Nonlinear maps preserving the elementary symmetric functions

Constantin Costara

Abstract


Let ℳn be the algebra of all n × n complex matrices, and for a natural number 2 ≤ k ≤ n denote by Ek(x) the kth elementary symmetric function on the eigenvalues of x ∈ ℳn. For two maps φ, ψ: ℳn → ℳn, one of them being surjective, we prove that if Ek(λx + y) = Ek(λφ(x) + ψ(y)) for each λ ∈ C and x, y ∈ ℳn, then φ = ψ on ℳn, the common value being a linear map from ℳn into itself. In particular, for 3 ≤ k ≤ n the general form of φ and ψ can be computed explicitly.


Keywords


Elementary symmetric function, nonlinear, preserver

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DOI: https://doi.org/10.26493/1855-3974.2488.f5c

ISSN: 1855-3974

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