Strictly sub row Hadamard majorization

‎Let \textbf{M}_{m,n} be the set of all m-by-n real matrices‎. ‎A matrix R in \textbf{M}_{m,n} with nonnegative entries is called strictly sub row stochastic if the sum of entries on every row of R is less than ۱‎. ‎For A,B\in\textbf{M}_{m,n}‎, ‎we say that A is strictly sub row Hadamard majorized by B (denoted by A\prec_{SH}B) if there exists an m-by-n strictly sub row stochastic matrix R such that A=R\circ B where X \circ Y is the Hadamard product (entrywise product) of matrices X,Y\in\textbf{M}_{m,n}‎. ‎In this paper‎, ‎we introduce the concept of strictly sub row Hadamard majorization as a relation on \textbf{M}_{m,n}‎. ‎Also‎, ‎we find the structure of all linear operators T:\textbf{M}_{m,n} \rightarrow \textbf{M}_{m,n} which are preservers (resp‎. ‎strong preservers) of strictly sub row Hadamard majorization‎.

Keywords:

Linear preserver‎ , ‎Strong linear preserver‎ , ‎Strictly sub row stochastic‎ matrices