Norm preservers of Jordan products

Kuzma, Bojan; Lešnjak, Gorazd; Li, Chi-Kwong; Petek, Tatjana; Rodman, Leiba

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Title:	Norm preservers of Jordan products
Authors:	ID Kuzma, Bojan (Author) ID Lešnjak, Gorazd (Author) ID Li, Chi-Kwong (Author) ID Petek, Tatjana (Author) ID Rodman, Leiba (Author)
Files:	http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol22_pp959-978.pdf
Language:	English
Work type:	Not categorized
Typology:	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
PID:	20.500.12556/RUP-765
ISSN:	1081-3810
UDC:	512.643
COBISS.SI-ID:	16068441
Publication date in RUP:	15.10.2013
Views:	4068
Downloads:	239
Metadata:
:	KUZMA, Bojan, LEŠNJAK, Gorazd, LI, Chi-Kwong, PETEK, Tatjana and RODMAN, Leiba, 2011, Norm preservers of Jordan products. [online]. 2011. Vol. 22, p. 959–978. [Accessed 5 April 2025]. Retrieved from: http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol22_pp959-978.pdf Copy citation

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Secondary language

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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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