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 ‖g(x)‖=∥x∥ for all g ϵ G, x ϵ V . The concept of G-invariant norm specializes to many interesting particular cases such as the absolute norms on F n , symmetric gauge functions on R n , unitarily invariant norms on F m× 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 r G( c) (·) on V defined by r G(c)(x) = max{|〈x, g(c)〉|:gϵ G}, where c ϵ 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.