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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On subordinated holomorphic semigroups

Authors: Alfred S. Carasso and Tosio Kato
Journal: Trans. Amer. Math. Soc. 327 (1991), 867-878
MSC: Primary 47D03; Secondary 60J35
MathSciNet review: 1018572
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Abstract: If $ [{e^{ - tA}}]$ is a uniformly bounded $ {C_0}$ semigroup on a complex Banach space $ X$, then $ - {A^\alpha },$, $ 0 < \alpha < 1$, generates a holomorphic semigroup on $ X$, and $ [{e^{ - t{A^\alpha }}}]$ is subordinated to $ [{e^{ - tA}}]$ through the Lévy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on the measure's Laplace transform for that to be the case. We then construct probability measures that do not have this property.

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Keywords: Subordinated semigroups, holomorphic extensions
Article copyright: © Copyright 1991 American Mathematical Society

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