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The growth of a C_0-semigroup characterised by its cogenerator

We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomee and show that polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial boundedness of the cogenerators implies polynomial boundedness of the semigroup. In addition, we show by simple examples in (C^2,\|\cdot\|_p), p \neq 2, that our results on the characterisation of contractivity are sharp. These examples also show that the famous Foias-Sz.-Nagy theorem on cogenerators of contraction semigroups on Hilbert spaces fails in (C^2,\|\cdot\|_p) for p\neq 2.