### On symmetry of Birkhoff-James orthogonality of linear operators

#### 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 $\mathbb{X}$. Turnšek [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 $(\mathbb{R}^n, \Vert \cdot \Vert_{\infty})$ and characterize the right symmetric and left symmetric operators on $(\mathbb{R}^n, \Vert \cdot \Vert_{\infty})$.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 428-434.

Dates
Accepted: 12 June 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431719

Digital Object Identifier
doi:10.22034/aot.1703-1137

Mathematical Reviews number (MathSciNet)
MR3730038

Zentralblatt MATH identifier
06804219

#### Citation

Ghosh, Puja; Sain, Debmalya; Paul, Kallol. On symmetry of Birkhoff-James orthogonality of linear operators. Adv. Oper. Theory 2 (2017), no. 4, 428--434. doi:10.22034/aot.1703-1137. https://projecteuclid.org/euclid.aot/1512431719

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