The Sylvester equation in Banach algebras (Q2238857)

Let \({\mathcal A}\) be a commutative unital complex semisimple Banach algebra, and \(M_{\mathcal A}\) denote its maximal ideal space. Let \(n\) and \(m\) be positive integers and \({\mathcal A}^{n\times m}\) denote the set of all \(n\times m\) matrices with entries from \({\mathcal A}\). Let \(A\in {\mathcal A}^{n\times n}\) and \(B\in {\mathcal A}^{m\times m}\) be two matrices, and, for any \(\varphi\in M_{\mathcal A}\), let \(\widehat{A}(\varphi)\) be the matrix obtained by applying \(\varphi\) to all entries of \(A\). The Sylvester matrix and operator equations, and variants of them, have been widely studied in the literature. In this paper, the author proves that if the eigenvalues of \(\widehat{A}(\varphi)\) and \(\widehat{B}(\varphi)\) are distinct for any \(\varphi\in M_{\mathcal A}\), then 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.