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POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

Part of: Measures, integration, derivative, holomorphy Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  22 March 2011

MICHAEL G. COWLING  and
MICHAEL LEINERT
MICHAEL G. COWLING*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, Australia (email: m.cowling@unsw.edu.au)
MICHAEL LEINERT
Affiliation:
Institut für angewandte Mathematik, Im Neuenheimer Ruprechts-Karl-Universität Heidelberg, D-69120 Heidelberg, Germany (email: leinert@math.uni-heidelberg.de)
*
✉For correspondence; e-mail: m.cowling@unsw.edu.au
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Abstract

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A submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

Keywords

semigroupsvector-valuedpointwise convergence

MSC classification

Secondary: 47D06: One-parameter semigroups and linear evolution equations 46G20: Infinite-dimensional holomorphy

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 44 - 48
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

It is a pleasure for the first-named author to acknowledge the generous support of an Alexander von Humboldt Foundation Research Prize, and the hospitality of the University of Heidelberg.

References

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