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

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 $B({\mathrm{\ell }}^2(X,\mathcal{H}))$ 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 ${\mathrm{\ell }}_{\omega }^p({\mathbb{Z}}^{\mathbb{N}})$ for $0<p\le 1$, and establish inverse-closedness of infinite variable analogue of BGS-type algebra in $B({\mathrm{\ell }}^2({\mathbb{Z}}^{\mathbb{N}},\mathcal{H}))$.