Les opérateurs semi-Fredholm sur des espaces de Hilbert non séparables

The aim of this paper is to study the α-semi-Fredholm operators in a nonseparable Hilbert space H for all cardinals α with ${\displaystyle {\aleph }_0\le \alpha \le dimH}$. In the first part, we find the relation between ${\displaystyle {\gamma }_{\alpha }\left(T\right)}$ and ${\displaystyle c\left({\pi }_{\alpha }\left(T\right)\right)}$ for all ${\displaystyle {\aleph }_0}$-regular cardinals α, where ${\displaystyle {\gamma }_{\alpha }}$ is the reduced minimum modulus of weight α, c is the reduced minimum modulus (in a C*-algebra) and ${\displaystyle {\pi }_{\alpha }}$ is the canonical surjection from B(H) onto ${\displaystyle C_{\alpha }\left(H\right)=B\frac{H}{K_{\alpha }}\left(H\right)}$. We study the continuity points of the maps ${\displaystyle c_{\alpha }:T\to c\left({\pi }_{\alpha }\left(T\right)\right)}$ and ${\displaystyle {\gamma }_{\alpha }:T\to {\gamma }_{\alpha }\left(T\right)}$. In the second part, we prove some approximation results for semi-Fredholm operators. We show that all connected components of semi-Fredholm operators of at most countable index have the same topological boundary. We show that this is not true for indices strictly greater than ${\displaystyle {\aleph }_0}$.