$\gamma$-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space $\text{B}\mathrm{.}$ If we denote by $H$ the Hilbert space $L^2((0,\mathrm{\infty }),dt\mathrm{/}t),\gamma (H,\text{B})$ represents the space of $\gamma$-radonifying operators from $H$ into $\text{B}\mathrm{.}$ We prove that the Hermite square function defines bounded operators from $BM{O}_{\mathcal{L}}(\mathbb{R},\text{B})$ (respectively, $H_{\mathcal{L}}^1(\mathbb{R},\text{B})$) into $BM{O}_{\mathcal{L}}(\mathbb{R},\gamma (H,\text{B}))$ (respectively, $H^1_$\mathcal{L}$(\R, \gamma(H,\B))$),where$BMO_$\mathcal{L}$$and$H^1_$\mathcal{L}$$denote$BMO$ and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in $BM{O}_{\mathcal{L}}(\mathbb{R},\text{B})$ and $H_{\mathcal{L}}^1(\mathbb{R},\text{B})$ by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.