On the B-Fredholm Alternative

The aim of this paper is to give an extended version of the Fredholm alternative, by including the B-Fredholm case. Thus if T a B-Fredholm operator acting on a Banach space, we study equations of the type T(x) =  y, for x, y elements of the range R(T ⁿ ) of T for ${n \in \mathbb{N}}$ . In this study we include the case when 0 is a pole of infinite rank of the resolvent of T. Preliminarily, we define unbounded closed B-Fredholm operators on Banach spaces.