Linear mappings of local preserving-majorization on matrix algebras

Let $\M_{n\times n}$ be the algebra of all $n\times n$ matrices. For $x,y\in {R}^{n}$ it is said that $x$ is majorized by $y$ if there is a double stochastic matrix $A\in {M}_{n\times n}$ such that $x=Ay$ (denoted by $x\prec y$). Suppose that $\Phi$ is a linear mapping from ${R}^{n}$ into ${R}^{n}$, which is said to be strictly isotone if $\Phi(x)\prec \Phi(y)$ whenever $x\prec y$. We say that an element $\alpha\in {R}^{n}$ is a strictly all-isotone point if every strictly isotone $\varphi$ at $\alpha$ (i.e. $\Phi(\alpha)\prec\Phi(y)$ whenever $x\in {R}^{n}$ with $\alpha\prec x$, and $\Phi(x)\prec\Phi(\alpha)$ whenever $x\in {R}^{n}$ with $x\prec \alpha$) is a strictly isotone. In this paper we show that every $\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{n})\in {R}^{n}$ with $\alpha_{1}>\alpha_{2}>...>\alpha_{n}$ is a strictly all-isotone point.