Stable subnorms

Abstract

Let f be a real-valued function defined on a nonempty subset of an algebra over a field , either or , so that is closed under scalar multiplication. Such f shall be called a subnorm on if f(a)>0 for all , and f(αa)=∣α∣f(a) for all and . If in addition, is closed under raising to powers, and f(am)=f(a)m for all and m=1,2,3,…, then f shall be called a submodulus. Further, a subnorm f shall be called stable if there exists a constant σ>0 so that f(am)⩽σf(a)m for all and m=1,2,3,… Our primary purpose in this paper is to study stability properties of continuous subnorms on subsets of finite dimensional algebras. If f is a subnorm on such a set , and g is a continuous submodulus on the same set, then our main results state that g is unique, f(am)1/m→g(a) as m→∞, and f is stable if and only if it majorizes g. In particular, if f is a subnorm on a subset of , the algebra of n×n matrices over , and if has the above properties but no nilpotent elements, then we show that f is stable if and only if it is spectrally dominant, i.e., f(A)⩾ρ(A) for all , where ρ is the spectral radius. Part of the paper is devoted to norms on algebras, where the above findings hold almost verbatim. We illustrate our results by discussing certain subnorms on matrix algebras, as well as on the complex numbers, the quaternions, and the octaves, where these number systems are viewed as algebras over the reals.

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