Further study of growth of fractional-power semigroups

AbstractIf the resolvent of a closed linear operator A in Banach space is defined and decays suitably in a region asymptotic, in a special sense, to a half-plane, then fractional powers −(−A)a generate semigroups, holomorphic in sectors of the complex plane. We show that the growth of such semigroups near the origin, or with angular approach to the edges of their sectors of definition, corresponds to the rate of decay of the resolvent of A and the extent of the resolvent set of A