# Invertible completions of FLI and FRI upper triangular operator matrices

## TL;DR

This paper investigates the conditions under which upper triangular operator matrices can be completed to be invertible in the Fredholm sense, providing new insights and perturbation results for such operator completions.

## Contribution

It introduces novel criteria for completing operator matrices to be Fredholm invertible and extends existing results with new perturbation theorems and spectral filling techniques.

## Key findings

- Established conditions for Fredholm invertibility of operator completions.
- Derived perturbation results related to operator matrix completions.
- Solved the 'filling in holes' problem for Fredholm spectra.

## Abstract

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.