Continuous and holomorphic semicocycles in Banach spaces

TL;DR

This paper investigates the properties of semicocycles over semigroups in Banach spaces, establishing continuity, differentiability conditions, and growth bounds, with a focus on holomorphic semicocycles and their generators.

Contribution

It introduces the concept of generators for semicocycles, providing conditions for differentiability and growth bounds, and explores the relationship between continuity and boundedness in holomorphic cases.

Findings

Semicocycles over continuous semigroups are themselves continuous.

Differentiability of semicocycles is characterized by a generator and linear evolution equations.

Holomorphic semicocycles' continuity relates directly to boundedness of their generators.

Abstract

We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator' of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.