## Annals of Functional Analysis

### 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 $M_{X,Y}:=[\begin{smallmatrix}A&B\\X&Y\end{smallmatrix}]$, 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 $Y\in \mathcal{B}(\mathcal{K})$.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 2 (2019), 229-241.

Dates
Accepted: 17 July 2018
First available in Project Euclid: 19 March 2019

https://projecteuclid.org/euclid.afa/1552960865

Digital Object Identifier
doi:10.1215/20088752-2018-0018

Mathematical Reviews number (MathSciNet)
MR3941384

Zentralblatt MATH identifier
07083891

#### Citation

Wu, Xiufeng; Huang, Junjie; Chen, Alatancang. On Weyl completions of partial operator matrices. Ann. Funct. Anal. 10 (2019), no. 2, 229--241. doi:10.1215/20088752-2018-0018. https://projecteuclid.org/euclid.afa/1552960865

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