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arXiv · 0809.3184

Compact operators that commute with a contraction

Karim Kellay·Mohamed Zarrabi
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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 $σ(T)\cap \TT$, where $σ(T)$ 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 $\lim_{n\to \infty} T^nf(T) =0$.

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Karim Kellay, Mohamed Zarrabi. 2008-09-18. Compact operators that commute with a contraction. https://arxiv.org/abs/0809.3184

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