On Sylvester equations in Banach subalgebras

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.