This is an announcement for the paper “On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces” by Debmalya Sainhttps://arxiv.org/find/math/1/au:+Sain_D/0/1/0/all/0/1, Puja Ghoshhttps://arxiv.org/find/math/1/au:+Ghosh_P/0/1/0/all/0/1, Kallol Paulhttps://arxiv.org/find/math/1/au:+Paul_K/0/1/0/all/0/1.
Abstract: 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 \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 $T\in B(X)$ 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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.03663