On the inverse-closedness of operator-valued matrices with polynomial off-diagonal decay

TL;DR

This paper provides a proof that operator-valued matrices with polynomial off-diagonal decay form an inverse-closed Banach algebra within the space of all bounded operators on a Hilbert space-valued sequence space.

Contribution

It offers a self-contained proof of the inverse-closedness of the operator-valued Jaffard algebra, extending classical results to the setting of operator-valued matrices.

Findings

The Jaffard algebra of operator-valued matrices is a Banach algebra.

This algebra is inverse-closed in the space of bounded operators on the Hilbert space-valued sequence space.

The proof is self-contained and extends known scalar results to the operator-valued setting.

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.