  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.
