# Journal of Operator Theory

Volume 69, Issue 1, Winter 2013 pp. 59-85.

Katznelson-Tzafriri type theorem for integrated semigroups**Authors**: Jose E. Gale (1), Maria M. Martinez (2), and Pedro J. Miana (3)

**Author institution:**(1) Departamento de Matematicas e IUMA, Universidad de Zaragoza, 50009, Zaragoza, Spain

(2) Departamento de Matematicas e IUMA, Universidad de Zaragoza, 50009, Zaragoza, Spain

(3) Departamento de Matematicas e IUMA, Universidad de Zaragoza, 50009, Zaragoza, Spain

**Summary:**Y. Katznelson and L. Tzafriri proved that if $T$ is a power-bounded operator and $f$ is an analytic function, in the Wiener algebra, of spectral synthesis with respect to its peripheral spectrum then $\lim\limits_{n\rightarrow\infty} \|T^{n}f(T) \| = 0$. Here $f(T)$ is given by the usual functional calculus associated with $T$. The analogous version for bounded $C_0$-semigroups of operators was obtained by J.~Esterle, E. Strouse and F. Zouakia and independently by Q.P. V$\tilde{\rm u}$. We extend this result to $\al$-times integrated semigroups. The proof is based on an analysis of homomorphisms from convolution Banach algebras of Sobolev type.

**Keywords:**convolution, Sobolev algebra, fractional calculus, spectral synthesis, integrated semigroup, asymptotic behaviour

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