Wiener pairs of Banach algebras of operator-valued matrices
Lukas Köhldorfer, Peter Balazs
TL;DR
This work extends Wiener's lemma to a broad family of Wiener pairs ${\mathcal{A}}(X) \subseteq {\mathcal{B}}(\ell^2(X; \mathcal{H}))$, where ${\mathcal{A}}(X)$ consists of operator-valued matrices indexed by a relatively separated set $X$. It develops several concrete inverse-closed algebras—weighted Schur-type ${\mathcal{S}}_{\nu}^1$, Jaffard-type ${\mathcal{J}}_{s}$, Baskakov-Gohberg-Sjöstrand-type ${\mathcal{C}}_{\nu}$, and anisotropic variants—by leveraging Hulanicki's lemma, Brandenburg's trick, Barnes' Lemm, and derivation methods. The paper proves symmetry and spectral-invariance results for these algebras, including their inverse-closedness in ${\mathcal{B}}(\ell^2(X; \mathcal{H}))$, and introduces anisotropic decay classes via commuting derivations to extend the Wiener-pair framework to irregular matrix-structures. These findings provide a versatile toolkit for operator theory, Fourier analysis of operator-valued matrices, and localization phenomena in g-frames and related time-frequency analyses, with potential applications to pseudodifferential operators and quantum harmonic analysis.
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-Sjöstrand 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.