Continuous and holomorphic semicocycles in Banach spaces

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 condit...

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Bibliographic Details
Published in:Journal of evolution equations 2019-12, Vol.19 (4), p.1199-1221
Main Authors: Elin, Mark, Jacobzon, Fiana, Katriel, Guy
Format: Article
Language:English
Subjects:
Analysis
Banach spaces
Mathematical analysis
Mathematics
Mathematics and Statistics
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Summary: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 differentiability 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.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-019-00509-5