On the Right (Left) Invertible Completions for Operator Matrices |
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| Authors: | Guojun Hai Alatancang Chen |
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| Institution: | 1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, People’s Republic of China
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| Abstract: | Let ${\mathcal {H}_{1}}Let H1{\mathcal {H}_{1}} and H2{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H1), B ? B(H2){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H2, H1){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H1, H2){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained. |
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