Additive matrix maps that are monotone with respect to the orders induced by group inverse

M. A. Efimov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 23-40.

Résumé

We characterize additive maps on the matrix algebra over an arbitrary field with three or more elements that are monotone with respect to the $\overset\sharp\leq$- and $\overset{\mathrm{cn}}\leq$-orders and build some examples of nonadditive monotone maps.

Bibliographie

[1] Bogdanov I. I., Guterman A. E., “Monotonnye otobrazheniya matrits, zadannye gruppovoi obratnoi, i odnovremennaya diagonalizuemost”, Mat. sb., 198:1 (2007), 3–20 | DOI | MR | Zbl

[2] Efimov M. A., “Lineinye otobrazheniya matrits, monotonnye otnositelno poryadkov $\overset\sharp\leq$ i $\overset{\mathrm{cn}}\leq$”, Fundament. i prikl. mat., 13:4 (2007), 53–66 | MR | Zbl

[3] Pirs R., Assotsiativnye algebry, Mir, M., 1986 | MR

[4] Alieva A., Guterman A., “Monotone linear transformations on matrices are invertible”, Commun. Algebra, 33 (2005), 3335–3352 | DOI | MR | Zbl

[5] Baksalary J. K., Pukelsheim F., Styan G. P. H., “Some properties of matrix partial orderings”, Linear Algebra Appl., 119 (1989), 57–85 | DOI | MR | Zbl

[6] Ben-Israel A., Greville T., Generalized Inverses: Theory and Applications, John Wiley and Sons, New York, 1974 | MR | Zbl

[7] Englefield M. J., “The commuting inverses of a square matrix”, Proc. Cambridge Philos. Soc., 62 (1966), 667–671 | DOI | MR | Zbl

[8] Erdelyi I., “On the matrix equation $Ax=\lambda Bx$”, J. Math. Anal. Appl., 17 (1967), 117–132 | DOI | MR

[9] Guterman A., “Linear preservers for Drazin star partial order”, Commun. Algebra, 29:9 (2001), 3905–3917 | DOI | MR | Zbl

[10] Guterman A., “Linear preservers for matrix inequalities and partial orderings”, Linear Algebra Appl., 331:1–3 (2001), 75–87 | DOI | MR | Zbl

[11] Hartwig R. E., “How to partially order regular elements”, Math. Japon., 25:1 (1980), 1–13 | MR | Zbl

[12] Hartwig R. E., Mitra S. K., “Partial orders based on outer inverses”, Linear Algebra Appl., 176 (1982), 3–20 | MR

[13] Legiša P., “Automorphisms of $\mathrm M_n$, partially ordered by rank subtractivity ordering”, Linear Algebra Appl., 389 (2004), 147–158 | DOI | MR | Zbl

[14] Legiša P., “Automorphisms of $\mathrm M_n$, partially ordered by the star order”, Linear and Multilinear Algebra, 54:3 (2006), 157–188 | DOI | MR | Zbl

[15] Mitra S. K., “A new class of g-inverse of square matrices”, Sankhyā Ser. A, 30 (1963), 323–330 | MR

[16] Mitra S. K., “On group inverses and the sharp order”, Linear Algebra Appl., 92 (1987), 17–37 | DOI | MR | Zbl

[17] Nambooripad K. S. S., “The natural partial order on a regular semigroup”, Proc. Edinburgh Math. Soc., 23 (1980), 249–260 | DOI | MR | Zbl

[18] Ovchinnikov P. G., “Automorphisms of the poset of skew projections”, J. Funct. Anal., 115 (1993), 184–189 | DOI | MR | Zbl

[19] Pierce S. et al., “A survey of linear preserver problems”, Linear and Multilinear Algebra, 33 (1992), 1–119 | DOI | Zbl

[20] De Pillis J., “Linear transformations which preserve Hermitian and positive semidefinite operators”, Pacific J. Math., 23 (1967), 129–137 | DOI | MR | Zbl

[21] Rao C. R., Mitra S. K., Generalized Inverse of Matrices and its Applications, Wiley, New York, 1971 | MR | Zbl

[22] Robert P., “On the group-inverse of a linear transformation”, J. Math. Anal. Appl., 22 (1968), 658–669 | DOI | MR | Zbl

[23] Šemrl P., “Order-preserving maps on the poset of idempotent matrices”, Acta Sci. Math. (Szeged), 69 (2003), 481–490 | MR | Zbl