Mappings preserving spectra of products of matrices

Let Mn be the set of n × n complex matrices, and for every A ε Mn, let Sp(A) denote the spectrum of A. For various types of products A1* • • • * Aκk on Mn, it is shown that a mapping η:Mn → M n satisfying Sp(A1* • • • * Aκ) = Sp(ηp(A1) * • • • * η(Aκ)) for all A1,.,Aκ ε Mn has the form X →ψ.S-1XS or A →η S-1X tS for some invertible S ε Mn and scalar ε. The result covers the special cases of the usual product A1 * • • • * Aκ = A1 • • •Aκ, the Jordan triple product A1 *A 2 = A1 *A2 *A1, and the Jordan product A1 *A2 = (A1A2 +A2A1/2. Similar results are obtained for Hermitian matrices.