arXiv subjects
Publications and source records attributed to Abdelmalek Ouahab.
We prove that a bounded linear operator $T$ is a direct sum of an invertible operator and an operator with at most countable spectrum iff $0\notin\mbox{acc}^{ω_{1}}\,σ(T),$ where $ω_{1}$ is the smallest uncountable ordinal and $\mbox{acc}^{ω_{1}}\,σ(T)$ is the $ω_{1}$-th Cantor-Bendixson derivative of $σ(T).$
This paper treatises the preservation of some spectra under perturbations not necessarily commutative and generalizes several results which have been proved in the case of commuting operators.
We study the stability of certain spectra under some algebraic conditions weaker than the commutativity and we generalize many known commutative perturbation results.
The Kato's decomposition \cite[Theorem 4]{kato} is generalized to semi-B-Fredholm operators.
The index of a pseudo B-Fredholm operator will be defined and generalize the usual index of a B-Fredholm operator. This concept will be used to extend some known results in Fredholm's theory. Among other results, the nullity, the deficiency, the ascent and the descent will be extended and defined for a pseudo-Fredholm operator.
The main purpose of this paper, is to introduce and study the classes $(ab_{e})$ and $(aw_{e})$ which are strongly related to what has been recently studied in \cite{aznay-zariouh}. Furthermore, we give the connection between these classes and those that have been studied in \cite{berkani-zariouh0}. We also give an affirmative answer to a question asked in \cite{aznay-zariouh}.