Two-sided sgut-majorization and its linear preservers

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 all $n$-by-$m$ real matrices, and let $\mathbb{R}^{n}$ be  the set of all $n$-by-$1$ real vectors. An $n$-by-$m$ matrix $R=[r_{ij}]$ is called g-row substochastic if $\sum_{k=1}^{m} r_{ik}\leq 1$  for all $i\
    (1\leq i \leq n)$.  For $x$, $y \in \mathbb{R}^{n}$, it is said that $x$ is $\textit{sgut-majorized}$ by $y$, and we write  $ x
    \prec_{sgut}y$  if there exists an $n$-by-$n$ upper triangular g-row substochastic matrix $R$ such that $x=Ry$. Define the relation $\sim_{sgut}$ as follows. $x\sim_{sgut}y$ if and only if $x$ is   sgut-majorized  by $y$ and $y$ is sgut-majorized  by $x$.  This paper characterizes all (strong)  linear preservers   of  $\sim_{sgut}$ on $\mathbb{R}^{n}$.


[1] T. Ando, Majorization, doubly stochastic matrices, and comparision of eigenvalues, Linear Algebra Appl., 118 (1989), pp. 163-248.
[2] T. Ando, Majorization and inequalities in matrix theory, Linear Algebra Appl., 199 (1994), pp. 17-67.
[3] A. Armandnejad and Z. Gashool, Strong linear preservers of g-tridiagonal majorization on Rn, Electronic. J. Linear Algebra, 23 (2012), pp. 115-121.
[4] A. Armandnejad and A. Ilkhanizadeh Manesh, Gut-majorization and its linear preservers, Electronic. J. Linear Algebra, 23 (2012), pp. 646-654.
[5] H. Chiang and C. K. Li, Generalized doubly stochastic matrices and linear preservers, Linear and Multilinear Algebra, 53 (2005), pp. 1-11.
[6] G.H. Hardy, J.E. Littlewood, and G. Polya, Some simple inequalities satisfed by convex functions., Messenger of Mathematics , 58 (1929), pp. 145-152.
[7] A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Electron. J. Linear Algebra, 15 (2006), pp. 260-268.
[8] A. M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majorization, Linear Algebra Appl, 423 (2007), pp. 255-261.
[9] A. Ilkhanizadeh Manesh, On linear preservers of sgut-majorization on Mn;m, Bull. Iranian Math. Soc., 42 (2016), pp. 470-481.
[10] A. Ilkhanizadeh Manesh, Right gut-Majorization onMn;m, Electron. J. Linear Algebra, 31 (2016), pp. 13-26.
[11] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of majorization and its applications, Springer, New York, 2011.
[12] I. Schur, Uber enie klasse von mittelbildungen mit anwendungen auf die determinan-tentheorie, Sitzungsberichte der Berliner Mathematischen Gesellschaft, 22 (1923), pp.9-20.