Application of topological radicals to calculation of joint spectral radii

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.