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

Lukas Köhldorfer, Peter Balazs

Session: General

Keywords: Jaffard’s Lemma, Wiener’s Lemma, inverse- closed, spectral invariance, polynomial off-diagonal decay, operator-valued matrices, matrix algebras

TL;DR: We give a self-contained proof of Jaffard's Lemma for operator-valued matrices.

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.

Submission Number: 5