A spectral bound for asymptotically norm-continuous semigroups

We introduce a new growth bound for C-o-semigroups giving information about the absence of norm-continuity of the semigroup and we give a correspondin g spectral bound. For semigroups on general Banach spaces we prove an inequ ality between these bounds and we give a version of the spectral mapping th eorem in terms of the new growth bound. For semigroups on Hilbert space Re show that the bounds are equal and hence obtain new characterizations of as ymptotically norm-continuous semigroups and semigroups norm-continuous for t > 0 in terms of the resolvent of the infinitesimal generator. In the last section we prove that versions of the spectral mapping theorem holds for t hree different definitions of the essential spectrum and give nice relation ships between the new growth bound and the essential growth bound of the se migroup.