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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

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