G-invariant norms and G(c)-radii

Abstract

Let V be a finite dimensional inner product space over F(=R or C), and let G be a closed subgroup of the group of unitary operators on V. A norm or a seminorm ∥·∥ on V is said to be G-invariant if {norm of matrix}g(x){norm of matrix}=∥x∥ for all g ε{lunate} G, x ε{lunate} V. The concept of G-invariant norm specializes to many interesting particular cases such as the absolute norms on Fn, symmetric gauge functions on Rn, unitarily invariant norms on Fm×n, etc., which are of wide research interest. In this paper, we study the general properties of G-invariant norms. Our main strategy is to study G-invariant norms via the G(c)-radius rG(c)(·) on V defined by rG(c)(x) = max{|〈x, g(c)〉|:gε{lunate} G}, where c ε{lunate} V. It is shown that the G(c)-radii are very important G-invariant seminorms because every G-invariant norm or seminorm admits a representation in terms of them. As a result, one may focus attention on G(c)-radii in order to get results on G-invariant norms. We study the norm properties of G(c)-radii and obtain various results relating G-invariant norms and G(c)-radii. The linear operators on V that preserve G-invariant norms, G-invariant seminorms, or G(c)-radii are also investigated. Several open questions are mentioned. © 1991.