C0-semigroup and Operator Ideals

Let T (t), 0 ≤ t < ∞, be a one parameter c0-semigroup of bounded linear operators on a Banach space X with infinitesimal generator A and R(λ, A) be the resolvent operator of A. The Hille-Yosida Theorem for c0-semigroups asserts that the resolvent operator of the infinitesimal generator A satisfies ‖R(λ, A)‖ ≤ M λ−ω for some constants M > 0 and λ ∈ R (the set of real numbers), λ > ω. The object of this paper is to investigate when a c0-semigroup can be in some operator ideals and to show that the previous inequality may not hold for certain ideal norms, including the p-summing, nuclear and the Schatten norms.