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Linear mappings preserving square-zero matrices

Published online by Cambridge University Press:  17 April 2009

Peter Šemrl
Peter Šemrl
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19 61000 Ljubljana, Slovenia
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Let sln denote the set of all n × n complex matrices with trace zero. Suppose that ø: sln → sln is a bijective linear mapping preserving square-zero matrices. Then ø is either of the form ø(A) = cUAU-1 or ø(A) = cUAtU-1 where U is an invertible n × n matrix and c is a nonzero complex number. The same result holds if we assume that ø is a linear mapping preserving square-zero matrices in both directions. Applying this result we prove that a linear mapping ø defined on the algebra of all n × n matrices is an automorphism if and only if it preserves zero products in both directions and satisfies ø(I) = I. An extension of this last result to the infinite-dimensional case is considered.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 3 , December 1993 , pp. 365 - 370
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Botta, P., Pierce, S., and Watkins, W., ‘Linear transformations that preserve the nilpotent matrices’, Pacific J. Math. 104 (1983), 39–46.CrossRefGoogle Scholar
[2]Dixon, J., ‘Rigid embeddings of simple groups in the general linear group’, Canad. J.Math. 29 (1977), 384–391.CrossRefGoogle Scholar
[3]Howard, R., ‘Linear maps that preserve matrices annihilated by a polynomial’, Linear Algebra Appl. 30 (1980), 167–176.CrossRefGoogle Scholar
[4]Li, C.K. and Tsing, N.K., ‘Linear preserver problems: A brief introduction and some special techniques’, Linear Algebra Appl. 162–164 (1992), 217–235.Google Scholar
[5]Marcus, M. and Moyls, B.N., ‘Linear transformations on algebras of matrices’, Canad. J.Math. 11 (1959), 61–66.Google Scholar
[6]Omladič, M., ‘On operators preserving commutativity’, J. Funct. Anal. 66 (1986), 105–122.CrossRefGoogle Scholar