functional Analysis

Abstract: Let $T$ be a $C 0$-contraction on a separable Hilbert space. We assume that$I H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$and continuous on $\bar\DD$, we show that $fT$ is compact if and only if $f$vanishes on $\sigma T\cap \TT$, 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 $fT$ is compact if and only if $\lim {n\to \infty} T^nfT =0$.

Author: Karim Kellay LATP, Mohamed Zarrabi IMB

Source: https://arxiv.org/