Linear k-power preservers and trace of power-product preservers

TL;DR

This paper characterizes linear maps that preserve matrix powers and trace relations involving powers, extending existing results and providing new insights with applications in quantum information theory.

Contribution

It provides a systematic characterization of linear maps preserving matrix powers and trace relations, generalizing previous results and applying to various matrix classes.

Findings

Linear maps preserving matrix powers are necessarily $k$-potent preservers.

Characterizations include maps satisfying trace relations involving powers.

Results extend and unify existing literature on matrix power preservers.

Abstract

Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $\psi:V\to V$ that satisfy $\psi(A^k)=\psi(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $\phi,\psi:V\to V$ that satisfy "$ \operatorname{tr}(\phi(A)\psi(B)^k)=\operatorname{tr}(AB^k)$ for all $A\in V$, $B\in S$, and $\psi$ is linear" or "$ \operatorname{tr}(\phi(A)\psi(B)^k)=\operatorname{tr}(AB^k)$ for all $A, B\in S$ and both $\phi$ and $\psi$ are linear." The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.