  In this paper we find new equivalent norms in $L^p(\mathbb{R}^n,\mathbb{B})$
by using multivariate Littlewood-Paley functions associated with Poisson
semigroup for the Hermite operator, provided that $\mathbb{B}$ is a UMD Banach
space with the property ($\alpha$). We make use of $\gamma$-radonifying
operators to get new equivalent norms that allow us to obtain
$L^p(\mathbb{R}^n,\mathbb{B})$-boundedness properties for (vector valued)
multivariate spectral multipliers for Hermite operators. As application of this
Hermite multiplier theorem we prove that the Banach valued Hermite Sobolev and
potential spaces coincide.
