GROWTH RATES AND THE PERIPHERAL SPECTRUM OF POSITIVE OPERATORS

摘要

Let T be a positive operator on a complex Banach lattice. It is a long open problem whether the peripheral spectrum sigma(per) (T) of T is always cyclic. We consider several growth conditions on T, involving its eigenvectors or its resolvent, and show that these conditions provide new sufficient criteria for the cyclicity of the peripheral spectrum of T. Moreover, we give an alternative proof of the recent result that every (WS)-bounded positive operator has cyclic peripheral spectrum. We also consider irreducible operators T. If such an operator is Abel bounded, then it is known that every peripheral eigenvalue of T is algebraically simple. We show that the same is true if T only fulfils the weaker condition of being (WS)-bounded.