Invertible completions of FLI and FRI upper triangular operator matrices

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 $$M_C=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)\in \mathcal{B}(\mathcal{H}\oplus \mathcal{K})$$ acting on a direct sum of infinite dimensional separable Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, where $C\in \mathcal{B}(\mathcal{K},\mathcal{H})$ 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.