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arXiv · 1805.01744

On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

Muddappa Gowda·Juyoung Jeong
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Abstract

Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to R^n takes any element x in V to the vector of eigenvalues of x written in the decreasing order. A spectral set in V is the inverse image of a permutation set in R^n under the eigenvalue map. If the permutation set is also a convex cone, the spectral set is said to be a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit is arcwise connected, we show that if a permutation invariant set is connected (arcwise connected), then the corresponding spectral set is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.

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Muddappa Gowda, Juyoung Jeong. 2018-05-04. On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras. https://arxiv.org/abs/1805.01744

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