Authors: Đorđević, Bogdan 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Singular Sylvester equation in Banach spaces and its applications: Fredholm theory approach
Journal: Linear Algebra and Its Applications
Volume: 622
First page: 189
Last page: 214
Issue Date: 1-Aug-2021
Rank: ~M21
ISSN: 0024-3795
DOI: 10.1016/j.laa.2021.03.035
Abstract: 
We prove that, by assuming the existence of at least one left upper semi-Fredholm operator, then under some natural conditions, the singular operator equation AX−XB=C is solvable if the appropriate matrix equation is solvable. This characterization is convenient because the matrix version of the problem has been closed in [14] and [17]. In addition, we obtain sufficient conditions for A, B and X such that the generalized derivation AX−XB is a compact operator. A connection is established with Fréchet derivatives and commutators of idempotents. Applications to Schur coupling and linear time-invariant systems are mentioned.
Keywords: Fredholm theory | Operator algebras | Sylvester equations
Publisher: Elsevier
Project: Functional analysis, stochastic analysis and applications 

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