6533b855fe1ef96bd12b1ac1

RESEARCH PRODUCT

On the operators which are invertible modulo an operator ideal

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Atkinson [3] studied the operators which are left invertible $i(X, Y) or right invertible $T{X, Y) modulo /C, with K. the compact operators. He proved that an operator T € C(X, Y) belongs to <£/ or $ r if and only if the kernel and the range of T are complemented and additionally, the kernel is finite dimensional or the range is finite codimensional, respectively. Yood [19] obtained some perturbation results for these classes and Lebow and Schechter [12] proved that the inessential operators form the perturbation class for $,(A") and $r{X). Yang [18] extended some results of ^3, 19] to operators invertible modulo W, with W the weakly compact operators. His aim was to study a generalised Fredholm theory in which the reflexive spaces played the role of the finite dimensional spaces. Moreover, Astala and Tylli [4] compared the left-invertible operators modulo W with the Tauberian operators and other classes of operators defined in terms of weak compactness. In this paper we study the classes Ai and AT of operators which are left and right invertible respectively, modulo an operator ideal A. We show that these classes are open semigroups in the sense of [2], and that there is a close connection between Ai, AT and the radical A (in the sense of [15]) of the operator ideal A. In fact, if A, B are operator ideals, then the equalities A = 5 a d , Ai = Bi and AT — BT are equivalent. We obtain characterisations of A simpler than the original definition in [15] and we show that »4 is the perturbation class for Ai, Ai and Ai f~l Ar.