Evolution Semigroups and Sums of Commuting Operators: A New Approach to the Admissibility Theory of Function Spaces

Abstract

This paper is concerned with conditions for the admissibility of a translation invariant function space M with respect to a well posed linear evolution equation du/dt=Au+f(t), t∈R (*). We propose a new approach to this problem by considering the sum of two commuting operators −d/dt≔−DM and the operator of multiplication by A on M. On the one hand, the closure of this operator is the infinitesimal generator of the so-called evolution semigroup associated with (*). On the other hand, the generator G of this semigroup relates a mild solution u of (*) to the forcing term f by the rule Gu=−f. Consequently, various spectral criteria of the type σ(DM)∩σ(A)=∅ for the admissibility of the function space M with respect to (*) can be proved in an elegant manner. Moreover, they can be naturally extended to general classes of differential equations, including higher order and abstract functional differential equations. Applications and examples are provided to illustrate the obtained results.