Row and column-majorization on \(\mathbf M_{n,m}\) (Q426094)

For two vectors \(x,y\in\mathcal{R}^{n}\) we say that \(x\) is majorized by \(y\) (\(x\prec_{ls}y\)) if there is a doubly stochastic matrix \(D\) such that \(x=Dy\). For two real matrices \(A\) and \(B\) of size \(n\times m\) (\(A,B\in {\mathbb M}_{n,m}\)) we say that \(A\) is multivariate majorized by \(B\) (\(A\prec_{ls}B\)) if \(A=DB\) for some doubly stochastic matrix \(D\). Several generalizations of this concept are considered. For example, \(A\) is \(ls\)-row majorized by \(B\) (\(A\prec_{ls}^{row}B\)) if every row of \(A\) is majorized by the corresponding row of \(B\). Linear operators \(T: {\mathbb M}_{n,m}\to {\mathbb M}_{n,m}\) which (strongly) preserve some of these majorization relation are studied in this paper. Several types of full characterizations of these operators \(T\) are presented.