The modulus semigroup for linear delay equations II

Abstract

The main purpose of this paper is describing the generator of the modulus semigroup of the $C₀$-semigroup associated with the delay equation $$\begin{cases} u'(t)=Au(t)+Lu_t & (t≥ 0)\spe, \\ u(0)=x∈ X, & u₀ =f∈ L_p(-h,0; X)\spe,\end{cases} $$ in the Banach lattice $X? L_p(-h,0; X)$, where X is a Banach lattice with order continuous norm. As a preparation it is shown that $W¹_p(a,b; X)$ is a sublattice of $L_p(a,b; X)$, for $1 ≤ p < ∈fty$. A further preparation is the computation of the modulus of the operator L appearing above. Also, we establish a result concerning the existence of the modulus semigroup for $C₀$-semigroups acting in $KB$-spaces.