Monotone Additive Matrix Transformations

A. È. Guterman

Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 681-692.

Résumé

We investigate additive transformations on the space of real or complex matrices that are monotone with respect to any admissible partial order relation. A complete characterization of these transformations is obtained. In the real case, we show that such transformations are linear and that all nonzero monotone transformations are bijective. As a corollary, we characterize all additive transformations that are monotone with respect to certain classical matrix order relations, in particular, with respect to the Drazin order, left and right $*$-orders, and the diamond order.

Détail

Keywords: matrix partial order, partially ordered set, Lewner order, Hartwig order, Drazin order, diamond order.
Mots-clés : monotone transformation

Bibliographie

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

[2] K. S. S. Nambooripad, “The natural partial order on a regular semigroup”, Proc. Edinb. Math. Soc. (2), 23:3 (1980), 249–260 | DOI | MR | Zbl

[3] G. Frobenius, “Über die Darstellung der endlichen Gruppen durch lineare Substitutionen”, Sitzungsber, Preuss. Akad. Wiss. Berlin, Berlin, 1897, 994–1015 | Zbl

[4] S. Pierce, “General introduction”, A survey of linear preserver problems, Linear Multilinear Algebra, 33{, 1–2}, Gordon and Breach Science Publ., Yverdon, 1992, 3–5 | MR

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

[6] A. A. Alieva, “Obratimost lineinykh otobrazhenii, sokhranyayuschikh $f$-poryadki”, Fundam. i prikl. matem., 9:3 (2003), 3–11 | MR | Zbl

[7] A. Alieva, A. Guterman, “Monotone linear transformations on matrices are invertible”, Comm. Algebra, 33:9 (2005), 3335–3352 | DOI | MR | Zbl

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

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

[10] P. G. Ovchinnikov, “Automorphisms of the poset of skew projections”, J. Funct. Anal., 115:1 (1993), 184–189 | DOI | MR | Zbl

[11] P. Šemrl, “Order-preserving maps on the poset of idempotent matrices”, Acta Sci. Math. (Szeged), 69:3–4 (2003), 481–490 | MR | Zbl

[12] P. Šemrl, “Hua's fundamental theorem of the geometry of matrices”, J. Algebra, 272:2 (2004), 801–837 | DOI | MR | Zbl

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

[14] W.-L. Chooi, M.-H. Lim, “Additive preservers of rank-additivity on matrix spaces”, Linear Algebra Appl., 402 (2005), 291–302 | DOI | MR | Zbl

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

[16] M. P. Drazin, “Natural structures on semigroups with involution”, Bull. Amer. Math. Soc., 84:1 (1978), 139–141 | DOI | MR | Zbl

[17] B. Kuzma, “Additive mappings decreasing rank one”, Linear Algebra Appl., 348:1–3 (2002), 175–187 | DOI | MR | Zbl

[18] J. K. Baksalary, S. K. Mitra, “Left-star and right-star partial orderings”, Linear Algebra Appl., 149 (1991), 73–89 | DOI | MR | Zbl

[19] J. K. Baksalary, J. Hauke, “A further algebraic version of Cochran's theorem and matrix partial orderings”, Linear Algebra Appl., 127 (1990), 157–169 | DOI | MR | Zbl

[20] W. G. Cochran, “The distribution of quadratic forms in a normal system, with applications to the analysis of covariance”, Proc. Cambridge Philos. Soc., 30 (1934), 178–191 | DOI | Zbl

[21] J. K. Baksalary, J. Hauke, “Partial orderings on matrices referring to singular values or eigenvalues”, Linear Algebra Appl., 96 (1987), 17–26 | DOI | MR | Zbl

[22] J. Groß, “A note on rank-subtractivity ordering”, Linear Algebra Appl., 289:1–3 (1999), 151–160 | DOI | MR | Zbl

[23] J. Hauke, A. Markiewicz, T. Szulc, “Inter- and extrapolatory properties of matrix partial orderings”, Linear Algebra Appl., 332–334 (2001), 437–445 | DOI | MR | Zbl