Solving the Sylvester equation in Banach modules

TL;DR

This paper characterizes the solvability, uniqueness, and solutions of the Sylvester equation within Banach modules over complex Banach algebras, providing necessary and sufficient conditions.

Contribution

It offers a complete characterization of the Sylvester equation's solvability and solutions in the setting of Banach modules over Banach algebras, extending previous results.

Findings

Established necessary and sufficient conditions for solvability.

Provided formulas for particular solutions when solvable.

Characterized the uniqueness of solutions.

Abstract

For given unital complex Banach algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, let $\mathfrak{M}$ be a Banach module acting between them. Let $a\in \mathcal{A}_1$, $b\in\mathcal{A}_2$, and $c\in\mathfrak{M}$ be provided such that $\sigma_{\mathcal{A}_1}(a)\cap\sigma_{\mathcal{A}_2}(b) \neq\emptyset$. In this paper we completely characterize the consistency of the Sylvester equation $$ax-xb=c.$$ Precisely, we establish verifiable sufficient and necessary solvability conditions, and we provide some formulas for particular solutions $x\in\mathfrak{M}$ when the equation is solvable. Moreover, we characterize the uniqueness of the solutions.