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 and for which there exists some such that the operator is surjective/with dense range, for some . We completely solve the former and give some partial results regarding the latter one.
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Dragana S. Cvetković-Ilić, Vladimir Pavlović, On surjectivity and denseness of range of the operator . J. Spectr. Theory 8 (2018), no. 3, pp. 871–881DOI 10.4171/JST/215