Fixed points of holomorphic mappings for domains in Banach spaces

We discuss the Earle‐Hamilton fixed‐point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle‐Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer‐Phillips theorem estimating resolvents to dissipative holomorphic functions.