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Abstract

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