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We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space X has finite dimension if and only if the essential ascent of every operator on X is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some finite rank power if and only if ${\sigma }_{asc}^e(T+F)={\sigma }_{asc}^e\left(T\right)$ for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued extension...