标题： FLI和FRI上三角算子矩阵的可逆补全 显示英文标题

标题： Invertible completions of FLI and FRI upper triangular operator matrices

摘要： 如果$A\in\mathcal{B}(\mathcal{H})$和$B\in\mathcal{B}(\mathcal{K})$是给定的算子，记$M_C$为一个形式为$$M_C=\begin{pmatrix} A & C\\ 0 & B \end{pmatrix}\in\mathcal{B}(\mathcal{H}\oplus\mathcal{K})$$的算子矩阵，作用于无限维可分希尔伯特空间$\mathcal{H}$和$\mathcal{K}$的直和，且$C\in\mathcal{B}(\mathcal{K},\mathcal{H})$是未知的。 在本文中，我们解决了$M_C$到弗雷德霍姆左可逆和弗雷德霍姆右可逆的补全问题，并由此获得了适当的扰动结果。我们通过解决弗雷德霍姆左谱和弗雷德霍姆右谱的“填补空洞”问题来说明我们的结果。最后，我们考虑了一些特殊的算子类。 throughout the article, 我们恢复了一些已知的结果。 显示英文摘要

摘要： If $A\in\mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B}(\mathcal{K})$ are given operators, denote by $M_C$ an operator matrix of the form $$M_C=\begin{pmatrix} A & C\\ 0 & B \end{pmatrix}\in\mathcal{B}(\mathcal{H}\oplus\mathcal{K})$$ acting on a direct sum of infinite dimensional separable Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, and $C\in\mathcal{B}(\mathcal{K},\mathcal{H})$ is unknown. In this article we solve the completion problem of $M_C$ to Fredholm left and Fredholm right invertibility, and we obtain appropriate perturbation results as consequences. We illustrate our results by solving the 'filling in holes' problem for Fredholm left and Fredholm right spectra. Finally, we consider some special classes of operators. Throughout the article, we recover some known results.