Asymptotic behaviour of C-0-semigroups with bounded local resolvents

Let {T(t)}(t greater than or equal to0) be a C-0-semigroup on a Banach spac e X with generator A, and let H-T(infinity) be the space of all is an eleme nt of X such that the local resolvent lambda bar right arrow R(lambda ,A)x has a bounded holomorphic extension to the right half-plant. For the class of integrable functions phi on [0, infinity) whose Fourier transforms are i ntegrable, we construct a functional calculus phi bar right arrow T phi, as operators on H-T(infinity). We show that each orbit T((.))T(phi)x is bound ed and uniformly continuous, and T(t)T(phi)x --> 0 weakly as t --> infinity , and we give a new proof that \\T(t)R(mu, A)x\\ = O(t) We also show that \ \T(t)T(phi)x\\ --> 0 when T is sun-reflexive, and that \\T(t)R(mu ,A)x\\ = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H-T(infinity).