Nonlinear maps preserving the elementary symmetric functions - Costara, Constantin (Ars mathematica contemporanea, 2021)
Let ?$\mathcal{M}_n$? be the algebra of all ?$n × n$? complex matrices, and for a natural number ?$2 \le k \le n$? denote by ?$Ek(x)$? the ?$k$?th elementary symmetric function on the eigenvalues of ?$x \in \mathcal{M}_n$?. For two maps ?$φ,ψ \colon \mathcal{M}_n → \mathcal{M}_n$?, one of them being surjective, we prove that if ?$E_k(λ x + y) = E_k(λ φ(x) + ψ(y))$? for each ?$λ \in C$? and ?$x, y \in \mathcal{M}_n$?, then ?$φ =\ psi$? on ?$\mathcal{M}_n$?, the common value being a linear map from ?$\mathcal{M}_n$? into itself. In particular, for ?$3 \le k \le n$? the general form of ?$φ$? and ?$ψ$? can be computed explicitly.
Keywords: elementarne simetrične funkcije; elementary symmetric function; nelinearen; nonlinear; ohranjevalec; preserver