On symmetricity of orthogonality in function spaces and space of operators on Banach spaces
Shamim Sohel, Debmalya Sain, Kallol Paul
TL;DR
The paper extends symmetry analyses from Birkhoff–James orthogonality to $\rho$-orthogonality in Banach spaces, focusing on $C(K,\mathbb{X})$ and operator spaces. It develops characterizations of $(\rho_+)$-, $(\rho_-)$-, and $\rho$-symmetric points, linking them to maximal and extreme points and to evaluation functionals, with special attention to when the compact space $K$ is perfectly normal. In operator spaces, it shows that left symmetry of a compact operator $T$ imposes corresponding symmetry on $T$'s action at maximal points, and provides necessary/sufficient conditions that lead to nonexistence results in several settings. The work culminates in concrete criteria for function and operator spaces, including $C(K)$, $\ell_{\infty}^n$, and $\mathbb{K}(\mathbb{X},C(K))$, and analyzes rank-one operators, thereby contributing to the geometric understanding of isometries and orthogonality in Banach spaces.
Abstract
We study symmetric points with respect to $(ρ_+)$-orthogonality, $(ρ_{-})$-orthogonality and $ρ$-orthogonality in the space $C(K, \mathbb{X}),$ where $K$ is a perfectly normal, compact space and $ \mathbb X$ is a Banach space. We characterize left symmetric points and right symmetric points in $C(K, \mathbb{X})$ with respect to $(ρ_{+})$-orthogonality and $(ρ_{-})$-orthogonality, separately. Furthermore, we provide necessary conditions for left symmetric and right symmetric points with respect to $ρ$-orthogonality. As an application of these results we also study these symmetric points in the space of operators defined on some special Banach spaces.