## Advances in Operator Theory

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

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

Puja Ghosh, Debmalya Sain, and Kallol Paul

#### 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**

Received: 15 March 2017

Accepted: 12 June 2017

First available in Project Euclid: 4 December 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.22034/aot.1703-1137

**Mathematical Reviews number (MathSciNet)**

MR3730038

**Zentralblatt MATH identifier**

06804219

**Subjects**

Primary: 46B20: Geometry and structure of normed linear spaces

Secondary: 47A30: Norms (inequalities, more than one norm, etc.)

**Keywords**

Birkhoff-James Orthogonality left syemmetric operator right symmetric operator

#### 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