Wiener pairs of Banach algebras of operator-valued matrices

Abstract

In this article we consider several new examples of Wiener pairs $A\subseteq B$, where $B=B\left({\ell }^2\right(X;H\left)\right)$ is the Banach algebra of bounded operators acting on the Hilbert space-valued Bochner sequence space ${\ell }^2(X;H)$ and $A=A\left(X\right)$ is a Banach algebra consisting of operator-valued matrices indexed by some relatively separated set $X\subset R^d$. In particular, we consider $B\left(H\right)$-valued versions of the Jaffard algebra, of certain weighted Schur-type algebras, of Banach algebras which are defined by more general off-diagonal decay conditions than polynomial decay, of weighted versions of the Baskakov-Gohberg-Sjöstrand algebra, and of anisotropic variations of all of these matrix algebras, and show that they are inverse-closed in $B\left({\ell }^2\right(X;H\left)\right)$. In addition, we obtain that each of these Banach algebras is symmetric.