Ascent spectrum and essential ascent spectrum
We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach spaceX has nite dimension if and only if the essential ascent of every operator on X is nite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator F on X has some nite rank power if and only if e asc(T +F ) = 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 nite essential ascent. 1. Introduction. Throughout this paper X will be an innite-dimen- sional 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 nite rank and compact operators on X, respectively. For an operator T 2 L (X), write T for its adjoint, N(T ) for its kernel and R(T ) for its range. Also, denote by (T ), ap(T ) and su(T ) its spectrum, approximate point spectrum and surjective spectrum, respectively. An operator T 2 L (X) is upper semi- Fredholm (respectively lower semi-Fredholm) if R(T ) is closed and dim N(T ) (respectively codim R(T )) is nite. 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 nite we say that T is Fredholm. Recall that for T 2 L (X), the ascent, a(T ), and the descent, d(T ), are dened by a(T ) = inffn 0: N(T n ) = N(T n+1 )g; d(T ) = inffn 0: R(T n ) = R(T n+1 )g;