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In this paper we find new equivalent norms in $L^p(\mathbb{R}^n,\mathbb{B})$by using multivariate Littlewood-Paley functions associated with Poissonsemigroup for the Hermite operator, provided that $\mathbb{B}$ is a UMD Banachspace with the property ($\alpha$). We make use of $\gamma$-radonifyingoperators 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 thisHermite multiplier theorem we prove that the Banach valued Hermite Sobolev andpotential spaces coincide.

Vector valued multivariate spectral multipliers, Littlewood-Paley functions, and Sobolev spaces in hte Hermite setting