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• W2051209089 abstract "If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket e Superscript minus t upper A Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mi>t</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[{e^{ - tA}}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a uniformly bounded <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{C_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> semigroup on a complex Banach space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus upper A Superscript alpha Baseline comma> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>α<!-- α --></mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>- {A^alpha },</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than alpha greater-than 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0 > alpha > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, generates a holomorphic semigroup on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket e Superscript minus t upper A Super Superscript alpha Superscript Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>α<!-- α --></mml:mi> </mml:msup> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[{e^{ - t{A^alpha }}}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is subordinated to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket e Superscript minus t upper A Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mi>t</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[{e^{ - tA}}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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." @default.
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• W2051209089 date "1991-01-01" @default.
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• W2051209089 title "On subordinated holomorphic semigroups" @default.
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• W2051209089 doi "https://doi.org/10.1090/s0002-9947-1991-1018572-4" @default.
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