Domain of Fractional power of Operator

Question

I have some questions about the Domain of the fractional power of operator as follows. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ with regular boundary. Consider the operator $C=\Delta_N- e^2$, where $\Delta_N$ is the Neumann realization of Laplace operator in $H=L^2(\Omega,\mathbb{R})$ and $e\in H$ is Lipschitz function. Let $E=C(\bar\Omega, \mathbb{R})$ and $P_\alpha$ be the Domain of $(-C_E)^{\alpha}$, for some $\alpha \in (0,1).$ Then some following properties is true or not, if it is true how can we prove that?

1. Let $\varphi \colon \mathbb{R} \to \mathbb{R}$ is continuously differentiable and $f:E \to E$, $f(u)=\varphi(u(x))$ then $f(P_\alpha)$ is contained by $P_\alpha$ ?
2. If $u,v \in P_\alpha$, what about $uv$?

Thank you so much for your attention.