Generalized Fredholm operators (Q800611)

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 \(F\subset A\) and \(Fi\subset Op(A)\), Fredholm operators are particular A-Fredholm operators. \textit{K. W. Yang} [Trans. Am. Math. Soc. 216, 313-326 (1976; Zbl 0297.47027) and Pac. J. Math. 71, 559-564 (1977; Zbl 0359.47019)] 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 \textit{T. C. Wu} [Proc. Am. Math. Soc. 65, 252-254 (1977; Zbl 0365.47015)] and Yang [loc.cit.] are particular cases of some results presented here.