On the operator equation \(AXB - XD = E\)
type: Item
instance of: scholarly article
title: On the operator equation \(AXB - XD = E\)
zbMATH Open document ID: 1254.47015
DOI: 10.1134/S1995080209030056
author: Juan-Miguel Gracia
published in: Lobachevskii Journal of Mathematics
publication date: 2011-09-21T00:00:00Z
review text: 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).
reviewed by: Edward L. Pekarev
Mathematics Subject Classification ID: 47A62
zbMATH DE Number: 5948813
zbMATH Keywords: operator equation
zbMATH Keywords: generalized commutator
zbMATH Keywords: Putnam-Fuglede's theorem
MaRDI profile type: Publication
cites work: Q5690490
cites work: On the matrix equations $AX - XB = C$ and $AX - YB = C$
cites work: Commutator Approximants
cites work: On the operator equation \(BX - XA = Q\)
cites work: Q4090924
cites work: The operator equation AX-XB=C with normal A and B
cites work: Finite Operators
cites work: Über die Unbeschränktheit der Operatoren der Quantenmechanik
full work available at URL: s1995080209030056
OpenAlex ID: W2017301281
description: scientific article; zbMATH DE number 5948813