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The Sylvester equation in Banach algebras

Sasane, Amol (2021) The Sylvester equation in Banach algebras. Linear Algebra and Its Applications, 631. pp. 1-9. ISSN 0024-3795

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Identification Number: 10.1016/j.laa.2021.08.015


Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M∈A n×n, Mˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M∈C n×n, σ(M)⊂C denotes the set of eigenvalues of M. It is shown that if A∈A n×n and B∈A m×m are such that for all φ∈M A, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then for all C∈A n×m, the Sylvester equation AX−XB=C has a unique solution X∈A n×m. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra.

Item Type: Article
Official URL:
Additional Information: © 2021 Elsevier
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Aug 2021 09:15
Last Modified: 21 Jun 2024 05:12

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