Abstract: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 $TP\in W$.
Keywords: inessential operator, Fredholm operator, generalized inverse
AMS Subject Classification: Primary 47A05, 47A55