Operator norms, multiplicativity factors, and C-numerical radii

Abstract

Let V be a normed vector space over C, let B ( V) denote the algebra of linear bounded operators on V, and let N be an arbitrary seminorm or norm on B ( V). In this paper we discuss multiplicativity factors for N, i.e., constants μ>0 for which N μ ≡ μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C-numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.