A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call {sl pseudo-contractivity}, is introduced by means of the Abel averages of such m
aps. We show that the studied maps are dissipative in the spirit of the classical Lumer-Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.
Necessary and sufficient conditions are presented for the Abel averages of discrete and strongly continuous semigroups, $T^k$ and $T_t$, to be power convergent in the operator norm in a complex Banach space. These results cover also the case where $T
$ is unbounded and the corresponding Abel average is defined by means of the resolvent of $T$. They complement the classical results by Michael Lin establishing sufficient conditions for the corresponding convergence for a bounded $T$.
