A resolvent test for admissibility of Volterra observation operators

Necessary and sufficient conditions are given for finite-time admissibility of a linear system defined by a Volterra integral equation when the underlying semigroup is equivalent to a contraction semigroup, in terms of a pointwise bound on the resolvent of the infinitesimal generator. This generalizes an analogous result known to hold for the standard Cauchy problem. For infinite-time admissibility, however, it is shown by means of an example that the natural generalization of the Weiss resolvent test is no longer valid.