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arXiv · 0805.0209

Application of topological radicals to calculation of joint spectral radii

Victor S. Shulman·Yuri V. Turovskii
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Abstract

It is shown that the joint spectral radius $ρ(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $ρ_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \[ r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{ρ(a):a\in M^{n}\right\} ^{1/n}) . \] Some more general Banach-algebraic results of this kind are also proved. The proofs are based on the study of special radicals on the class of Banach algebras.

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Victor S. Shulman, Yuri V. Turovskii. 2008-05-02. Application of topological radicals to calculation of joint spectral radii. https://arxiv.org/abs/0805.0209

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