The Sylvester equation in Banach algebras

Abstract

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