Vector valued Beurling algebra analogues of Wiener's Theorem

Abstract

Let $0<p\le 1$, $\omega$ be a weight on $\mathbb{Z}$, and let $\mathcal{A}$ be a unital Banach algebra. If $f$ is a continuous function from the unit circle $\mathbb{T}$ to $\mathcal{A}$ such that ${\sum }_{n\in \mathbb{Z}}\Vert \hat{f}(n){\Vert }^p\omega (n{)}^p<\infty$ and $f(z)$ is left invertible for all $z\in \mathbb{T}$, then there is a weight $\nu$ on $\mathbb{Z}$ and a continuous function $g:\mathbb{T}\to \mathcal{A}$ such that $1\le \nu \le \omega$, $\nu$ is constant if and only if $\omega$ is constant, $g$ is a left inverse of $f$ and ${\sum }_{n\in \mathbb{Z}}\Vert \hat{g}(n){\Vert }^p\nu (n{)}^p<\infty$. We shall obtain a similar result when $\omega$ is an almost monotone algebra weight and $1<p<\infty$. We shall obtain an analogue of this result on the real line. We shall apply these results to obtain $p-$power weighted analogues of the results of off diagonal decay of infinite matrices of operators.