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

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Abstract

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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References

  1. 1.

    Ando, T.: Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl. 118, 163–248 (1989)

  2. 2.

    Beasley, L.B., Lee, S.G., Lee, Y.H.: A characterization of strong preservers of matrix majorization. Linear Algebra Appl. 367, 341–346 (2003)

  3. 3.

    Bhatia, R.: Matrix Analysis. Springer-Verlag, New York (1997)

  4. 4.

    Dahl, G.: Matrix majorization. Linear Algebra Appl. 288, 53–73 (1999)

  5. 5.

    Hasani, A.M., Radjabalipour, M.: On linear preservers of (right) matrix majorization. Linear Algebra Appl. 423, 255–261 (2007)

  6. 6.

    Hasani, A.M., Radjabalipour, M.: The structure of linear operators strongly preserving majorizations of matrices. Electron. J. Linear Algebra 15, 260–268 (2006)

  7. 7.

    Hasani, A.M., Vali, M.A.: Linear maps which preserve or strongly preserve weak majorization, J. Inequal. Appl., 2007, Article ID 82910 (2007)

  8. 8.

    Li, C.K., Poon, E.: Linear operators preserving directional majorization. Linear Algebra Appl. 325, 15–21 (2001)

  9. 9.

    Li, C.K., Tam, B.S., Tsing, N.K.: Linear maps preserving permutation and stochastic matrices. Linear Algebra Appl. 341, 5–22 (2002)

  10. 10.

    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorizations and its Applications, 2nd edn. Springer, New York (2001)

  11. 11.

    Martínez Pería, F.D., Massey, P.G., Silvestre, L.E.: Weak matrix-majorization. Linear Algebra Appl. 403, 343–368 (2005)

  12. 12.

    Motlaghian, S.M., Armandnejad, A., Hall, F.J.: Linear preservers of Hadamard majorization. Electron. J. Linear Algebra 31, 593–609 (2016)

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Acknowledgements

Thanks to the referee and the editor for their comments.

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Correspondence to Ahmad Mohammadhasani.

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Communicated by Abbas Salemi.

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Mohammadhasani, A. Linear Preservers of Ultraweak, Row, and Weakly Hadamard Majorization on \(M_{mn}\). Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00361-1

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Keywords

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

Mathematics Subject Classification

  • 15A04
  • 15A21
  • 15A51