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The aim of this paper is to study the α-semi-Fredholm operators in a nonseparable Hilbert space H for all cardinals α with ${\aleph }_0\le \alpha \le dimH$. In the first part, we find the relation between ${\gamma }_{\alpha }\left(T\right)$ and $c\left({\pi }_{\alpha }\left(T\right)\right)$ for all ${\aleph }_0$-regular cardinals α, where ${\gamma }_{\alpha }$ is the reduced minimum modulus of weight α, c is the reduced minimum modulus (in a C*-algebra) and ${\pi }_{\alpha }$ is the canonical surjection from B(H) onto $C_{\alpha }\left(H\right)=B\left(H\right)/K_{\alpha }\left(H\right)$. We study the continuity points of the maps $c_{\alpha }:T\to c\left({\pi }_{\alpha }\left(T\right)\right)$ and ${\gamma }_{\alpha }:T\to {\gamma }_{\alpha }\left(T\right)$. In the second part, we prove some approximation results for semi-Fredholm operators....