Abstract
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.
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This work was completed with the support of the Specialized Research Foundation for the Doctoral Program of Higher Education (No. 20070126002), the National Natural Science Foundation of China (No. 10962004) and The Scientific Research Foundation for the Returned Overseas Chinese Scholars.
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Hai, G., Chen, A. On the Right (Left) Invertible Completions for Operator Matrices. Integr. Equ. Oper. Theory 67, 79–93 (2010). https://doi.org/10.1007/s00020-010-1771-1
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DOI: https://doi.org/10.1007/s00020-010-1771-1