Let $T$ be a closed unbounded (in my case also symmetric) operator on a Hilbert space $\mathcal{H}$ with dense domain $\mathcal{D}(T)$, and let $f\in \mathcal{D}(T)$. Suppose there is a dense subspace $Y$ of $\mathcal{H}$ contained in $\mathcal{D}(T)$. If I choose a sequence $f_n$ in $Y$ converging to $f$, under what conditions, if any, can I ensure that $Tf_n\to Tf$? The convergence of $Tf_n$ may be taken to be strong or weak. I know that $Tf$ is defined, but is it the limit of $Tf_n$?
Indeed, supposing my Hilbert space is $L^2(\mathbb{R}^n)$, I'm even interested in the $w^*$ convergence of the functionals $(Tf_n,\dot\:)$, where $(\dot\:,\dot\:)$ is the $L^2$-inner product. (These functionals are of course bounded on $\mathcal{D}(T)$ and so extend by continuity to $L^2(\mathbb{R}^n)$, which, by Riesz's lemma means they still take the form $(Tf_n,\dot\:)$.)
A related (and equivalent, by Banach-Steinhaus) question is, whether the sequence $Tf_n$ is uniformly bounded.