Norm properties of C-numerical radii

Given n×n complex matrices A, C, the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)≡ma{|tr(CU}{}^1\text{AU)|:U unitary}}$

. For C=diag(1,0,…,0) it reduces to the classical numerical radius $\text{r(A)= max{|x}{}^1\text{Ax|:x}{}^1\text{x=1}}$. We show that r_c is a generalized matrix norm if and only if C is nonscalar and trC≠0. Next, we consider an arbitrary generalized matrix norm and characterize all constants v?0 for which vN is multiplicative. A technique to obtain such v is then applied to C-numerical radii with Hermitian C. In particular we find that vr is a matrix norm if and only if v?4.