Let $T$ be a unbounded positive self adjoint linear operator ($T: \mathcal{D} (T) \to \mathcal{H}, \mathcal{D} (T) \subset \mathcal{H}$, where $\mathcal{H}$ is a Hilbert space) such that $\mathcal{N} (T) = \{ 0 \}$, where $\mathcal{N} (T)=\{ x \in \mathcal{D} (T): T x = 0 \}$.
My problem: For $x \in \mathcal{H} \setminus \mathcal{D} (T)$ show that $$\| T^{-\frac{1}{2}} E_n x \| \substack{\longrightarrow\\n\to\infty} \infty,$$ where $E_n=E_T ((\frac{1}{n},n)), n \in \mathbb{N}$ ($E_T$ is the spectral measure of $T$).
This fact is used but not proved in the proof of the Lemma 10.11 "Unbounded Self-adjoint Operators on Hilbert Space", K. Schmudgen, Springer, 2012. I have no idea how to prove it.