# Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 545-565.

A Katznelson-Tzafriri theorem for analytic Besov functions of operators

**Authors**:
Charles Batty (1), David Seifert (2)

**Author institution:** (1) St. John's College,
University of Oxford,
Oxford, OX1~3JP, U.K.

(2) School of Mathematics, Statistics and Physics,
Newcastle University,
Newcastle upon Tyne, NE1 7RU, U.K.

**Summary: ** 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 $\lim\limits_{n\to\infty} \|T^n f(T)\| = 0$ 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.

**DOI: **http://dx.doi.org/10.7900/jot.2022jun21.2390

**Keywords: **Katznelson-Tzafriri theorem, analytic Besov function, functional calculus, power-bounded operator, unitary spectrum, resolvent condition

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