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...

Based on conjugate duality we construct several gap functions for general variational inequalities and equilibrium problems, in the formulation of which a so-called perturbation function is used. These functions are written with the help of the Fenchel-Moreau conjugate of the functions involved. In case we are working in the convex setting and a regularity condition is fulfilled, these functions become gap functions. The techniques used are the ones considered in [Altangerel L., Boţ R.I., Wanka...