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In this paper we characterize Birkhoff-James orthogonality of linearoperators defined on a finite dimensional real Banach space $ \mathbb{X}. $ Wealso explore the symmetry of Birkhoff-James orthogonality of linear operatorsdefined on $ \mathbb{X}. $ Using some of the related results proved in thispaper, 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 andonly if $ T $ is the zero operator. We conjecture that the result holds for anyfinite dimensional strictly convex and smooth real Banach space $ \mathbb{X}, $in particular for the Banach spaces $ l_{p}^{n} (p > 1, p \neq \infty). $

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