Linear mappings of local preserving-majorization on matrix algebras

Abstract

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 $\phi$ 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.

8 pages