A Katznelson-Tzafriri theorem for analytic Besov functions of operators

arXiv2201.12076MaRDI QIDQ6505404

David Seifert, Charles J. K. Batty

Abstract: Let $T$ be a power-bounded operator on a Banach space $X$, $mathcalA$ be a Banach algebra of bounded holomorphic functions on the unit disc $mathbbD$, and assume that there is a bounded functional calculus for the operator $T$, so there is a bounded algebra homomorphism mapping functions $finmathcalA$ to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type establish that $li{m}_{noinfty}|T^{n}f(T)|=0$ for functions $finmathcalA$ whose boundary functions vanish on the unitary spectrum $sigma(T)capmathbbT$ of $T$, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when $mathcalA$ is the Banach algebra $mathcalB(mathbbD)$ of analytic Besov functions on $mathbbD$. We prove a Katznelson-Tzafriri theorem for the $mathcalB(mathbbD)$-calculus which extends several previous results.

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