Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on M_mn

Let X,Y∈Mmn. We say that X is ultraweak Hadamard majorized by Y, denoted by X≺uwHY, if there exists a matrix D=[dij]∈Mmn, where 0≤dij≤1, such that X=D∘Y. Also, we say that X is row Hadamard majorized (resp. weakly Hadamard majorized) by Y, denoted by X≺rHY (resp. X≺wHY), if there exists a row stochastic matrix R (resp. doubly substochastic matrix D), such that X=R∘Y(resp. X=D∘Y). In this paper, some properties of ≺uwH,≺rH and ≺wH on Mmn are first obtained, and then, the (strong) linear preservers of ≺uw H,≺r H and ≺w H on Mmn are characterized.