ON SYMMETRY OF BIRKHOFF-JAMES ORTHOGONALITY OF LINEAR OPERATORS ON FINITE-DIMENSIONAL REAL BANACH SPACES

We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space X, which answers a question raised recently by one of the authors in [7] [D. Sain, Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl. 447 (2017) 860-866]. We prove that T is an element of B(X) is left symmetric if and only if T is the zero operator. If 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.