Adjoint Semigroup Theory for a Class of Functional Differential Equations

Abstract

We consider the semigroup and adjoint semigroup for a class of linear functional differential equations with infinite delays, which includes certain linear Volterra integro-differential equations. In particular, we show that by an appropriate choice of the state space the semigroup constructed by Miller [13] can be considered the adjoint semigroup of the semigroup constructed by Barbu and Grossman [2]. This provides a useful characterization of Miller’s semigroup which can be applied to obtain additional information about the semigroup defined by Barbu and Grossman.