Some results of Katznelson–Tzafriri type

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For semigroups S=R+,Z+ and Z+k, we show that if T is a representation of S by contractions on a Hilbert space then inft∈S‖T(t)f̂(T)‖=sup{|f(λ)|,λ∈Spu(T,S)}, where f∈L1(S) and Spu(T,S) is the unitary spectrum of T with respect to S. For T a representation of a suitable semigroup S by contractions on a Banach space, we give sharp conditions on Spu(T,S) which guarantee that the equality above holds. These conditions concern the thinness of Spu(T,S) in the harmonic analysis sense. These results are related to theorems of Katznelson–Tzafriri type, which give conditions guaranteeing that inft∈S‖T(t)f̂(T)‖ vanishes.