Multipliers and imaginary powers of the schr\"odinger operators characterizing UMD Banach spaces

In this paper we establish $L^p$-boundedness properties for Laplace type transform spectral multipliers associated with the Schr"odinger operator $\mathcal{L}=-\mathrm{\Delta }+V$. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the $L^p$-boundedness of the imaginary powers ${\mathcal{L}}^{i\gamma }$, $\gamma \in \mathbb{R}$, of $\mathcal{L}$.