Mappings preserving spectra of products of matrices

Abstract

Let M n be the set of n × n complex matrices, and for every A ε M n , let Sp(A) denote the spectrum of A. For various types of products A 1 * • • • * Aκk on M n , it is shown that a mapping η:M n → M n satisfying Sp(A 1 * • • • * Aκ) = Sp(ηp(A 1 ) * • • • * η(Aκ)) for all A 1 ,.,Aκ ε M n has the form X →ψ.S -1 XS or A →η S -1 X t S for some invertible S ε M n and scalar ε. The result covers the special cases of the usual product A 1 * • • • * A κ = A 1 • • •A κ , the Jordan triple product A 1 *A 2 = A 1 *A 2 *A 1 , and the Jordan product A 1 *A 2 = (A 1 A 2 +A 2 A 1 /2. Similar results are obtained for Hermitian matrices. © 2006 American Mathematical Society.