Invertibility of Operator Matrices on a Banach Space

Let $mathcal X $ and $mathcal Y $ be Banach spaces, and let $Ain mathcal B (mathcal X )$ and $Cin mathcal B (mathcal Y , mathcal X )$ be given operators. A necessary and sufficient condition is given for $left[ begin{array}{cc} A&C X&Y end{array} right]$ to be invertible (respectively, left invertible) for some $Xin mathcal B (mathcal X , mathcal Y )$ and $Yin mathcal B (mathcal Y )$ . Furthermore, some related results are obtained.