Compact operators that commute with a contraction

TL;DR

This paper characterizes when functions of a specific class of contraction operators are compact, linking the compactness of $f(T)$ to the behavior of $f$ on the spectrum and the operator's powers.

Contribution

It provides a complete characterization of compactness for functions of $C_0$-contractions with compact defect operators, extending known results to broader classes of functions.

Findings

$f(T)$ is compact iff $f$ vanishes on $\sigma(T)igcap ext{unit circle}$ for continuous $f$.

For bounded holomorphic $f$, $f(T)$ is compact iff $ ext{lim}_{n oty} T^n f(T)=0$.

The results connect spectral properties of $T$ with the compactness of operator functions.

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 $f(T)$ 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 $f(T)$ is compact if and only if $\lim_{n\to \infty} T^nf(T) =0$.