Existence of spectral values for irreducible \(C_ 0\)-semigroups (Q1100416)

A \(C_ 0\)-semigroup \((T_ t)\) of positive linear operators on a Banach lattice E is called irreducible if it leaves no closed lattice ideals \(\neq \{0\}\), E invariant. Let us denote by A the generator of \((T_ t)\) and s(A) the spectral bound sup\(\{\) Re \(\lambda\) : \(\lambda\in \sigma (A)\}\). It is proved that each of the following conditions on E and \(T_ t\) implies that \(s(A)>-\infty:\) (i) \(E=C_ 0(U)\), U locally compact. (ii) The positive cone \(E_+\) contains an extreme ray. (iii) For some \(t>0\), \(T_ t\) is compact. (iii') The generator A has compact resolvent. (iv) E is order complete and for some \(t>0\), \(T_ t\) is an abstract kernel operator. In particular, in cases (iii) and (iii') the generator A of \(T_ t\) has the eigenvalue s(A) with corresponding positive eigenvector. Analogous results are established for the band irreducible \(C_ 0\)- semigroups.