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Review | Open Access

Another special role of L∞-spaces-evolution equations and Lotz' theorem

University of the Free State, Department of Mathematics and Applied Mathematics, Faculty of Natural and Agriculture Sciences, PO Box 339, Bloemfontein 9300, South Africa
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Abstract

In this paper, we review the underappreciated theorem by Lotz that tells us that every strongly continuous operator semigroup on a Grothendieck space with the Dunford-Pettis property is automatically uniformly continuous. A large class of spaces that carry these geometric properties are L∞(Ω,Σ,μ) for non-negative measure spaces. This shows once again that L∞-spaces have to be treated differently.

Keywords

L∞-spacesLotz' theoremGrothendieck spacesDunford-Pettis property
CLC number: 47D03, 47D06

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AIMS Mathematics
Volume 9 Issue 12,
December 2024
Pages 36158-36166
DOI: 10.3934/math.20241716

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Budde C. Another special role of L∞-spaces-evolution equations and Lotz' theorem. AIMS Mathematics, 2024, 9(12): 36158-36166. https://doi.org/10.3934/math.20241716

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Received: 15 August 2024
Revised: 16 December 2024
Accepted: 19 December 2024
Published: 15 December 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)