Unitarily Invariant Norms under Which the Map $A\to |A|$ Is Lipschitz Continuous

Abstract

We will characterize the unitarily invariant norms (for compact operators) under which the map $A\to |A|=(A^{\ast }A{)}^{1/2}$ is Lipschitz-continuous. Although the map is not Lipschitz-continuous for the trace class norm, we will obtain a certain Lipschitz-type estimate by making use of the Macaev ideal.