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

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