Ghosh, P., Sain, D., Paul, K. (2017). On symmetry of Birkhoff-James orthogonality of linear operators. Advances in Operator Theory, 2(4), 428-434. doi: 10.22034/aot.1703-1137

Puja Ghosh; Debmalya Sain; Kallol Paul. "On symmetry of Birkhoff-James orthogonality of linear operators". Advances in Operator Theory, 2, 4, 2017, 428-434. doi: 10.22034/aot.1703-1137

Ghosh, P., Sain, D., Paul, K. (2017). 'On symmetry of Birkhoff-James orthogonality of linear operators', Advances in Operator Theory, 2(4), pp. 428-434. doi: 10.22034/aot.1703-1137

Ghosh, P., Sain, D., Paul, K. On symmetry of Birkhoff-James orthogonality of linear operators. Advances in Operator Theory, 2017; 2(4): 428-434. doi: 10.22034/aot.1703-1137

On symmetry of Birkhoff-James orthogonality of linear operators

Receive Date: 15 March 2017,
Revise Date: 06 June 2017,
Accept Date: 12 June 2017

Abstract

A bounded linear operator $T$ on a normed linear space $\mathbb{X}$ is said to be right symmetric (left symmetric) if $A\perp_{B} T \Rightarrow T \perp_B A $ ($T \perp_{B} A \Rightarrow A \perp_B T $) for all $ A \in B(\mathbb{X}),$ the space of all bounded linear operators on $\mathbb{X}$. Turnsek [Linear Algebra Appl., 407 (2005), 189-195] proved that if $\mathbb{X}$ is a Hilbert space then $T$ is right symmetric if and only if $T$ is a scalar multiple of an isometry or coisometry. This result fails in general if the Hilbert space is replaced by a Banach space. The characterization of right and left symmetric operators on a Banach space is still open. In this paper we study the orthogonality in the sense of Birkhoff-James of bounded linear operators on $ (\mathbb{R}^n, |\cdot|_{\infty}) $ and characterize the right symmetric and left symmetric operators on $(\mathbb{R}^n,|\cdot|_{\infty}).$