Wiener pairs of Banach algebras of operator-valued matrices

TL;DR

This paper introduces new classes of operator-valued matrix Banach algebras, demonstrating their inverse-closedness and symmetry within the algebra of bounded operators on Hilbert space-valued sequence spaces.

Contribution

It constructs several novel examples of Wiener pairs of Banach algebras of operator-valued matrices and proves their inverse-closedness and symmetry properties.

Findings

New operator-valued matrix Banach algebras are inverse-closed in bounded operators.

These algebras are shown to be symmetric.

The paper generalizes classical decay conditions to operator-valued settings.

Abstract

In this article we introduce several new examples of Wiener pairs $\mathcal{A} \subseteq \mathcal{B}$, where $\mathcal{B} = \mathcal{B}(\ell^2(X;\mathcal{H}))$ is the Banach algebra of bounded operators acting on the Hilbert space-valued Bochner sequence space $\ell^2(X;\mathcal{H})$ and $\mathcal{A} = \mathcal{A}(X)$ is a Banach algebra consisting of operator-valued matrices indexed by some relatively separated set $X \subset \mathbb{R}^d$. In particular, we introduce $\mathcal{B}(\mathcal{H})$-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-Sj\"ostrand algebra, and of anisotropic variations of all of these matrix algebras, and show that they are inverse-closed in $\mathcal{B}(\ell^2(X;\mathcal{H}))$. In addition, we obtain that each of these Banach algebras is symmetric.