‎Linear Preserves of Logarithm Majorization

Abstract

Let $X‎, ‎Y\in \Bbb R^n‎, ‎X,Y>0$‎, ‎we say $X$ {\it logarithm majorized} by $Y$‎, ‎written $X\prec_{log} Y$ if $\log X\prec \log Y$‎. ‎Let $M_{nm}^+$ be the collection of matrices with positive entries‎. ‎For $X,Y\in M_{nm}^+$‎, ‎it is said that $X$ is {\it logarithm column (row) majorized} by $Y$‎, ‎and is denoted as $X\prec_{log}^{column} Y (X\prec_{log}^{row}Y)$‎, ‎if $X_{j}\prec_{log} Y_{j} (X_{i}\prec_{log} Y_{i}) $ for all $j=1,2,\cdots m (i=1,2,\cdots n)$‎, ‎where $X_{j}$ and $Y_{j}$ ($X_{i}$ and $Y_{i}$) are the ith column (row) of $X$ and $Y$ respectively‎.

‎In the present paper‎, ‎the relations column (row) logarithm majorization on $M_{nm}^+$ are studied and also all linear operators $T:M_{nm}^+\longrightarrow M_{nm}^+$ preserving column (row) logarithm majorization will be characterized‎.