On the structure of finite-dimensional paracontractions (Q2442360)

A complex matrix \(A\) is said to be a paracontraction with respect to a vector norm \( \|\cdot\|\) if, for every \(x \in \mathbb C^n\), either \(Ax=x\) or \(\|A x\|< \|x\| \). In the paper under review, the author gives a characterization of paracontractions with respect to a strictly convex norm on \(\mathbb C^n\), and then he characterizes linear maps on the algebra \(\mathcal M_n (\mathbb C)\) of complex matrices which preserve, in both directions, paracontractions with respect to \(p\)-norms (\(1 \leq p\leq \infty\)). This generalizes the results in [\textit{B. Kuzma}, Monatsh. Math. 139, No. 4, 271--274 (2003; Zbl 1047.47028)], where the author considered paracontractions with respect to the Euclidean norm.