Diagonals of positive semigroups

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL_r(E) onto the centerZ(E) ofE. Let {T(t)}_t0 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 >0 the diagonalD(R(, A)) of the resolvent operatorR(, A) is strictly positive, then (D(R(, A)))^–1 is a center valued Bernstein function. As an application of these results it follows that the order limit lim₀D(R(,A)) exists inZ(E) and equals the order limit lim_mD((R(, A))^m) for any >0.