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.