Weighted and local stability of semigroups of operators

We prove global and local weighted stability theorems for representations o f abelian semigroups on Banach spaces under countable spectral conditions a nd certain growth assumptions on the weights. In the global case, we use a limit semigroup construction together with an extension theorem for weighte d semigroup, representations. In the local case, we adapt a definition of A lbrecht to introduce a local spectrum of a representation which is no large r than the usual notion of local spectrum for representations of Z(+) and R +, and we establish the corresponding local stability theorem. We give an a pplication to the asymptotic theory of functions on abelian semigroups.