Geodesic
Topological Radicals and Joint Spectral Radius
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 4, pp. 61-82

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It is shown that the joint spectral radius $\rho(M)$ of a precompact set $M$ of operators on a Banach space equals the maximum of two numbers, the joint spectral radius $\rho_{e}(M)$ of the image of $M$ in the Calkin algebra and the BW-radius $r(M)$. Similar results related to general normed algebras are also obtained. The proofs are based on the theory of topological radicals of normed algebras.
Keywords: joint spectral radius, the Berger–Wang formula, topological radical, invariant subspace. 
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Yu. V. Turovskii; V. S. Shulman. Topological Radicals and Joint Spectral Radius. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 4, pp. 61-82. http://geodesic.mathdoc.fr/item/FAA_2012_46_4_a4/