Nittka's invariance criterion and Hilbert space valued parabolic equations in $L_p$

Nittka gave an efficient criterion on a form defined on $L_2(\Omega)$ which implies that the associated semigroup is $L_p$-invariant for some given $p \in (1,\infty)$. We extend this criterion to the Hilbert space valued~$L_2(\Omega,H)$. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is $L_p$-contractive for all $p \in [p_-,p_+]$ for some $1 < p_- < 2 < p_+ < \infty$.