Group invariant operators and some applications to norm-attaining theory (Q2102750)

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 \(L_G(X, Y )\).