The formula $\varrho \left(M\right)=max{\varrho }_{\chi }\left(M\right),r\left(M\right)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), ${\varrho }_{\chi }\left(M\right)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let ${\sigma }_F\left(T\right)=\sigma \left(T\right)\setminus {\sigma }_e\left(T\right)$ be the non-essential spectrum of T. We show that, for each connected component M of the manifold $Reg\left({\sigma }_F\left(T\right)\right)$ of all smooth points of ${\sigma }_F\left(T\right)$, there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups $H^p({(z-T)}^k,E)$ grow at least like the sequence ${\left(k^d\right)}_{k\ge 1}$ with d = dim M.