Vector valued Beurling algebra analogues of Wiener's Theorem

Let $0<p\leq 1$, $\omega$ 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 $\sum_{n\in \mathbb Z} \|\widehat f(n)\|^p \omega(n)^p<\infty$ 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 $1\leq \nu \leq \omega$, $\nu$ is constant if and only if $\omega$ is constant, $g$ is a left inverse of $f$ and $\sum_{n\in \mathbb Z}\|\widehat g(n)\|^p\nu(n)^p<\infty$. 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.