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

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.