Ascent and Essential Ascent Spectrum of Linear Relations

Abstract

In the present paper, we study the ascent of a linear relation everywhere defined on a Banach space X and the related essential ascent spectrum. Some properties and characterization of such spectra are given. In particular, we show that a Banach space X is finite dimensional if and only if the ascent and the essential ascent of every closed linear relation in X is finite. As an application, we focus on the stability of the ascent and the essential ascent spectrum under perturbations. We prove that an operator F in X has some finite rank power, if and only if, σeasc(T + F ) = σeasc(T ), for every closed linear relation T commuting with F.

Keywords

• Ascent
• essential ascent
• perturbation
• spectrum
• linear relations