On Pseudo-Inverses of Fredholm Operators

Suppose that A is a Fredholm operator on a Banach space. We prove that A has index = 0 (resp. ≥ 0, resp. ≤ 0) if and only if A has pseudo-inverse which is invertible (resp. Fredholm and left invertible, resp. Fredholm and right invertible). Furthermore, we determine the interior points of some classes of linear operators.