C0-semigroups on a locally convex space

AbstractA necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C0-semigroup on X is that there exist a real number β ⩾ 0 such that, for each λ > β, the resolvent (λI − A)−1 exists and the family {(λ − β)k(λI − A)−k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI − A)−1, λ > β, of the given operator A and all exponentials exp(tA), t ⩾ 0, of the operator A belong to a Banach algebra Bг(X) which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ⩾ 0 and for each x ϵ X, one has limk → z (I − k−1tA)−k x = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C0-semigroup by an operator which is an element of Bг(X) is obtained