This is the abstract of the paper
  "A Spectral bound for Asymptotically Norm-Continuous Semigroups"
by Mark Blake.

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Abstract: We introduce a new growth bound for $C_0$-semigroups giving
information
about the absence of norm-continuity of the semigroup and we give a
corresponding spectral bound. For semigroups on general Banach spaces we
prove an inequality between these bounds and we give a version of the
spectral mapping theorem in terms of the new growth bound. For semigroups on
Hilbert space we show that the bounds are equal and hence obtain new
characterizations of asymptotically 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 three different definitions of the
essential spectrum and give nice relationships between the new growth bound
and the essential growth bound of the semigroup.