CFI upper triangular operator matrices

Abstract

Let $\mathcal{H}$ and $\mathcal{K}$ be complex infinite dimensional separable Hilbert spaces. For given operators $A\in \mathcal{B}(\mathcal{H})$ and $B\in \mathcal{B}(\mathcal{K})$, we provide necessary and sufficient conditions which make a $2\times 2$ upper triangular operator matrix $M_C=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$ a CFI operator for some (or every) $C\in \mathcal{B}(\mathcal{K}\mathcal{,}\mathcal{H})$.

Keywords:

• Range
• Fredholm operator
• Weyl operator
• CFI
• upper triangular operator matrix

2010 Mathematics Subject Classifications:

• Primary 47A53
• Secondary 47A05
• 47A10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 12101081) and Fundamental Research Program of Shanxi Province (Grant No. 20210302124079).