On linear preservers of sgut-majorization on $\textbf{M}_{n,m}$

Document Type : Research Paper


Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7713936417, Rafsanjan, Iran.


‎Let $\textbf{M}_{n,m}$ be the set of $n$-by-$m$‎ ‎matrices with entries in the field of real numbers‎. ‎A matrix $R$ in $\textbf{M}_{n}=\textbf{M}_{n,n}$ is a generalized row substochastic matrix (g-row substochastic‎, ‎for short) if $Re\leq e$‎, ‎where $e=(1,1,\ldots,1)^t$‎. ‎For $X,$ $Y \in \textbf{M}_{n,m}$‎, ‎$X$ is said to be sgut-majorized by $Y$ (denoted by $X‎ ‎\prec_{sgut} Y$) if there exists an $n$-by-$n$ upper triangular g-row substochastic matrix $R$ such that $X=RY$‎. ‎This paper characterizes all‎ ‎linear preservers and strong linear preservers of $\prec_{sgut}$ on $\mathbb{R}^{n}$ and $\textbf{M}_{n,m}$ respectively‎.


Main Subjects

T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118 (1989) 163--248.
A. Armandnejad, Right gw-majorization on Mn;m, Bull. Iranian math. Soc. 35 (2009), no. 2, 69--76.
A. Armandnejad and H. Heydari, Linear preserving gd-majorization functions from Mn;m to Mn;k, Bull. Iranian math. Soc. 37 (2011), no. 1, 215--224.
A. Armandnejad and A. Ilkhanizadeh Manesh, Gut-majorization and its linear preservers, Electron. J. Linear Algebra 23 (2012) 646--654.
A. Armandnejad and A. Salemi, On linear preservers of lgw-majorization on Mn;m, Bull. Malays. Math. Soc. (2) 35 (2012), no. 3, 755--764.
A. Armandnejad and A. Salemi, The structure of linear preservers of gs- majorization, Bull. Iranian Math. Soc. 32 (2006), no. 2, 31--42.
H. Chiang and C. K. Li, Generalized doubly stochastic matrices and linear preservers, Linear Multilinear Algebra 53 (2005), no. 1, 1--11.
A. M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majorization, Linear Algebra Appl. 423 (2007), no. 2-3, 255--261.
A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Electron. J. Linear Algebra 15 (2006) 260--268.
A. Ilkhanizadeh Manesh, Linear functions preserving sut-majorization on Rn, Iran. J. Math. Sci. Inform., (submission).
A. Ilkhanizadeh Manesh, Right gut-majorization on Mn;m, Electron. J. Linear Algebra, Accepted.
A. Ilkhanizadeh Manesh and A. Armandnejad, Ut-majorization on Rn and its Linear Preservers, Operator theory, operator algebras and applications, 253--259, Oper. Theory Adv. Appl., 242, Birkhäuser-Springer, Basel, 2014.
A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of majorization and its applications, Springer, New York, 2011.
B. Y. Wang, Foundations of Majorization Inequalities, Beijing Normal Univ. Press, Beijing, 1990.