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 $\sum _{n\in \mathbb Z} \Vert \widehat{f}(n)\Vert ^p \omega (n)^p<\infty$ 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 $g:\mathbb T \rightarrow \mathcal A$ such that $1\le \nu \le \omega$, $\nu$ is constant if and only if $\omega$ is constant, $\nu$ is admissible if and only if $\omega$ is admissible, g is a left inverse of f and $\sum _{n\in \mathbb Z}\Vert \widehat{g}(n)\Vert ^p\nu (n)^p<\infty$. We shall obtain a similar result when $\omega$ is a p-almost monotone algebra weight and $1<p<\infty$. We shall obtain an analogue of this result on the real line.