Uniform stability of (a,k)-regularized families

Abstract

In this article we study the uniform stability of an (a,k)-regularized family {S(t)}_t≥0 generated by a closed operator A. We give sufficient conditions, on the scalar kernels a, k and the operator A, to ensure the uniform stability of the family {S(t)}_t≥0 in Hilbert spaces. Our main result is a generalization of Theorem 1 in [Proc. Amer. Math. Soc. 132(1) (2004), 175–181], concerning the stability of resolvent families, and can be seen as a substantial generalization of the Gearhart–Greiner–Prüss characterization of exponential stability for strongly continuous semigroups.