A Katznelson-Tzafriri theorem for analytic Besov functions of operators

Let T be a power-bounded operator on a Banach space X, A be a Banach algebra of bounded holomorphic functions on the unit disc D, and assume that there is a bounded functional calculus for the operator T, so there is a bounded algebra homomorphism mapping functions f ∈ A to bounded operators f(T) on X. Theorems of Katznelson-Tzafriri type establish that limn→∞ kT n f(T)k = 0 for functions f ∈ A whose boundary functions vanish on the unitary spectrum σ(T) ∩ T of T, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when A is the Banach algebra B(D) of analytic Besov functions on D. We prove a Katznelson-Tzafriri theorem for the B(D)-calculus which extends several previous results