  In this paper we consider Littlewood-Paley functions defined by the
semigroups associated with the operator $\mathcal{A}=-\frac{\Delta}{2}-x\nabla$
in the inverse Gaussian setting for Banach valued functions. We characterize
the uniformly convex and smooth Banach spaces by using
$L^p(\mathbb{R}^n,\gamma_{-1})$- properties of the
$\mathcal{A}$-Littlewood-Paley functions. We also use Littlewood-Paley
functions associated with $\mathcal{A}$ to characterize the K\"othe function
spaces with the UMD property.
