Vector valued weighted analogue of converse of Wiener-Lévy theorem and applications to modulation spaces

We shall prove a strong converse of the Wiener-Lévy theorem in weighted setting for Banach algebra valued functions. In fact, it is shown that if $ω$ is a weight on a discrete abelian group $G$, $\mathcal X$ is a unital commutative Banach algebra and if $F$ is a $\mathcal X-$valued function defined on $\mathbb C$ such that $T_F(f)=F\circ f \in A^q(Γ,\mathcal X)$ for all $f\in A^1_ω(Γ,\mathcal X)$, then $F$ must be real analytic. Its multivariate analogue and analogue for locally compact abelian $G$ are also established. Lastly, similar results are obtained for modulation and amalgam spaces as an application.