Fredholm spectrum and growth of cohomology groups

Jörg Eschmeier¹

¹ Fachrichtung Mathematik Universität des Saarlandes Postfach 15 11 50 D-66041 Saarbrücken, Germany

Studia Mathematica, Tome 186 (2008) no. 3, pp. 237-249

Résumé

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 $d = \dim M.$

Détail

DOI : 10.4064/sm186-3-3

Keywords: commuting tuple bounded linear operators complex banach space sigma sigma setminus sigma non essential spectrum each connected component manifold reg sigma smooth points sigma there number ldots each point dimensions cohomology groups grow least sequence geq dim