On symmetric functions and symmetric operators on Banach spaces

TL;DR

This paper analyzes left and right symmetry with respect to Birkhoff–James orthogonality in spaces of vector-valued bounded and continuous functions, and extends these insights to operator spaces. It provides explicit one-point support characterizations for left symmetry in $\ell_{\infty}(K, \mathbb{X})$ and $C_0(K, \mathbb{X})$ under suitable topological conditions on $K$ (e.g., locally compact and perfectly normal), and analogous criteria for right symmetry. It then develops operator-space results, giving complete characterizations of left/right symmetric operators in spaces such as $\mathbb{K}(\mathbb{X}, C(K))$, $\mathbb{L}(\ell_1^n, \mathbb{X})$, and $\mathbb{L}(\mathbb{X}, \ell_{\infty}^n)$, with consequences for onto isometries. Overall, the work unifies and generalizes existing findings on symmetric points in function and operator spaces, enriching the geometry of Banach spaces through symmetry considerations.

Abstract

We study left symmetric and right symmetric elements in the space $\ell_{\infty}(K, \mathbb{X}) $ of bounded functions from a non-empty set $K$ to a Banach space $\mathbb{X}.$ We prove that a non-zero element $ f \in\ell_{\infty}(K, \mathbb{X}) $ is left symmetric if and only if $f$ is zero except for an element $k_0 \in K$ and $f(k_0)$ is left symmetric in $\mathbb{X}.$ We characterize left symmetric elements in the space $C_0(K, \mathbb{X}),$ where $K$ is a locally compact perfectly normal space. We also study the right symmetric elements in $\ell_{\infty}(K, \mathbb{X}).$ Furthermore, we characterize right symmetric elements in $C_0(K, \mathbb{X}),$ where $K$ is a locally compact Hausdorff space and $\mathbb{X}$ is real Banach space. As an application of the results obtained in this article, we characterize the left symmetric and right symmetric operators on some special Banach spaces. These results improve and generalize the existing ones on the study of left and right symmetric elements in operator spaces.

Theorems & Definitions (63)

• Definition 1.1
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Corollary 2.4
• Proposition 2.5
• proof
• Theorem 2.6
• proof
• Corollary 2.7