Linear maps on $M_n(\mathbb{R})$ preserving Schur stable matrices

Abstract

An $n\times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection $\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map such that $L(\mathcal{S})\subseteq \mathcal{S}$, then $\rho (L)$ (the spectral radius of $L$) is at most $1$ and when $L(\mathcal{S})=\mathcal{S}$, we have $\rho (L)=1$. In the latter case, the map $L$ preserves the spectral radius function and using this, we characterize such maps on both $M_n(\mathbb{R})$ as well as on ${\mathcal{S}}^n$.