On the operator equation \(AXB - XD = E\)
The aim of the authors of the paper under review is to generalize the results of \textit{W. E. Roth} [Proc. Am. Math. Soc. 3, 392--396 (1952; Zbl 0047.01901)] and \textit{A. Schweinsberg} [Pac. J. Math. 102, 447--453 (1982; Zbl 0524.47012)] in order to give necessary and sufficient conditions for the solvability of the operator equations \[ AXB-X=C \tag{1} \] and \[ AXB-XD=E, \tag{2} \] where \(A,B,C,D\) are (bounded linear) normal operators in a Hilbert space \(\mathfrak{H}\), \(E\in\mathcal{L}(\mathfrak{H})\). Reviewer's remark. Considering (1), the authors have proved (Theorem 2.1) that, given \(A\) normal and the pair \((A,B)\) with Fuglede-Putnam's property (if \(AT=TB\) for some \(T \in \mathcal{L}(\mathfrak{H})\), then \(A^{*}T=TB^{*}\), where \(A^{*}, B^{*}\) denote the adjoints of \(A,B\) correspondingly), then this equation has a solution if \(\left[\begin{smallmatrix} A-\lambda I & O\\ O & I-\lambda B\end{smallmatrix}\right] \) and \( \left[\begin{smallmatrix} A-\lambda I & C\\ O& I-\lambda B\end{smallmatrix}\right] \) are \textit{similar} matrix operators in \(\mathfrak{H}\oplus\mathfrak{H}\) for any \(\lambda \in\mathbb{C}\) (not ``equivalent'', as stated in Theorem 2.1) and only if these matrix operators are equivalent. Analogously, the corresponding remark needs to be made with regard to equation (2).
scientific article; zbMATH DE number 5948813