Inverse-closedness of weighted Schur and BGS type quasi-Banach algebras

Abstract

We prove that the weighted quasi-Banach algebras of operator valued matrices satisfying Schur and Baskakov-Gohberg-Sjöstrand (BGS) conditions are inverse-closed in the Banach algebra $B(\ell^2(X,\mathcal{H}))$ whenever the weight is admissible, where $\mathcal{H}$ is a Hilbert space and $X$ is a relatively separated subset of $\mathbb{R}^d$. Furthermore, we identify the Gel'fand space of weighted infinite variable group algebra $\ell^p_ω(\mathbb{Z^N})$ for $0<p\leq1$, and establish inverse-closedness of infinite variable analogue of BGS-type algebra in $B(\ell^2(\mathbb{Z^N},\mathcal{H}))$.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Theorem 2.2: Hulanicki’s lemma for quasi algebras
• Theorem 2.3: Vector-valued weighted Wiener's theorem
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3