Abel averages and holomorphically pseudo-contractive maps in Banach spaces

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 pseudo-contractivity, is introduced by means of the Abel averages of such maps. 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.