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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