Theorems of Katznelson-Tzafriri type for semigroups of operators

Abstract

Let S be a locally compact abelian semigroup and T a bounded representation of S by linear bounded operators in a Banach space X, with spectrum Sp( T). Let Sp u ( T) be the intersection of Sp( T) with the set of unitary characters of S. We prove that if S= R + or if T is norm-continuous and if f is a function in L 1( S) which is of spectral synthesis with respect to Sp u ( T), then inf t ϵ S ∥∝ S f( s) T( t + s) ds∥ = 0.