Non-surjective Spectral Isometries on Matrix Spaces

Abstract

We prove that if $\varphi \,{:}\, {\mathcal {M}}_{m}\rightarrow {\mathcal {M}}_{n}$ is a linear map such that the spectral radius of $x \in {\mathcal {M}}_{m}$ equals the spectral radius of $\varphi (x) \in {\mathcal {M}}_{n}$ for each $x \in {\mathcal {M}}_{m}$, there exists then a unimodular constant $\xi$ such that the spectrum of $\varphi (x)$ in ${\mathcal {M}}_{n}$ contains the spectrum of $\xi x \in {\mathcal {M}}_{m}$ for each x. Structural informations on the map $\varphi$ in a particular case are also obtained.