Norms that are invariant under unitary similarities and theC-numerical radii
Chi-Kwong Li, Nam Kiu Tsing · Linear and Multilinear Algebra · 1989
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.