Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces

TL;DR

This paper characterizes Birkhoff-James orthogonality of linear operators on finite-dimensional Banach spaces, explores its symmetry, and identifies conditions under which operators are left symmetric, notably proving the zero operator is the only left symmetric operator on certain spaces.

Contribution

It provides a characterization of Birkhoff-James orthogonality for operators on finite-dimensional Banach spaces and establishes a key result about left symmetry in specific spaces.

Findings

Zero operator is the only left symmetric operator on l_p^2 for p ≥ 2, p ≠ ∞.

Characterization of Birkhoff-James orthogonality for linear operators.

Conjecture that results extend to all finite-dimensional strictly convex and smooth Banach spaces.

Abstract

In this paper we characterize Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Banach space $ \mathbb{X}. $ We also explore the symmetry of Birkhoff-James orthogonality of linear operators defined on $ \mathbb{X}. $ Using some of the related results proved in this paper, we finally prove that $ T \in \mathbb{L}(l_{p}^2) (p \geq 2, p \neq \infty) $ is left symmetric with respect to Birkhoff-James orthogonality if and only if $ T $ is the zero operator. We conjecture that the result holds for any finite dimensional strictly convex and smooth real Banach space $ \mathbb{X}, $ in particular for the Banach spaces $ l_{p}^{n} (p > 1, p \neq \infty). $