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

We give a self-contained proof of a recently established $\mathcal{B}\left( \mathcal{H} \right)$-valued version of Jaffards Lemma. That is, we show that the Jaffard algebra of $\mathcal{B}\left( \mathcal{H} \right)$-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 $\mathcal{B}\left( {{\ell ^2}(X;\mathcal{H})} \right)$ of all bounded linear operators on ${\ell ^2}(X;\mathcal{H})$, the Bochner-space of square-summable $\mathcal{H}$-valued sequences.