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 $ℵ_0 ≤ α ≤ dim H$. In the first part, we find the relation between $γ_α(T)$ and $c(π_α(T))$ for all $ℵ_0$-regular cardinals α, where $γ_α$ is the reduced minimum modulus of weight α, c is the reduced minimum modulus (in a C*-algebra) and $π_α$ is the canonical surjection from B(H) onto $C_α (H) = B(H)/K_α(H)$. We study the continuity points of the maps $c_α : T → c(π_α(T))$ and $γ_α : T → γ_α(T)$. 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 $ℵ_0$.