Diagonals of positive semigroups

Abstract

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceLr(E) onto the centerZ(E) ofE. Let {T(t)}t≥0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for λ>0 the diagonalD(R(λ, A)) of the resolvent operatorR(λ, A) is strictly positive, then (D(R(λ, A)))−1 is a center valued Bernstein function. As an application of these results it follows that the order limit limγ↓0γD(R(γ,A)) exists inZ(E) and equals the order limit limm→∞D((λR(λ, A))m) for any λ>0.