On the inverse-closedness of operator-valued matrices with polynomial off-diagonal decay
Lukas Köhldorfer, Peter Balazs
TL;DR
The paper extends Jaffard's Lemma to operator-valued matrices with polynomial off-diagonal decay by formulating the Jaffard class ${\mathcal{J}}_s(X;{\mathcal{B}}(\mathcal{H}))$ within the Bochner-space setting $\ell^2(X;\mathcal{H})$ and proving inverse-closedness in ${\mathcal{B}}(\ell^2(X;\mathcal{H}))$ for $s>d$. It provides a self-contained proof by adapting weighted Schur-type estimates and Hulanicki's Lemma to ${\mathcal{B}}(\mathcal{H})$-valued matrices. Key contributions include establishing that ${\mathcal{J}}_s$ is a unital Banach *-algebra embedded in ${\mathcal{B}}(\ell^2(X;\mathcal{H}))$ and proving spectral-invariance (inverse-closedness) with a sharp radii inequality. This result enables rigorous analysis of localized operator-valued systems, with potential applications to g-frames, Fourier analysis of operators, and harmonic analysis in operator theory.
Abstract
We give a self-contained proof of a recently established $\mathcal{B}(\mathcal{H})$-valued version of Jaffards Lemma. That is, we show that the Jaffard algebra of $\mathcal{B}(\mathcal{H})$-valued matrices, whose operator norms of their respective entries decay polynomially off the diagonal, is a Banach algebra which is inverse-closed in the Banach algebra $\mathcal{B}(\ell^2(X;\mathcal{H}))$ of all bounded linear operators on $\ell^2(X;\mathcal{H})$, the Bochner-space of square-summable $\mathcal{H}$-valued sequences.