Unitarily Invariant Norms under Which the Map <i>A</i> → |A| Is Lipschitz Continuous

Abstract

We will characterize the unitarily invariant norms (for compact operators) under which the map A → |A| = (A*A)/sup> 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.