Advances in Operator Theory

On symmetry of Birkhoff-James orthogonality of linear operators

Puja Ghosh, Debmalya Sain, and Kallol Paul

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 47A30: Norms (inequalities, more than one norm, etc.)

Birkhoff-James Orthogonality left syemmetric operator right symmetric operator


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.

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