Strictly sub row Hadamard majorization

Let Mm,nMm,n be the set of all mm-by-nn real matrices‎. ‎A matrix RR in Mm,nMm,n with nonnegative entries is called strictly sub row stochastic if the sum of entries on every row of RR is less than 1‎. ‎For A,B∈Mm,nA,B∈Mm,n‎, ‎we say that AA is strictly sub row Hadamard majorized by BB (denoted by A≺SHB)A≺SHB) if there exists an mm-by-nn strictly sub row stochastic matrix RR such that A=R∘BA=R∘B where X∘YX∘Y is the Hadamard product (entrywise product) of matrices X,Y∈Mm,nX,Y∈Mm,n‎. ‎In this paper‎, ‎we introduce the concept of strictly sub row Hadamard majorization as a relation on Mm,nMm,n‎. ‎Also‎, ‎we find the structure of all linear operators T:Mm,n→Mm,nT:Mm,n→Mm,n which are preservers (resp‎. ‎strong preservers) of strictly sub row Hadamard majorization‎.