Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem

Christian Budde; Christian Budde

doi:10.3934/math.20241716

AIMS Mathematics

2024, Volume 9, Issue 12: 36158-36166. doi: 10.3934/math.20241716

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Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem

Christian Budde ^,

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

Received: 15 August 2024 Revised: 16 December 2024 Accepted: 19 December 2024 Published: 27 December 2024
MSC : 47D03, 47D06

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 $ \mathrm{L}^\infty(\Omega, \Sigma, \mu) $ for non-negative measure spaces. This shows once again that $ \mathrm{L}^\infty $-spaces have to be treated differently.
- $ \mathrm{L}^\infty $-spaces,
- Lotz' theorem,
- Grothendieck spaces,
- Dunford-Pettis property
Citation: Christian Budde. Another special role of $ \mathrm{L}^\infty $-spaces-evolution equations and Lotz' theorem[J]. AIMS Mathematics, 2024, 9(12): 36158-36166. doi: 10.3934/math.20241716

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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 $ \mathrm{L}^\infty(\Omega, \Sigma, \mu) $ for non-negative measure spaces. This shows once again that $ \mathrm{L}^\infty $-spaces have to be treated differently.

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