Abstract. This paper deals with approximation---in the Banach algebra of all bounded linear operators on an infinite-dimensional (possibly nonseparable) real or complex Hilbert space---by the semi-$\alpha $-Fredholm operators. These are the operators which are either left or right invertible modulo the closed two-sided ideal of the algebra above which is associated to an infinite cardinal number $\alpha $, less than or equal to the Hilbert dimension of the space. The boundaries of all semi-$\alpha $-Fredholm components, as well as the boundaries of their closures, are characterized in terms of the approximate nullities of their elements and of the respective adjoints. The distances from a bounded linear operator $T$ to several sets related to the semi-$\alpha $-Fredholm operators are also computed. In particular, formulas for the distances to the semi-$\alpha $-Fredholm components, to their boundaries and to the boundaries of their closures are given. For each distance, a formula in terms of the weighted reduced minimi moduli of $T$ and a formula in terms of the minima of the weighted spectra of the modulus of $T$ are given.

AMS Subject Classification (1991): 47A58, 47A53

Keyword(s): \alpha, semi-Fredholm and semi--Fredholm operators, nonseparable Hilbert spaces, \alpha, semi--Fredholm components and boundaries, distance formulas, \alpha, reduced minimum modulus of weight

Received April 15, 1998 and in revised form November 26, 1998. (Registered under 2683/2009.)