Invertibility of operator matrices on a Banach space (Q2440926)

Let \(\mathcal X\) and \(\mathcal Y\) be Banach spaces, \(A\) a bounded operator on \(\mathcal X\) and \(B\) a bounded operator from \(\mathcal Y\) to \(\mathcal X\). Let \(M(C,D)\) be the operator matrix defined by \(M(C,D)(x,y)=(Ax+By,Cx+Dy)\) for \(x\) in \(\mathcal X\) and \(y\) in \(\mathcal Y\), where \(C\) is some bounded linear operator from \(\mathcal X\) to \(\mathcal Y\) and \(D\) is a bounded linear operator from \(Y\) to itself. The authors give necessary and sufficient conditions for the operator \(M(C,D)\) defined on \(\mathcal X\oplus \mathcal Y\) to be invertible (respectively, left invertible)