On Weyl completions of partial operator matrices

Abstract

Let $\mathcal{H}$ and $\mathcal{K}$ be complex separable Hilbert spaces. Given the operators $A\in \mathcal{B}(\mathcal{H})$ and $B\in \mathcal{B}(\mathcal{K},\mathcal{H})$, we define , where $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$ are unknown elements. In this article, we give a necessary and sufficient condition for $M_{X,Y}$ to be a (right) Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$. Moreover, we show that if $dim\mathcal{K}<\infty$, then $M_{X,Y}$ is a left Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$ if and only if $[AB]$ is a left Fredholm operator and $ind([AB])\le dim\mathcal{K}$; if $dim\mathcal{K}=\infty$, then $M_{X,Y}$ is a left Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$.