The generalized Fredholm operators

Abstract

Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if T ( X ) T(X) is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces ↔ \leftrightarrow finite-dimensional spaces, weakly compact operators ↔ \leftrightarrow compact operators, generalized Fredholm operators ↔ \leftrightarrow Fredholm operators, Tauberian operators with closed range ↔ \leftrightarrow semi-Fredholm operators.