Generalized invertibility of operator matrices (Q692553)

Let \(Z\) be a Banach space, such that \(Z=X\oplus Y\) for some closed and complementary subspaces \(X\) and \(Y\). Then each operator \(M\in \mathbb B(Z)\) which is invariant on \(X\), has a decomposition into some operators \(A\in\mathbb B(X)\), \(B\in\mathbb B(Y)\) and \(C\in\mathbb B(Y, X)\). In this paper, the authors discuss under which conditions on \(A\) and \(B\), there exists some \(C\) such that \(M\) has a generalized inverse, or is left Browder (that is, left Fredholm with finite ascent), or has some other similar properties.