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Abstract: We characterise contractivity, boundedness and polynomial boundedness for aC 0-semigroup on a Banach space in terms of its cogenerator V or the Cayleytransform of the generator or its resolvent. In particular, we extend resultsof Gomilko and Brenner, Thomee and show that polynomial boundedness of asemigroup implies polynomial boundedness of its cogenerator. As is shown by anexample, the result is optimal. For analytic semigroups we show that theconverse holds, i.e., polynomial boundedness of the cogenerators impliespolynomial boundedness of the semigroup.In addition, we show by simple examples in C^2,\|\cdot\| p, p eq 2, thatour results on the characterisation of contractivity are sharp. These examplesalso show that the famous Foias-Sz.-Nagy theorem on cogenerators of contractionsemigroups on Hilbert spaces fails in C^2,\|\cdot\| p for p eq 2.



Author: Tanja Eisner, Hans Zwart

Source: https://arxiv.org/







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