The behavior of iterates of multipliers in commutative Banach algebras

Let $A$ be a complex, commutative, and semisimple Banach algebra and let $T$ be a power bounded multiplier on $A$. We prove a Katznelson–Tzafriri type theorem for $T$. As an application, we give some results concerning convergence of the sequence $\{ T^{n}a\} $ $(a\in A) $. Some related problems are also discussed.