On the invertible completions for relation matrices

Abstract

Let $H$ and $K$ be separable Hilbert spaces.
In this paper, for $A\in\mathcal{BR}(H)$, $B\in\mathcal{BR}(K)$ and $C\in\mathcal{BR}(K,H)$, a necessary and sufficient condition is given for relation matrices $M_{X}=(A C;X B)$ to be right (left) invertible and invertible relation for some $X\in \mathcal{B}(H,K)$ $(X\in \mathcal{BR}(H,K)).$ Moreover, some relevant properties and illustrating examples are also given.