On the invertibility of elementary operators

Let \(\mathscr{X}\) be a complex Banach space and \(\mathcal{L}(\mathscr{X})\) be the algebra of all bounded linear operators on \(\mathscr{X}\). For a given elementary operator \(\Phi\) of length \(2\) on \(\mathcal{L}(\mathscr{X})\), we determine necessary and sufficient conditions for the existen...

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Bibliographic Details
Published in:arXiv.org 2013-12
Main Authors: Boudi, Nadia, Bračič, Janko
Format: Article
Language:English
Subjects:
Banach spaces
Economic models
Linear operators
Operators (mathematics)
Online Access:Get full text
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Summary:Let \(\mathscr{X}\) be a complex Banach space and \(\mathcal{L}(\mathscr{X})\) be the algebra of all bounded linear operators on \(\mathscr{X}\). For a given elementary operator \(\Phi\) of length \(2\) on \(\mathcal{L}(\mathscr{X})\), we determine necessary and sufficient conditions for the existence of a solution of the equation \({\rm X} \Phi=0\) in the algebra of all elementary operators on \(\mathcal{L}(\mathscr{X})\). Our approach allows us to characterize some invertible elementary operators of length \(2\) whose inverses are elementary operators.
ISSN:2331-8422