Linear Preserves of Logarithm Majorization
DOI:
https://doi.org/10.30495/jme.v17i0.2348Keywords:
doubly stochastic, linear preserver, logarithm majorizationAbstract
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.
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