Symmetry of Birkhoff-James orthogonality of operators defined between infinite dimensional Banach spaces

TL;DR

This paper characterizes when bounded linear operators between infinite dimensional Banach spaces are symmetric with respect to Birkhoff-James orthogonality, showing that non-zero symmetry occurs only under specific convexity conditions.

Contribution

It provides a complete characterization of left symmetric operators between strictly convex Banach spaces, establishing they must be zero, and explores non-zero cases when convexity conditions are relaxed.

Findings

Left symmetric operators are zero between strictly convex Banach spaces.

Non-zero left symmetric operators exist when spaces are not strictly convex.

The paper analyzes right symmetric operators in the same context.

Abstract

We study left symmetric bounded linear operators in the sense of Birkhoff-James orthogonality defined between infinite dimensional Banach spaces. We prove that a bounded linear operator defined between two strictly convex Banach spaces is left symmetric if and only if it is zero operator when the domain space is reflexive and Kadets-Klee. We exhibit a non-zero left symmetric operator when the spaces are not strictly convex. We also study right symmetric bounded linear operators between infinite dimensional Banach spaces.