Growth properties of semigroups generated by fractional powers of certain linear operators

AbstractCertain semigroups are generated by powers −(−A)a, for closed operators A in Banach space and 0 < a < 1. Properties of extent of the resolvent set and size of the resolvent operator of A correspond to properties relating to the sectors of holomorphy of the semigroups, and their growth near the origin and infinity. In this paper, we deal with semigroups having two different types of growth properties. In the first instance, the semigroup grows near the origin as r−t, 0 < t < 1. We show that such semigroups are fractional-power semi-groups of operators A, whose resolvents decay as r−s, 0 < s < 1, in subsectors of the right-hand half-plane. In the second instance, the semigroups are bounded near the origin, and admit special estimates on growth at the periphery of their sectors of definition. We show that for the corresponding A, the resolvent is defined and admits special growth estimates in a region which contains every subsector of the right half-plane; and in these subsectors, the resolvent decays as r−1