# Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 93-106.

Completions of upper-triangular matrices to left-Fredholm operators with non-positive index

Authors:  Dragana S. Cvetkovic-Ilic
Author institution: Department of Mathematics, Faculty of Science and Mathematics, University of Nis, 18000 Nis, Serbia

Summary:  In this paper, for given operators $A\in\mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B}(\mathcal{K})$, where $\mathcal{H}$, $\mathcal{K}$ are infinite-dimensional complex separable Hilbert spaces, we describe the set of all $C\in\mathcal{B}(\mathcal{K},\mathcal{H})$ such that, the operator matrix $M_C=%\bmatrix{cc} \left[\begin{smallmatrix} A&C\\0&B\end{smallmatrix}\right]$ %\endbmatrix$belongs to$\Phi_+^-(\mathcal{H}\oplus\mathcal{K})$, which means that it is a left-Fredholm operator with non-positive index. As an application of our results, in the case when at least one of the operators$A\in\mathcal{B}(\mathcal{H})$,$B\in\mathcal{B}(\mathcal{K})$is compact we obtain some interesting corollaries pertaining to intersections of the spectra$\sigma_{\Phi_+^-}(M_C)$where$C\$ runs through certain classes of operators.

DOI: http://dx.doi.org/10.7900/jot.2015sep07.2078
Keywords:  Fredholm operator, left-Fredholm operator with non-positive index, index of operator, upper-triangular operator matrix