Year: 2025
Type: preprint
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 $\mathcal{K}$, 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.
Source: arXiv (Cornell University)
Author: Sarajlija, Nikola
Institution: University of Novi Sad
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414450437
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