The growth of a C ₀-semigroup characterised by its cogenerator

Abstract.

We characterise contractivity, boundedness and polynomial growth for a C ₀-semigroup 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, Thomée and show that polynomial growth of a semigroup implies polynomial growth 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 growth of the cogenerators implies polynomial growth of the semigroup. In addition, we show by simple examples in (${\mathbb{C}}^2, || · ||_{p}$), $p \neq 2$, that our results on the characterization of contractivity are sharp. These examples also show that the famous Foiaş-Sz.-Nagy theorem on cogenerators of contractive C ₀-semigroups on Hilbert spaces fails in (${\mathbb{C}}^2, || · ||_{p}$) for $p \neq 2$.