On spectral properties of positive operators

AbstractLet T be an invertible positive operator on a Banach lattice E such that the number 0 belongs to the unbounded connected component of the resolvent set of T, then some power of T dominates a positive multiple of the identity operator I on E, i.e., there exist a positive number a and a positive integer k such that Tk≥a·I. As consequences, some theorems on the peripheral spectrum of such operators are deduced. In particular, it is proved that any positive operator with spectrum contained in the spectral circle has a cyclic spectrum. Various applications of the main theorem are given