Positive operators acting on a sequence of vectors Let $A$ be self-adjoint, unbounded operator with domain $\mathcal{D}\subset \mathcal{H}$ ($\mathcal{H}$ - Hilbert space). We assume that the spectrum of $A$ is absolutely continuous and is the set $[0,\infty[$. We take any sequence $\mathbb{N}\ni n \mapsto \psi_n\in\mathcal{D}$. Is it true that $\lim_{n\rightarrow \infty}\|\psi_n\|=0$ if $\lim_{n\rightarrow \infty}\|A\psi_n\|=0$.

No. If $\lim_{n} \|A\psi_{n}\|=0$ implies $\lim_{n}\|\psi_{n}\|=0$, then the inverse of $A$ is continuous. It's easy to construct a counterexample. For example, on $L^{2}[0,\infty)$, $$ Af = xf,\;\;\; f \in \mathcal{D}(A)=\\{ f \in L^{2} : xf \in L^{2} \\}. $$ In this case, $\psi_{n}=\sqrt{n}\chi_{[0,1/n]}$ is a unit vector in $L^{2}[0,\infty)$, and $$ \|A\psi_{n}\|^{2}=n\int_{0}^{1/n}x^{2}dx \rightarrow 0 \mbox{ as } n \rightarrow\infty. $$