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Title:Norm preservers of Jordan products
Authors:Kuzma, Bojan (Author)
Lešnjak, Gorazd (Author)
Li, Chi-Kwong (Author)
Petek, Tatjana (Author)
Rodman, Leiba (Author)
Work type:Not categorized
Tipology:1.01 - Original Scientific Article
Organization:IAM - Andrej Marušič Institute
Abstract:V članku klasificiramo surjektivne preslikave, ki na algebri kompleksnih matrik ohranjajo Frobeniusovo normo Jordanskega produkta. Izkaže se, da so do unitarne podobnosti in množenja s skalarnim večkratnikom vse tovrstne preslikave le štirih možnih tipov: (i) preslikava, ki je lokalno adjungiranje na normalnih matrikah in identiteta izven normalnih matrik, (ii) transponiranje, (iii) kompleksna konjugacija in (iv) adjungiranje. Do podobnih zaključkov pridemo tudi v primeru nekaterih drugih unitarno invariantnih norm, kjer pokažemo, da preslikava bodisi normalne matrike množi s skalarji, bodisi jih adjungira in množi s skalarji.
Keywords:matematika, linearna algebra, jordanski produkt, matrična norma, ohranjevalci
Year of publishing:2011
Number of pages:str. 959-978
Numbering:Vol. 22
COBISS_ID:16068441 Link is opened in a new window
Categories:Document is not linked to any category.
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Secondary language

Abstract:Norm preserver maps of Jordan product on the algebra ▫$M_n$▫ of ▫$n times n$▫ complex matrices are studied, with respect to various norms. A description of such surjective maps with respect to the Frobenius norm is obtained: Up to a suitable scaling and unitary similarity, they are given by one of the four standard maps (identity, transposition, complex conjugation, and conjugate transposition) on ▫$M_n$▫, except for a set of normal matrices; on the exceptional set they are given by another standard map. For many other norms, it is proved that, after a suitable reduction, norm preserver maps of Jordan product transform every normal matrix to its scalar multiple, or to a scalar multiple of its conjugate transpose.
Keywords:mathematics, linear algebra, Jordan product, matrix norm, nonlinear preservers


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