Linear Preservers of Majorization

For vectors $X, Y\in \mathbb{R}^{n}$, we say $X$ is left matrix majorized by $Y$ and write $X \prec_{\ell} Y$ if for some row stochastic matrix $R,~X=RY.$ Also, we write $X\sim_{\ell}Y,$ when $X\prec_{\ell}Y\prec_{\ell}X.$ A linear operator $T\colon \mathbb{R}^{p}\to \mathbb{R}^{n}$ is said to be a linear preserver of a given relation $\prec$ if $X\prec Y$ on $\mathbb{R}^{p}$ implies that $TX\prec TY$ on $\mathbb{R}^{n}$. In this note we study linear preservers of $\sim_{\ell}$ from $\mathbb{R}^{p}$ to $\mathbb{R}^{n}.$ In particular, we characterize all linear preservers of $\sim_{\ell}$ from $\mathbb{R}^{2}$ to $\mathbb{R}^{n},$ and also, all linear preservers of $\sim_{\ell}$ from $\mathbb{R}^{p}$ to $\mathbb{R}^{p}.$