Linear preservers of two sided matrix majorization

For vectors X, Y ∈ Rn, it is said that X is left matrix majorizedby Y if for some row stochastic matrix R X = RY. The relationX ∼` Y, is defined as follows: X ∼` Y if and only if X is leftmatrix majorized by Y and Y is left matrix majorized by X. Alinear operator T : Rp → Rn is said to be a linear preserver ofa given relation ≺ if X ≺ Y on Rp implies that T X ≺ TY onRn. The linear preservers of ≺` from Rp to Rn are characterizedbefore. In this parer we characterize the linear preservers of ∼`from Rp to Rn, p ≥ 3. In fact we show that the linear preserversof ∼` from Rp to Rn are the same as the linear preservers of ≺`from Rp to Rn, but for p = 2, they are not the same.