Xiufeng Wu, Junjie Huang, Alatancang Chen. Source: Annals of Functional Analysis, Volume 10, Number 2, 229--241.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 $M_{X,Y}:=[\begin{smallmatrix}A&B\\X&Y\end{smallmatrix}]$ , 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}\lt \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 $[A\ B]$ is a left Fredholm operator and $\operatorname{ind}([A\ B])\leq \dim \mathcal{K}$ ; 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 $Y\in \mathcal{B}(\mathcal{K})$ .