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