Symmetric properties of orthogonality of linear operators on $(\mathbb{R}^n, \|.\|_{1}) $

Ghosh, Puja ; Paul, Kallol ; Sain, Debmalya

Novi Sad Journal of Mathematics, Tome 46 (2017), / Harvested from Faculty of Science, Novi Sad

Résumé

In [4] we studied symmetric properties of orthogonality of linear operators on a finite dimensional real Hilbert space $\mathbb{H}$ and proved that for  a bounded linear operator $T, ~ T\perp_B A \Rightarrow A \perp_B T$ for all $A \in B(\mathbb{H}) $ if and only if $T$ is the zero operator, we also proved that $A\perp_B T \Rightarrow T\perp_B A$ for all $A \in B(\mathbb{H}) $ if and only if $T$ attains norm at all points of the unit sphere. These results fail in general if  the Hilbert space is replaced by a Banach space. 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. 

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Publié le : 2017-01-01