Essential Ascent of Closed Operator and Some Decomposition Theorems

Abstract

The aim of this work is to study the essential ascent and the related essential ascent spectrum of closed unbounded operators on a Banach space. Our approach is based on the concept of paracomplete subspaces of Banach spaces. We prove an unbounded spectral mapping theorem for the ascent spectrum and the essential ascent spectrum. A characterization of closed unbounded operators with finite essential ascent as direct sum of a suitable operators is proved. The new notion of a-essential index for closed unbounded operators with finite essential ascent is introduced. We also give some perturbations results for such operators. This paper extends some results proved in [1] to closed unbounded operators.