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# Journal of Operator Theory

Volume 45, Issue 1, Winter 2001  pp. 111-130.

A spectral bound for asymptotically norm-continuous semigroups

Authors Mark D. Blake
Author institution: St. John's College, Oxford, OX1 3JP, England

Summary:  We introduce a new growth bound for $C_0$-semigroups giving information about th e absence of norm-continuity of the semigroup and we give a corresponding spectral bound. For semigroups on gene ral Banach spaces we prove an %%@ inequality 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 we show that the bounds are equal and hence obtain new %%@ characterizations of asymptotically norm-continuous semigroups and semigroups no rm-continuous for $t>0$ in %%@ terms of the resolvent of the infinitesimal generator. In the last section we pr ove 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.

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