On Sylvester equations in Banach subalgebras

TL;DR

The paper develops a framework for solving the operator Sylvester equation $BX-XA=Q$ within inverse-closed Banach subalgebras that admit norm-controlled inversion. It proves existence and uniqueness of a solution $X\in\mathcal{A}$ under disjoint spectral assumptions on $A$ and $B$ in the ambient algebra ${\mathcal{B}}$, with explicit norm bounds when $A$ and $B$ are normal on a Hilbert space, depending on the spectral distance $d(A,B)$ and subalgebra norms. The authors apply the results to classes of localized operators, including Gröchenig-Schur, Baskakov-Gohberg-Sjöstrand, Beurling algebras, and unitalized localized integral-operator algebras, establishing solvability and norm-control estimates within these algebras. The proof relies on a contour-integral representation of the inverse Sylvester operator and the norm-controlled inversion property to translate resolvent estimates into subalgebra bounds, enabling stable solving of operator equations in these structured settings.

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.

Theorems & Definitions (7)

• Theorem 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1
• proof : Proof of Theorem \ref{['maintheorem']}