The Weiss Conjecture for Bounded Analytic Semigroups |
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Authors: | le Merdy Christian |
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Institution: | Département de Mathématiques, Université de Franche-Comté 25030 Besançon Cedex, France lemerdy{at}math.univ-fcomte.fr |
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Abstract: | New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let
be a bounded analytic semigroup withgenerator A on some Banach space X. It is proved thatif A1/2 is admissible for A, that is, if there is an estimate
then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate
.for , Re( )<0. This holds in particular if
is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given. |
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