Abstract
Let M be the complex linear space of all n×n complex matrices or the real linear space of all n×n hermitian matrices. A norm N on M is invariant under unitary similarities if for any AeM and for any unitary matrix U. For N′ equal to the numerical radius or the spectral norm, we study the best constants α and β, i.e., the largest α and the smallest β, such that The results are then applied to study the multiplicativity factors ν for N with respect to different products ○ on M, i.e., constants ν>0 satisfying The particular case when N is a C-numerical radius and ○ is the usual product is studied in detail. Some other multiplicativity properties of N are also considered.
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