A Katznelson-Tzafriri theorem for analytic Besov functions of operators

Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$, and assume that there is a bounded functional calculus for the operator $T$, so there is a bounded algebra homomorphism mapping functions $f \in \mathcal{A}$ to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type establish that $\lim_{n\to\infty} \|T^n f(T)\| = 0$ for functions $f \in \mathcal{A}$ whose boundary functions vanish on the unitary spectrum $\sigma(T)\cap \mathbb{T}$ of $T$, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when $\mathcal{A}$ is the Banach algebra $\mathcal{B}(\mathbb{D})$ of analytic Besov functions on $\mathbb{D}$. We prove a Katznelson-Tzafriri theorem for the $\mathcal{B}(\mathbb{D})$-calculus which extends several previous results.