Cross-post from [math.sx](https://math.stackexchange.com/questions/4198077/convergence-criterion-in-the-domain-of-an-unbounded-operator).

My question is somewhat close to [this](https://math.stackexchange.com/questions/1653867/is-a-self-adjoint-operator-continuous-on-its-domain) one, but the counterexamples given there do not apply here.

<b>Setup.</b> Given a Hilbert space $\mathcal H$, a closed operator $A$ and a convergent sequence $(x_n)_{n\in\mathbb N}\subset \mathcal D(A)$, I want to ensure that $\lim_{n\to\infty}x_n \in \mathcal D(A)$.<br>It is <i>not</i> necessary that $\lim_{n\to\infty} Ax_n = Ax$.

My intuition says that something like a uniform upper bound on $\|Ax_n\|$ should be a sufficient condition. Let me illustrate that in the following setting for <i>normal</i> operators.

<b>Lemma.</b> Let $A$ be a normal operator on $\mathcal H$, $x_n$ a convergent sequence in $\mathcal D(A)$ and assume $\liminf_{n\to\infty}\|Ax_n\|<\infty$. Then $\lim_{n\to\infty} x_n \in \mathcal D(A)$.<br>
<i>Proof.</i> We use the spectral theorem and [Fatou's lemma](https://en.wikipedia.org/wiki/Fatou's_lemma#Extensions_and_variations_of_Fatou's_lemma) for weakly convergent measures. Let $E_A$ be the projection valued measure associated to $A$. Then
$$ \int_{\mathbb C} |\lambda|^2 d\langle x,E_A(\lambda)x\rangle \le \liminf_{n\to\infty} \int_{\mathbb C} |\lambda|^2 d\langle x_n,E_A(\lambda)x_n\rangle = \liminf_{n\to\infty}\|Ax_n\|^2<\infty. $$
Hence, $x\in\mathcal D(A)$. $\square$

<b>Question.</b> What extensions of above lemma are known? Does anyone have a counterexample? References or own proofs both are warmly welcome.