Abstract
This paper is concerned first with the behaviour of differences T(t)-T(s) near the origin, where (T(t)) is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane (in which case it is assumed analytic). For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T(t)-T(2t) have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.
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Supported by EPSRC (EP/F020341/1); the third author is partially supported by the research project AHPI, funded by ANR
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Bendouad, Z., Chalendar, I., Esterle, J. et al. Distances between elements of a semigroup and estimates for derivatives. Acta. Math. Sin.-English Ser. 26, 2239–2254 (2010). https://doi.org/10.1007/s10114-010-8569-6
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DOI: https://doi.org/10.1007/s10114-010-8569-6