THE HILLE-YOSIDA INEQUALITY FOR C-SEMIGROUPS AND IDEAL NORMS

Abstract

Let T(t); 0 leq t leq infty be a one parameter C-semigroup of bounded linear operators on a Banach space X with generator A, and R(lambda,A) be the resolvent operator of A. The Hille-Yosida Theorem for C-semigroups asserts that the resolvent operator of the generator A satisfies |CR(lambda, A)|leq frac{M}{lambda-w} for some constants M > 0 and lambda in R (the set of real numbers), lambda>w. The object of this paper is study when can this inequality holds true for certain operator ideal and to investigate when can an operator T(t) from a C-semigroup belongs to some operator ideal.