Two-sided sgut-majorization and its linear preservers

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-۱ real vectors. An n-by-m matrix R=[r_{ij}] is called g-row substochastic if \sum_{k=۱}^{m} r_{ik}\leq ۱  for all i\     (۱\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}.

Keywords:

Generalized row substochastic matrix , (strong) linear pre- server , two-sided sgut-majorization