Growth rates and the peripheral spectrum of positive operators

Let $T$ be a positive operator on a complex Banach lattice. It is a long open problem whether the peripheral spectrum ${\sigma }_{\mathrm{per}}(T)$ 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.