Stable Norms on Complex Numbers and Quaternions

In this paper, we study stability properties of norms on the complex numbers and on the quaternions. Our main findings are that these norms are stable if and only if they majorize the modulus function and that not all stable norms are strongly stable. Part of the paper is devoted to the standard matrix representations of the above number systems, where we show that norms on the corresponding matrix algebras are stable if and only if they are spectrally dominant. We conclude by considering proper seminorms, observing that none are stable on the complex numbers or on the quaternions.