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Mat. Zametki, 2007, Volume 81, Issue 5, Pages 681–692 (Mi mz3713)  

This article is cited in 7 scientific papers (total in 7 papers)

Monotone Additive Matrix Transformations

A. È. Guterman

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

Keywords: matrix partial order, monotone transformation, partially ordered set, Lewner order, Hartwig order, Drazin order, diamond order

DOI: https://doi.org/10.4213/mzm3713

Full text: PDF file (478 kB)
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English version:
Mathematical Notes, 2007, 81:5, 609–619

Bibliographic databases:

UDC: 512.643
Received: 07.06.2006
Revised: 22.11.2006

Citation: A. È. Guterman, “Monotone Additive Matrix Transformations”, Mat. Zametki, 81:5 (2007), 681–692; Math. Notes, 81:5 (2007), 609–619

Citation in format AMSBIB
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\paper Monotone Additive Matrix Transformations
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  • http://mi.mathnet.ru/eng/mz/v81/i5/p681

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Orel M., Kuzma B., “Additive rank-one nonincreasing maps on Hermitian matrices over the field $\mathrm{GF}(2^2)$”, Electron. J. Linear Algebra, 18 (2009), 482–499  crossref  mathscinet  zmath  isi  elib
    2. A. E. Guterman, M. A. Efimov, “Monotone maps on matrices of index one”, J. Math. Sci. (N. Y.), 191:1 (2013), 36–51  mathnet  crossref  mathscinet
    3. Dolinar G., Guterman A., Marovt J., “Automorphisms of K(H) with Respect to the Star Partial Order”, Oper. Matrices, 7:1 (2013), 225–239  crossref  mathscinet  zmath  isi  elib  scopus
    4. Dolinar G., Guterman A., Marovt J., “Monotone Transformations on B(H) With Respect To the Left-Star and the Right-Star Partial Order”, Math. Inequal. Appl., 17:2 (2014), 573–589  crossref  mathscinet  zmath  isi  elib  scopus
    5. Burgos M., Marquez-Garcia A.C., Patricio P., “on Mappings Preserving the Sharp and Star Orders”, Linear Alg. Appl., 483 (2015), 268–292  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. E. Guterman, E. M. Kreines, Qing-Wen Wang, “Monotone linear transformations on matrices over semirings”, J. Math. Sci., 233:5 (2018), 675–686  mathnet  crossref
    7. Burgos M., Marquez-Garcia A.C., Morales-Campoy A., “Maps preserving the diamond partial order”, Appl. Math. Comput., 296 (2017), 137–147  crossref  mathscinet  isi  elib  scopus
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