Favard Spaces and Admissibility for Volterra Systems with Scalar Kernel

We introduce the Favard spaces for resolvent families, extending some well-known theorems for semigroups. Furthermore, we show the relationship between these Favard spaces and the Lp-admissibility of control operators for scalar Volterra linear systems in Banach spaces, extending some results in [22]. Assuming that the kernel a(t) is a creep function which satisfies a(0+) > 0, we prove an analogue version of the Weiss conjecture for scalar Volterra linear systems when p = 1. To this end, we also show that the finite-time and infinite-time (resp. finite-time and uniform finite-time) L1-admissibility coincide for exponentially stable resolvent families (reps. for reflexive state space), extending well-known results for semigroups.