Group invariant operators and some applications to norm-attaining theory
type: Item
instance of: scholarly article
title: Group invariant operators and some applications to norm-attaining theory
author: Sheldon Dantas
author: Javier Falcó
author: Mingu Jung
published in: Results in Mathematics
publication date: 2022-11-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/2110.02066
review text: For Banach spaces \(X\), \(Y\) over the same scalar field denote by \(L(X, Y)\) the space of all bounded linear operators acting from \(X\) to \(Y\). Let \(G \subset L(X,X)\) be a group of isometries. An operator \(T \in L(X, Y)\) is said to be \(G\)-invariant whenever \(T(g(x)) = T(x)\) for every \(x \in X\) and every \(g \in G\); \(K \subset X\) is \(G\)-invariant whenever \(g(K) \subset K\) for every \(g \in G\), and a point \(x \in X\) is \(G\)-invariant if \(\{x\}\) is \(G\)-invariant. Finally, \(L_G(X, Y) \subset L(X, Y)\) is the subset of all \(G\)-invariant operators. The authors demonstrate \(G\)-invariant versions of the Hahn-Banach separation theorems for group invariant mappings. In particular, they prove that, in the case of a compact group \(G\), for every convex \(G\)-invariant subset of \(C \subset X\) and a \(G\)-invariant point \(x_0 \in X\) such that \(\mathrm{dist} (x_0, C) > \delta > 0\) there exists a \(G\)-invariant functional \(x^* \in S_{X^*}\) such that \[ \mathrm{Re}\,x^*(x_0) > \sup \{\mathrm{Re}\,x^*(x) : x \in C\} + \delta. \] After that, the authors concentrate on the questions of approximation of \(G\)-invariant operators by norm-attaining \(G\)-invariant ones, which develops the line of research originated by the second author in [Proc. Am. Math. Soc. 149, No. 4, 1609--1612 (2021; Zbl 1465.46012)], where the case of functionals was considered. They transfer to the \(G\)-invariant setting several examples where such approximation does not work and extend to the \(G\)-invariant case a number of known positive results. In particular, they demonstrate that in a space \(X\) with the Radon-Nikodým property for every compact group of isometries \(G \subset L(X,X)\) and every non-empty closed, absolutely convex bounded \(G\)-invariant set \(C \subset X\), the subset of those \(G\)-invariant operators that attain their supremum on \(C\) is norm dense in \(L_G(X, Y )\).
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 47B01
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 7625273
zbMATH Keywords: Bishop-Phelps theorem
zbMATH Keywords: group invariant operator
zbMATH Keywords: norm attaining operators
zbMATH Keywords: Radon-Nikodým property
zbMATH Keywords: separation theorem
MaRDI profile type: Publication
OpenAlex ID: W3204179302
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DOI: 10.1007/S00025-022-01796-0
description: scientific article; zbMATH DE number 7625273