The Sylvester equation in Banach algebras

Abstract

Let $\mathcal{A}$ be a unital complex semisimple Banach algebra, and $M_{\mathcal{A}}$ denote its maximal ideal space. For a matrix $M\in {\mathcal{A}}^{n\times n}$, $\widehat{M}$ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix $M\in {\mathbb{C}}^{n\times n}$, $σ(M)\subset \mathbb{C}$ denotes the set of eigenvalues of $M$. It is shown that if $A\in {\mathcal{A}}^{n\times n}$ and $B\in {\mathcal{A}}^{m\times m}$ are such that for all $φ\in M_{\mathcal{A}}$, $σ(\widehat{A}(φ))\cap σ(\widehat{B}(φ))=\emptyset$, then for all $C\in {\mathcal{A}}^{n\times m}$, the Sylvester equation $AX-XB=C$ has a unique solution $X\in {\mathcal{A}}^{n\times m}$. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra.