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

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