On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces

TL;DR

This paper characterizes symmetric linear operators in finite-dimensional real Banach spaces, showing that left symmetry implies the operator is zero, and explores properties of right symmetric operators, especially in smooth and strictly convex spaces.

Contribution

It provides a complete characterization of left symmetric operators as zero operators in certain finite-dimensional Banach spaces and investigates the properties of right symmetric operators.

Findings

Left symmetric operators are exactly the zero operator in strictly convex and smooth spaces.

In two-dimensional spaces, the same characterization holds without smoothness.

Smooth operators cannot be right symmetric in these spaces.

Abstract

We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $ \mathbb{X},$ which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, $ 2016 $}]. We prove that $ T\in B(\mathbb{X}) $ is left symmetric if and only if $ T $ is the zero operator. If $ \mathbb{X} $ is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric.