Theorems of Katznelson-Tzafriri type for semigroups of operators (Q1185744)

The paper gives some extensions of results of Katznelson-Tzafiri to one parameter strongly continuous semigroups. Let \({\mathcal T}=\{T(t): t\geq 0\}\) be a contraction semigroup in a Banach space \(X\), with generator \(A\). If \(f\in L^ 1(\mathbb{R}_ +)\) is a function which is of spectral synthesis with respect to \((i\sigma(A))\cap\mathbb{R}\) then \(\lim_{t\to\infty} \| T(t)\widehat f({\mathcal T})\|=0\). The function \(\widehat f({\mathcal T})\) is the Laplace transformation of \(f\) with respect to the semigroup \({\mathcal T}\), i.e. \(\widehat f({\mathcal T})=\int^ \infty_ 0 f(t)T(t)dt\). Also the more general case of representations of locally compact Abelian semigroups is considered.