Preservers of Gershgorin Set of Jordan Product of Matrices

Manoj Joshi; Kota Nagalakshmi Rajeshwari; Kilambi Santaram; Sandeep Kanodia

doi:10.5666/KMJ.2018.58.4.589

Articles

Kyungpook Mathematical Journal 2018; 58(4): 589-597

Published online December 31, 2018

Preservers of Gershgorin Set of Jordan Product of Matrices

Manoj Joshi^∗, Kota Nagalakshmi Rajeshwari and Kilambi Santaram, Sandeep Kanodia

Department of Mathematics, Maharaja Ranjit Singh College of Professional Sciences, Indore (M. P.) 452001, India
e-mail : manojrjoshimrsc@yahoo.com

School of Mathematics, Devi Ahilya University, Indore (M. P.) 452001, India
e-mail : knr_k@yahoo.co.in and drksantaram@gmail.com

Department of Mathematics, Sri Aurobindo Institute of Technology, Indore (M. P.) 453555, India
e-mail : sandeepkanodia11@gmail.com

Received: July 24, 2017; Revised: January 29, 2018; Accepted: March 9, 2018

Abstract

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Abstract

For A, B ∈ M₂(ℂ), let the Jordan product be AB + BA and G(A) the eigenvalue inclusion set, the Gershgorin set of A. Characterization is obtained for maps φ : M₂(ℂ) → M₂(ℂ) satisfying G[φ(A)φ(B)+φ(B)φ(A)]=G(AB+BA)

for all matrices A and B. In fact, it is shown that such a map has the form φ(A) = ±(PD)A(PD)⁻¹, where P is a permutation matrix and D is a unitary diagonal matrix in M₂(ℂ).

Keywords: Eigenvalue, Inclusion sets, Jordan Product, Preservers, Gershgorin Set.