Linear maps on $M_n(\mathbb C)$ preserving the local spectral radius
Tom 188 / 2008
Studia Mathematica 188 (2008), 67-75
MSC: Primary 47B49; Secondary 47A10, 47A11.
DOI: 10.4064/sm188-1-4
Streszczenie
Let $x_0$ be a nonzero vector in $\mathbb C^n$. We show that a linear map ${\mit\Phi}:M_n(\mathbb C)\to M_n(\mathbb C)$ preserves the local spectral radius at $x_0$ if and only if there is $\alpha\in\mathbb C$ of modulus one and an invertible matrix $A\in M_n(\mathbb C)$ such that $Ax_0=x_0$ and ${\mit\Phi}(T)=\alpha ATA^{-1}$ for all $T\in M_n(\mathbb C)$.
