Minimizer of square root operator norm 
Let $A:D(A) \to \mathcal H$ be a positive self-adjoint operator and $\sqrt{A}$ defined by via the spectral theorem on $D(\sqrt{A}) = Q(A)$ where $Q(A)$ is the quadratic form domain. 
  Let $$E=\inf\{\lVert\sqrt{A}u\rVert^2 : u \in D(\sqrt{A}), \lVert u \rVert = 1\}.$$
  Assume there exists a minimizer $u_0 \in D(\sqrt{A})$ for $E$. Prove that $u_0\in D(A)$ and $Au_0 = E u_0$.

First I tried to show $u_0 \in D(A)$. Since $u_0 \in D(\sqrt{A}) = Q(A)$, it suffices to show $$\sup_{y \in Q(A)\\ \lVert y \rVert \leq 1}\lvert \langle u_0, Ay \rangle \rvert < \infty.$$
But I don't know how to proceed - We can write $$\langle u_0, A y \rangle = \langle \sqrt{A} u_0, \sqrt{A} y\rangle $$
but from here I don't know which estimate I can use. Any help appreciated, also any hint on the second part. 

Let $A=\int_0^{\infty} \lambda dP(\lambda)$ be the spectral decomposition of $A$. Then $u\in\mathcal{D}(\sqrt{A})$ iff
$$
     \|\sqrt{A}u\|^2=  \int_{0}^{\infty}\lambda d\|P(\lambda)u\|^2 < \infty.
$$
Suppose $\lambda_0 = \inf \{ \|\sqrt{\lambda}u\| : u\in\mathcal{D}(\sqrt{A}),\;\; \|u\|=1 \}$. If $u_0$ is a minimizer, meaning that $\|u_0\|=1$, $u_0\in\mathcal{D}(\sqrt{A})$, and $\|\sqrt{A}u_0\|=\lambda_0$, then it's not hard to see that $\mu(S)=\|P(S)u_0\|^2$ is a probability measure that must be concentrated at $\{\lambda_0\}$. Otherwise, the probability measure $\mu$, which is concentrated on $[\lambda_0,\infty)$, could not satisfy the following:
$$
        \lambda_0=\int_{\lambda_0}^{\infty}\lambda d\mu(\lambda).
$$
Therefore, $P\{\lambda_0\}u=u$ must hold and, hence, $\sqrt{A}u_0=\sqrt{\lambda_0}u_0$.
