# 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

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