Solvability of \(AXB - CXD = E\) in the operators algebra \(B(H)\)
type: Item
instance of: scholarly article
title: Solvability of \(AXB - CXD = E\) in the operators algebra \(B(H)\)
zbMATH Open document ID: 1300.47023
DOI: 10.1134/S1995080210030091
author: M. C. Fu
published in: Lobachevskii Journal of Mathematics
publication date: 2012-12-12T00:00:00Z
review text: This short note is a continuation of the paper published by the author and \textit{S. Mecheri} [Lobachevskii J. Math. 30, No. 3, 224--228 (2009; Zbl 1254.47015)]. It is proved that the equation \(AXB-CXD=E\), where \(A, B, C, D\in B(H)\) are normal operators on a separable (infinite-dimensional) complex Hilbert space \(H\) such that \(A\) and \(D\) commute and the pairs \((A,D)\) and \((B,C)\) possess the Fuglede-Putnam property, has a solution in \(B(H)\) if and only if, for every \(\lambda \in {\mathbb C}\), the block operator matrix which has \(A-\lambda I\) and \(D-\lambda I\) on the diagonal is equivalent to the block operator matrix which has \(A-\lambda I\) and \(D-\lambda I\) on the diagonal and \(E\) in the upper right corner.
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 47A62
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 6115135
zbMATH Keywords: generalised derivation
zbMATH Keywords: matrix and operator equations
zbMATH Keywords: Putnam-Fuglede's property
MaRDI profile type: Publication
cites work: Properties which normal operators share with normal derivations and related operators
cites work: Q3155739
cites work: Putnam-Fuglede theorem and the range-kernel orthogonality of derivations
cites work: On the matrix equations $AX - XB = C$ and $AX - YB = C$
cites work: Commutator Approximants
cites work: Another version of Maher's inequality
cites work: On the operator equation \(AXB - XD = E\)
cites work: On the operator equation \(BX - XA = Q\)
cites work: Über die Unbeschränktheit der Operatoren der Quantenmechanik
full work available at URL: s1995080210030091
OpenAlex ID: W2470020535
description: scientific article; zbMATH DE number 6115135