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Stable subnorms

Goldberg, Moshe and Luxemburg, W. A. J. (2000) Stable subnorms. Linear Algebra and its Applications, 307 (1-3). pp. 89-101. ISSN 0024-3795. doi:10.1016/s0024-3795(00)00011-2.

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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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Additional Information:© 2000 Elsevier. Under an Elsevier user license. Received 24 September 1999, Accepted 6 December 1999, Available online 2 October 2000. To Richard Brualdi on his 60th birthday.Submitted by M. Marcus. Research sponsored in part by the Fund for the Promotion of Research at the Technion, Grant 100-091.
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Issue or Number:1-3
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90103
Deposited By: George Porter
Deposited On:06 Oct 2018 22:24
Last Modified:16 Nov 2021 00:40

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