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Subordinate semigroups and order properties

Part of: Abstract harmonic analysis Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Akitaka Kishimoto  and
Derek W. Robinson
Akitaka Kishimoto
Affiliation:
Department of Pure Mathematics, University of New South Wales, Kensington 2033, Australia
Derek W. Robinson
Affiliation:
Department of Pure Mathematics, University of New South Wales, Kensington 2033, Australia
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Abstract

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Let St = exp{−tH}, Tt = exp{−tK}, be C0-semigroups on a Banach space . For appropriate f one can define subordinate semigroups Sft = exp{−tf(H)}, Ttf = exp{−tf(K)}, on and examine order properties of the pairs S, T, and Sf, Tf. If , = Lp(X;dv) we define St≽ Tt ≽ 0 if St − Tt and Tt map non-negative functions into non-negative functions. Then for p fixed in the range 1 > p > ∞ we characterize the functions for which St ≽ Tt ≽ 0 implies Sft ≽ Tft ≽ 0 for each Lp(X;dv) and the converse is true for all Lp(X;dv). Further we give irreducibility criteria for the strict ordering of holomorphic semigroups. This extends earlier results for L2-spaces.

MSC classification

Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 43A32: Other transforms and operators of Fourier type

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 1 , June 1981 , pp. 59 - 76
Copyright
Copyright © Australian Mathematical Society 1981

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