The Sylvester equation in Banach algebras

Abstract

Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M P A nˆn , x M denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M P C nˆn , σpM q Ă C denotes the set of eigenvalues of M . It is shown that if A P A nˆn and B P A mˆm are such that for all ϕ P M A , σp p Apϕqq X σp p Bpϕqq " H, then for all C P A nˆm , the Sylvester equation AX ´XB " C has a unique solution X P A nˆm . As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra.