Maps preserving Schatten \(p\)-norms of convex combinations
Summary: We study maps \(\phi\) of positive operators of the Schatten \(p\)-classes (\(1 < p < + \infty\)), which preserve the \(p\)-norms of convex combinations, that is, \(\| t \rho +(1 - t) \sigma \|_p = \| t \phi(\rho) +(1 - t) \phi(\sigma) \parallel_p\), \(\forall \rho, \sigma \in \mathcal{S}_p^+(H)_1\), \(t \in [0,1]\). They are exactly those carrying the form \(\phi(\rho) = U \rho U^*\) for a unitary or antiunitary \(U\). In the case \(p = 2\), we have the same conclusion whenever it just holds \(\| \rho + \sigma \|_2 = \| \phi(\rho) + \phi(\sigma) \|_2\) for all the positive Hilbert-Schmidt class operators \(\rho, \sigma\) of norm 1. Some examples are demonstrated.
positive Hilbert-Schmidt class operators
scientific article; zbMATH DE number 7022546
scientific article