The Sylvester equation in Banach algebras

TL;DR

This paper extends the classical Sylvester equation solution theory to matrices over unital complex semisimple Banach algebras, providing conditions for unique solutions and applications like Roth's removal rule.

Contribution

It introduces spectral conditions for the solvability of the Sylvester equation in Banach algebra matrices, generalizing known results from complex matrices.

Findings

Unique solution exists under spectral disjointness condition.

Spectral analysis via Gelfand transforms is key.

Application to Roth's removal rule in Banach algebra context.

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 $M\in {\mathbb{C}}^{n\times n}$, $\sigma(M)\subset \mathbb{C}$ 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 $\varphi \in M_{\mathcal{A}}$, $\sigma(\widehat{A}(\varphi))\cap \sigma(\widehat{B}(\varphi))=\emptyset$, 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.