On Sylvester equations in Banach subalgebras

TL;DR

This paper investigates the solvability and norm estimates of Sylvester equations within Banach subalgebras, extending results to operator algebras, localized matrices, and integral operators.

Contribution

It establishes conditions for unique solutions of Sylvester equations in Banach subalgebras with norm control, including explicit estimates under additional assumptions.

Findings

Unique solutions exist under spectral disjointness.

Explicit norm estimates are provided for solutions.

Applications to localized matrices and integral operators.

Abstract

Let ${\mathcal B}$ be a Banach algebra and ${\mathcal A}$ be a Banach subalgebra that admits norm-controlled inversion in ${\mathcal B}$. In this work, we take $A, B$ in the Banach subalgebra ${\mathcal A}$ with their spectra in the Banach algebra ${\mathcal B}$ being disjoint, and show that the operator Sylvester equation $ BX-XA=Q$ has a unique solution $X\in {\mathcal A}$ for every $Q\in {\mathcal A}$. Under the additional assumptions that ${\mathcal B}$ is the operator algebra ${\mathcal B}(H)$ on a Hilbert space $H$ and that $A$ and $B$ are normal in ${\mathcal B}(H)$, an explicit norm estimate for the solution $X$ of the above operator Sylvester equation is provided in this work. In addition, the above conclusion on norm control is applied to Banach subalgebras of localized infinite matrices and integral operators.