Abstract. The purpose of this paper is to introduce, by means of the extensions of almost closed operators, the notion of almost closable linear operator acting in a Hilbert or Banach space. This class of operators is strictly included in the class of all unbounded linear operators, it contains the set of all closable operators and that of all almost closed operators and is invariant under finite and countable sums, finite products, limits and integrals. We also present some fundamental properties relative to almost closability and we define a locally convex Hausdorff topology in the set of all almost closable operators.Mathematics subject classification (2010): 47A05, 47B33.

The paper present a survey of results concerning the fundamental properties of the Drazin inverse for bounded operators and an interesting study of the Drazin inverse for a closed operator in a Banach space. Some necessary and sufficient conditions for $A$ closed linear operator to possess a Drazin inverse $A^D$ are given, we obtain also a useful caracterization and explicit formula for the Drazin inverse $(A+B)^D$ and $(A B)^D$ if $A$ and $B$ are closed operators.

No abstract

In this paper, almost closed subspaces and almost closed linear operators are described in a Hilbert space. We show Neubauer’s lemma and we give necessary and sufficient conditions for an almost closed operator to be with closed range and we exhibit sufficient conditions under which it is either closed or closable.

The purpose of this paper is to study the relationship between spectral properties of a bounded operator and its left and right generalized Drazin inverses. The description of the associated spectral projections allows us to find some new representation results and certain generalizations on left and right generalized Drazin invertible bounded operators.Метою статтi є дослiдження спiввiдношення мiж спектральними властивостями обмеженого оператора i його лiвого та правого узагальненого оберненого в сенсi Дразiна. Опис вiдповiдних спектральних проєкторiв дозволяє знайти новi теореми представлення, а також певнi узагальнення класу операторiв, оборотних у сенсi Дразiна.2020 Mathematics Subject Classification. 47A10, 47A67.