Power-Preserving Linear Maps between Matrix Spaces

Abstract:

      Let $C$ be a complex field. Let $m$ and $n$ be positive integers. $k$ is a fixed positive integer and $k\geq 2$. Let $M_{m}(C)$ and $S_{n}(C)$ be the vector spaces of all $m\times m$ matrices and $n\times n$ symmetric matrices over $C$, respectively. We characterize the linear maps preserving $k$-power from $S_{n}(C)$ to $M_{m}(C)$ and $S_{n}(C)$ to $S_{m}(C)$, respectively.