Ascent spectrum and essential ascent spectrum

Abstract

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

Components of topological uniform descent resolvent set and local spectral theory

“…Directly from Theorem 3.1 we obtain the next corollary which generalizes the corresponding results of [8,12,13]. The TUD spectrum of an operator is not necessarily non-empty.…”

“…Several authors in [8,12,13] discuss the left Drazin spectrum, essential left Drazin spectrum, descent spectrum and essential descent spectrum of an operator. They get some characterizations of operators whose spectra of these kinds are at most countable.…”

On the ascent-descent spectrum

“…It is worth noting that the space of bounded linear operators B(X) is a particular case of the Banach algebra \scrA and Asc(B(X)) and Dsc(B(X)) are regularities on B(X). For more information, see [14,5,20]. Throughout our work, under suitable optimal hypothesis, we will prove that conditions (C 1 ) and (C 2 ) are approved properties for the ascent and descent spectra.…”

“…Recently, stability problems of operators under perturbation have attracted many researchers and undergone important contributions, see for instance [2,13,14,4,6,7,8,12,15,18,20]. Recall that, in [10], authors have proved nice relations between the left Browder spectrum and the left invertible spectrum (respectively, between the right The concept of Definition 1.3 is truly suitable since it comprises for example the left (right) spectrum, the left (right) approximate point spectrum, the Harte (the union of the left and right) spectrum.…”