On the essential spectrum of Banach-space operators

Let T and S be quasisimilar operators on a Banach space X. A well-known res ult of Herrero shows that each component of the essential spectrum of T mee ts the essential spectrum of S. Herrero used that, for an n-multicyclic ope rator, the components of the essential resolvent set with maximal negative index are simply connected. We give new and conceptually simpler proofs for both of Herrero's results based on the observation that on the essential r esolvent set of T the section spaces of the sheaves [GRAPHICS] are complete nuclear spaces that are topologically dual to each other. Othe r concrete applications of this result are given.