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

Text (The Sylvester equation in Banach algebras)
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## Abstract

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 |
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Official URL: | https://www.journals.elsevier.com/linear-algebra-a... |

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 |

URI: | http://eprints.lse.ac.uk/id/eprint/111787 |

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