Advances in Operator Theory

On symmetry of Birkhoff-James orthogonality of linear operators

Puja Ghosh, Debmalya Sain, and Kallol Paul

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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


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References

  • G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169–172.
  • R. C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947), 559–566.
  • R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61, 265–292 (1947 b) 69 (1958), 90–104.
  • P. Ghosh, D. Sain, and K. Paul, Orthogonality of bounded linear operators, Linear Algebra Appl. 500 (2016), 43–51.
  • K. Paul, D. Sain, and K. Jha, On strong orthogonality and strictly convex normed linear spaces, J Inequal. Appl. 2013, 2013:242.
  • D. Sain, Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl. 447 (2017), 860–866.
  • D. Sain and K. Paul, Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013), 2448–2452.
  • A. Turnšek,On operators preserving James' orthogonality, Linear Algebra Appl. 407 (2005), 189–195.