STUDIA MATHEMATICA 146 (1) (2001) Operator theoretic properties of semigroups in terms of their generators by S. Blunck and L. Weis (Karlsruhe) Abstract. Let (T t ) be a C 0 semigroup with generator A on a Banach space X and let A be an operator ideal, e.g. the class of compact, Hilbert–Schmidt or trace class operators. We show that the resolvent R(λ, A) of A belongs to A if and only if the integrated semigroup S t := t 0 T s ds belongs to A. For analytic semigroups, S t ∈A implies T t ∈A, and in this case we give precise estimates for the growth of the A-norm of T t (e.g. the trace of T t ) in terms of the resolvent growth and the imbedding D(A)  → X. 0. Introduction. In this paper we study how operator theoretic prop- erties of the generator and the resolvent of a C 0 semigroup on a Banach space X are reflected in the properties of the semigroup. Often operator theoretic properties of an operator T can be checked conveniently by showing that T belongs to a suitable operator ideal. If T belongs to the ideal of compact or strictly singular operators we know that its spectrum σ(T ) consists of a series of eigenvalues with possible limit point 0, that T is an admissible Fredholm perturbation, etc. We know about the summability of its eigenvalues and its trace if T belongs to the Hilbert– Schmidt class, the trace class or to one of the ideals extending the Schatten classes to the Banach space setting. To have a unified approach to many of these topics,we phrase our ques- tion as follows: Given an operator ideal A and a C 0 semigroup (T t ) with generator A and resolvent R(λ, A), how can we characterize “R(λ, A) ∈A” in terms of A and the semigroup T t ? Since the semigroup and the resolvent are connected by the Laplace transform (1) R(λ, A)= ∞ 0 e −λt T t dt the resolvent is a “smoothing” of the semigroup and one would generally 2000 Mathematics Subject Classification : 47A60, 47B10, 47D06. [35]