Symmetric properties of orthogonality of linear operators on $(R^n, ||.||_1)$


Puja Ghosh, Kallol Paul, Debmalya Sain




In this paper we study the orthogonality in the sense of Birkhoff-James of bounded linear operators on $ (\mathbb{R}^n, \|.\|_{1}).$ We prove that $T\perp_B A \Rightarrow A\perp_B T$ for all operators $A$ on $(\mathbb{R}^n, \|.\|_1 )$ if and only if $T$ attains norm at only one extreme point, image of which is a left symmetric point of $(\mathbb{R}^n, \|.\|_1)$ and images of other extreme points are zero. We also prove that $A\perp_B T \Rightarrow T\perp_B A$ for all operators $A$ on $(\mathbb{R}^n, \|.\|_1 )$ if and only if $T$ attains norm at all extreme points and images of the extreme points are scalar multiples of extreme points.