Existence of spectral values for irreducible C0-semigroups

AbstractA C0-semigroup (Tt) of positive linear operators on a Banach lattice E is called irreducible if it leaves no closed lattice ideals ≠ {0}, E invariant. It is shown that all known (and some recently discovered) conditions on E and/or an irreducible cyclic semigroup (Tn) implying that σ(T) ≠ {0}, have precise analogs for irreducible semigroups, to the effect that the generator A has nonvoid spectrum and, if (Tt) is eventually compact or A has compact resolvent, that the spectral bound s(A) is an eigenvalue of A with positive eigenvector