On Row-Sum Majorization

Let Mn,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{M}_{n,m}$$\end{document} be the set of all n × m real or complex matrices. For A, B ∈ Mn,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{M}_{n,m}$$\end{document}, we say that A is row-sum majorized by B (written as A ≺rsB) if R(A) ≺ R(B), where R(A) is the row sum vector of A and ≺ is the classical majorization on ℝn. In the present paper, the structure of all linear operators T:Mn,m→Mn,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T : \mathbb{M}_{n,m} \rightarrow \mathbb{M}_{n,m}$$\end{document} preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on ℝn and then find the linear preservers of row-sum majorization of these relations on Mn,m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{M}_{n,m}$$\end{document}.