Mappings preserving spectra of products of matrices

Let $M_n$ be the set of $n\times n$ complex matrices, and for every $A\in M_n$, let $\operatorname{Sp}(A)$ denote the spectrum of $A$. For various types of products $A_1* \cdots *A_k$ on $M_n$, it is shown that a mapping $\phi: M_n \rightarrow M_n$ satisfying $\operatorname{Sp}(A_1*\cdots*A_k) = \operatorname{Sp}(\phi(A_1)* \cdots*\phi(A_k))$ for all $A_1, \dots, A_k \in M_n$ has the form

$X \mapsto \xi S^{-1}XS \quad \hbox{ or } \quad A \mapsto \xi S^{-1}X^tS$

for some invertible $S \in M_n$ and scalar $\xi$. The result covers the special cases of the usual product $A_1* \cdots * A_k = A_1 \cdots A_k$, the Jordan triple product $A_1*A_2 = A_1*A_2*A_1$, and the Jordan product $A_1*A_2 = (A_1A_2+A_2A_1)/2$. Similar results are obtained for Hermitian matrices.