Geodesic
Linear inessential operators and generalized inverses
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 1, pp. 75-82.

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The space of inessential bounded linear operators from one Banach space $X$ into another $Y$ is introduced. This space, $I(X,Y)$, is a subspace of $B(X,Y)$ which generalizes Kleinecke's ideal of inessential operators. For certain subspaces $W$ of $\,I(X,Y)$, it is shown that when $T\in B(X,Y)$ has a generalized inverse modulo $W$, then there exists a projection $P\in B(X)$ such that $T(I-P)$ has a generalized inverse and $TP\in W$.
Classification : 47A05, 47A55
Keywords: inessential operator; Fredholm operator; generalized inverse 
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Barnes, Bruce A. Linear inessential operators and generalized inverses. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 1, pp. 75-82. http://geodesic.mathdoc.fr/item/CMUC_2009__50_1_a5/