On the Right (Left) Invertible Completions for Operator Matrices

Let \({\mathcal {H}_{1}}\) and \({\mathcal {H}_{2}}\) be separable Hilbert spaces, and let \({A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})}\) and \({C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})}\) be given operators. A necessary and sufficient condition is given for \({\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}\) to be a right (left) invertible operator for some \({X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}\). Furthermore, some related results are obtained.