A linear operator T: D(T) ⊂ X → Y, when X and Y are normed spaces, is called ubiquitously open (UO) if every infinite dimensional subspace M of D(T) contains another such subspace N for which T|N is open (in the relative sense). The following properties are shown to be equivalent: (i) T is UO, (ii) T is ubiquitously almost open, (iii) no infinite dimensional restriction of T is injective and precompact, (iv) either T is upper semi-Fredholm or T has finite dimensional range, (v) for each infinite...

Soit C(X,Y) l’ensemble des opérateurs fermés à domaines denses dans l’espace de Banach X à valeurs dans l’espace de Banach Y, muni de la métrique du gap. Soit $F_n=T\in C(X,Y):Tsemi-Fredholmavecind\left(T\right)=n$ et $C_{n,m}=T\in F_n:\alpha \left(T\right)=n+m$, où α (T) est la dimension du noyau de T. Nous montrons que ${\bigcup }_{m=0}^j{C}_{n,m}$ est un ouvert de $F_n$ (et donc ouvert dans C(X,Y)) et que $C_{n,m}$ est dense dans ${\bigcup }_{j\ge m}{C}_{n,j}$. Nous déduisons quelques résultats de densités. A la fin de se travail nous donnons un exemple d’espace de Banach X tel que, d’une part, $F_n$ n’est pas connexe dans B(X) et d’autre part, l’ensemble des...

Let T be a bounded linear operator acting on a Banach space X. For each integer n, define $T_n$ to be the restriction of T to $R\left(T^n\right)$ viewed as a map from $R\left(T^n\right)$ into $R\left(T^n\right)$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator $T_n$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where $T_n$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological...

The class of Rosenthal linear relations in normed spaces is introduced and studied in terms of their first and second conjugates. We investigate the relationship between a Rosenthal linear relation and its conjugate. In this paper, we also study the semi-Tauberian linear relations following the pattern followed for the study of the Tauberian linear relations. We prove that the semi-Tauberian linear relations share some of the properties of Tauberian linear relations and they are related to Rosenthal...