Group Invariant Operators and Some Applications to Norm-Attaining Theory

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualiz...

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Bibliographic Details
Published in:Resultate der Mathematik 2023-02, Vol.78 (1), Article 24
Main Authors: Dantas, Sheldon, Falcó, Javier, Jung, Mingu
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, which says that if X has the Radon–Nikodým property, then X has the G -Bishop–Phelps property for G -invariant operators whenever G ⊆ L ( X ) is a compact group of isometries on X .
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-022-01796-0