Nonlinear mappings preserving at least one eigenvalue
Volume 200 / 2010
Abstract
We prove that if $F$ is a Lipschitz map from the set of all complex $n\times n$ matrices into itself with $F(0)=0$ such that given any $x$ and $y$ we know that $F( x) -F( y) $ and $x-y$ have at least one common eigenvalue, then either $F( x) =uxu^{-1}$ or $F( x) =ux^{t}u^{-1}$ for all $x$, for some invertible $n\times n$ matrix $u$. We arrive at the same conclusion by supposing $F$ to be of class $\mathcal{C}^{1}$ on a domain in $\mathcal{M}_{n}$ containing the null matrix, instead of Lipschitz. We also prove that if $F$ is of class $\mathcal{C}^{1}$ on a domain containing the null matrix satisfying $F(0)=0$ and $\rho (F( x) -F( y) )=\rho (x-y)$ for all $x$ and $y$, where $\rho ( \cdot ) $ denotes the spectral radius, then there exists $\gamma \in \mathbb{C}$ of modulus one such that either $\gamma ^{-1}F$ or $\gamma ^{-1}\overline{F}$ is of the above form, where $\overline{F}$ is the (complex) conjugate of $F$.
