# On surjectivity and denseness of range of the operator $A+CX$

### Dragana S. Cvetković-Ilić

University of Niš, Serbia### Vladimir Pavlović

University of Niš, Serbia

## Abstract

Motivated by a result of Takahashi from *Invertible completions of operator matrices*, Integr. Equ. Oper. Theory 21 (1995) 355–361, in this paper we investigate the problems of characterization of all the pairs of operators $A\in\mathcal B(\mathcal H,\mathcal K)$ and $C\in\mathcal B(\mathcal L,\mathcal K)$ for which there exists some $X\in\mathcal B(\mathcal H,\mathcal L)$ such that the operator $A+CX$ is surjective/with dense range, for some $X\in\mathcal B(\mathcal H,\mathcal K)$. We completely solve the former and give some partial results regarding the latter one.

## Cite this article

Dragana S. Cvetković-Ilić, Vladimir Pavlović, On surjectivity and denseness of range of the operator $A+CX$. J. Spectr. Theory 8 (2018), no. 3, pp. 871–881

DOI 10.4171/JST/215