Theorems of Katznelson-Tzafriri type for semigroups of operators

AbstractLet 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 Spu(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 L1(S) which is of spectral synthesis with respect to Spu(T), then inft ϵ S∥∝Sf(s) T(t + s) ds∥ = 0