Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on \(M_{mn}\)


Let \(X,Y\in M_{mn}\). We say that X is ultraweak Hadamard majorized by Y, denoted by \(X\prec _H^{uw} Y\), if there exists a matrix \(D=[d_{ij}]\in M_{mn}\), where \(0\le d_{ij}\le 1\), such that \(X=D\circ Y\). Also, we say that X is row Hadamard majorized (resp. weakly Hadamard majorized) by Y, denoted by \(X \prec ^{r}_{H} Y\) (resp. \(X \prec ^{w}_{H} Y\)), if there exists a row stochastic matrix R (resp. doubly substochastic matrix D), such that \(X=R\circ Y\)(resp. \(X=D\circ Y\)). In this paper, some properties of \( \prec _H^{uw}, \prec ^{r}_{H}\) and \(\prec ^{w}_{H}\) on \(M_{mn}\) are first obtained, and then, the (strong) linear preservers of \( \prec _H^{uw}, \prec ^{r}_{H}\) and \(\prec ^{w}_{H}\) on \(M_{mn}\) are characterized.

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Thanks to the referee and the editor for their comments.

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Correspondence to Ahmad Mohammadhasani.

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Mohammadhasani, A. Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on \(M_{mn}\). Bull. Iran. Math. Soc. 47, 1–11 (2021).

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  • Ultraweak Hadamard majorization
  • Row Hadamard majorization
  • Weakly Hadamard majorization
  • linear preserver

Mathematics Subject Classification

  • 15A04
  • 15A21
  • 15A51