Maps preserving trace of products of matrices

TL;DR

This paper characterizes certain trace-preserving maps on matrix subsets, proving their linearity and injectivity, with applications to understanding maps that preserve the trace of matrix products across various matrix classes.

Contribution

It establishes the linearity and injectivity of maps satisfying trace conditions, extending to multiple matrix classes and providing real versions.

Findings

Maps satisfying trace conditions are linear and injective.

Characterization of trace-preserving maps for various matrix types.

Extension of results to real matrix versions.

Abstract

We prove the linearity and injectivity of two maps $\phi_1$ and $\phi_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $\phi_i:\mathcal{S}\to \mathcal{S}$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.