Generalized Fredholm operators

Abstract

0. Introduction. The classical Fredholm theory in Banach spaces studies normally solvable operators with null space or conull space in F, the ideal of all finite dimensional Banach spaces. The aim of this paper is to study normally solvable operators with null space or conull space in an arbitrary space ideal A. We use the operator ideal Op(A) to replace the operator ideal Fi := Op (F) of the classical theory in a natural way. Operators invertible modulo Op (A) are studied as well. Since F c A and Fi ~Op (A), Fredholm operators are particular A-Fredholm operators. Yang [10], [11] developed Fredholm theory relative to the ideal R of all reflexive Banach spaces with a functorial approach which does not allow to develop the theory in an arbitrary space ideal; theorems of [9], [ 10] and [ 11] are particular cases of some results presented here.