Stable subnorms Original Research Article

Let f be a real-valued function defined on a nonempty subset image of an algebra image over a field image, either image or image, so that image is closed under scalar multiplication. Such f shall be called a subnorm on image if f(a)>0 for all image, and f(αa)=midαmidf(a) for all image and image. If in addition, image is closed under raising to powers, and f(am)=f(a)m for all image 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)less-than-or-equals, slantσf(a)m for all image 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 image, 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 image of image, the algebra of n×n matrices over image, and if image 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)greater-or-equal, slantedρ(A) for all image, 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.