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Autumn 2017 On symmetry of Birkhoff-James orthogonality of linear operators
Puja Ghosh, Debmalya Sain, Kallol Paul
Adv. Oper. Theory 2(4): 428-434 (Autumn 2017). DOI: 10.22034/aot.1703-1137

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 $(\mathbb{R}^n, \Vert \cdot \Vert_{\infty})$.

Citation

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Puja Ghosh. Debmalya Sain. Kallol Paul. "On symmetry of Birkhoff-James orthogonality of linear operators." Adv. Oper. Theory 2 (4) 428 - 434, Autumn 2017. https://doi.org/10.22034/aot.1703-1137

Information

Received: 15 March 2017; Accepted: 12 June 2017; Published: Autumn 2017
First available in Project Euclid: 4 December 2017

zbMATH: 06804219
MathSciNet: MR3730038
Digital Object Identifier: 10.22034/aot.1703-1137

Subjects:
Primary: 46B20
Secondary: 47A30

Rights: Copyright © 2017 Tusi Mathematical Research Group

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Vol.2 • No. 4 • Autumn 2017
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