Abstract. 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 σ e asc (T + F ) = σ e asc (T ) for every operator T commuting with F . The quasi-nilpotent part, the analytic core and the single-valued extension property are also analyzed for operators with finite essential ascent.1. Introduction. Throughout this paper X will be an infinite-dimensional complex Banach space. We will denote by L (X) the algebra of all operators on X, and by F (X) and K (X) its ideals of finite rank and compact operators on X, respectively. For an operator T ∈ L (X), write T * for its adjoint, N(T ) for its kernel and R(T ) for its range. Also, denote by σ(T ), σ ap (T ) and σ su (T ) its spectrum, approximate point spectrum and surjective spectrum, respectively. An operator T ∈ L (X) is upper semiFredholm (respectively lower semi-Fredholm) if R(T ) is closed and dim N(T ) (respectively codim R(T )) is finite. If T is upper or lower semi-Fredholm, then T is called semi-Fredholm. The index of such an operator is given by ind(T ) = dim N(T ) − codim R(T ), and when it is finite we say that T is Fredholm. Recall that for T ∈ L (X), the ascent, a(T ), and the descent, d(T ), are defined by
In the present note, we study the problem of lifting poles in Calkin algebra on a separable infinite-dimensional complex Hilbert space H. We show by an example that such lifting is not possible in general, and we prove that if zero is a pole of the resolvent of the image of an operator T in the Calkin algebra, then there exists a compact operator K for which zero is a pole of T + K if and only if the index of T − λ is zero on a punctured neighbourhood of zero. Further, a useful characterization of poles in Calkin algebra in terms of essential ascent and descent is provided.
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