I want to prove that if an operator $A$ with domain $D(A)=\left\{u\in L^2\;\text{such that}\; Au\in L^2(\mathbb{R}^n) \right\}$ has a compact resolvant then there exist a constant $c>0$ such that for any $\alpha>0$
$$\langle u, (\alpha I-A )u\rangle \le c \quad\forall u\in \mathcal{C}_0^{\infty}.$$
Can someone help me?