Norm properties of C-numerical radii

Abstract Given n × n complex matrices A , C , the C -numerical radius of A is the nonnegative quantity r c (A)≡ma{|tr(CU ∗ AU)|:U unitary} . For C = diag (1,0,…,0) it reduces to the classical numerical radius r(A)= max{|x ∗ Ax|:x ∗ x=1} . We show that r c is a generalized matrix norm if and only if C is nonscalar and tr C ≠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.