Some results of Katznelson–Tzafriri type

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 inf t ∈ S ‖ T ( t ) f ̂ ( T ) ‖ = sup { | f ( λ ) | , λ ∈ Sp u ( T , S ) } , where f ∈ L 1 ( S ) and Sp u ( 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 Sp u ( T , S ) which guarantee that the equality above holds. These conditions concern the thinness of Sp u ( T , S ) in the harmonic analysis sense. These results are related to theorems of Katznelson–Tzafriri type, which give conditions guaranteeing that inf t ∈ S ‖ T ( t ) f ̂ ( T ) ‖ vanishes.