The Banach algebra generated by a contraction

Abstract:

Let $T$ be a contraction on a Banach space and $A_{T}$ the Banach algebra generated by $T$. Let $\sigma _{u}(T)$ be the unitary spectrum (i.e., the intersection of $\sigma (T)$ with the unit circle) of $T$. We prove the following theorem of Katznelson-Tzafriri type: If $\sigma _{u}(T)$ is at most countable, then the Gelfand transform of $R\in A_{T}$ vanishes on $\sigma _{u}(T)$ if and only if $\lim _{n\rightarrow \infty }\left \Vert T^{n}R\right \Vert =0.$