Application of topological radicals to calculation of joint spectral radii

Details Application of topological radicals to calculation of joint spectral radii Application of topological radicals to calculation of joint spectral radii

2008-05-02

arxiv.org/abs/0805.0209v1

Mathematics

Functional Analysis

Scientific paper

It is shown that the joint spectral radius $\rho(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $\rho_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \[ r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{\rho(a):a\in M^{n}\right\} ^{1/n}) . \] 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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