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 $Φ$ is a linear mapping from ${R}^{n}$ into ${R}^{n}$, which is said to be strictly isotone if $Φ(x)\prec Φ(y)$ whenever $x\prec y$. We say that an element $α\in {R}^{n}$ is a strictly all-isotone point if every strictly isotone $φ$ at $α$ (i.e. $Φ(α)\precΦ(y)$ whenever $x\in {R}^{n}$ with $α\prec x$, and $Φ(x)\precΦ(α)$ whenever $x\in {R}^{n}$ with $x\prec α$) is a strictly isotone. In this paper we show that every $α=(α_{1},α_{2},...,α_{n})\in {R}^{n}$ with $α_{1}>α_{2}>...>α_{n}$ is a strictly all-isotone point.

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