Monotonicity of unitarily invariant norms

Any matrix unitarily invariant norm gives rise to a symmetric gauge function of the singular values of its matrix argument, but the dependency on the singular values is not equally weighted among them in the sense that the norm may not increase with some of the singular values under sufficiently small increases while it always increases when some other singular values increase no matter how tiny the changes are. This paper introduces and characterizes (argument-)dependent classifications of unitarily invariant norms and in particular the class in which a unitarily invariant norm increases with each of the first k largest singular values.