Norms that are invariant under unitary similarities and theC-numerical radii

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Abstract
Let the set of all n× n complex matrices and let be the set of all n× n hermitian matrices. We study the norms on that are invariant under unitary similarities (abbreviate to u.s.i. norms), i.e., the norms N(⋅) that satisfy N (A) = N (UAU∗ )for all unitary U. An important subclass of the u.s.i. norms on is the collection of all unitarily invariant (abbreviate to u.i.) norms, i.e., the norms N(.) that satisfy N (A) = S (UA) = N (AV) for all unitary U.In this paper we extend a fundamental result of u.i. norms on to u.s.i. norms on . It turns out that the C-numerical radii play an important role in the theory. We also show that on the collection of all the C-numerical radii which are norms and the collection of all the u.i. norms are two disjoint subclasses ot u.s.i. norms. A characterization of u.s.i. norms on in terms of Schur-convex norm functions is given. Then we identify those u.s.i. norms on , which are induced by inner products. Finally, using the results obtained, we prove some inequalities related to u.s.i. norms of matrices, and give necessary and sufficient conditions for a matrix to be unitarily similar to a scalar multiple of another one.

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