  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.
