Linear mappings which preserve paracontractions (Q1408502)

A paracontraction (relative to a given norm \(\|\cdot \|\) on a vector space \({\mathbb C}^n\)) is a matrix \(A\in {\mathcal M}_n({\mathbb C})\) with the property that either \(Ax = x\) or \(\| Ax\| < \| x\|\). In this paper, the author proves that a linear surjective map \(\Phi :{\mathcal M}_n({\mathbb C})\rightarrow {\mathcal M}_n({\mathbb C})\), which preserves paracontractions in both directions, has one of the forms \(\Phi (A)=U^*AU\) (\(\forall A\in {\mathcal M}_n({\mathbb C})\)) or \(\Phi (A)=U^*A^{tr}U\) (\(\forall A\in {\mathcal M}_n({\mathbb C})\)), where \(U\) is a unitary matrix.