If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal X,\mathcal Y)$ such that $T_n x\rightarrow Tx$ for all $x\in \mathcal D (T)$?
If $\mathcal X$ and $\mathcal Y$ are Hilbert spaces, the polar decomposition of $T$ together with the Borel functional calculus allows one to answer the question in the affirmative, but I am not sure how to proceed in the Banach space setting, even if I assume separability.
Update: If the graph of $T$ has a Schauder basis, say $(x_n,Tx_n)_{n\in \mathbb N}$, then $(x_n)_{n\in \mathbb N}$ is a Schauder basis for $\mathcal X$ and we can consider the bounded projections $P_j( \sum_{n\geq 1}a_n x_n )= \sum_{j\geq n\geq 1}a_n x_n$. In this case $T_n=TP_n$ furnishes an approximation.
Update: If $\mathcal Y$ has a Schauder basis and $T$ has a bounded inverse, then the graph of T has a Schauder basis, so that $T$ can be approximated pointwise.
Assume more generally that $\mathcal Y$ has a Schauder basis and that there is a bounded operator $B\in \mathscr B (\mathcal X,\mathcal Y)$ such that $T+B$ has a bounded inverse. Then $T+B$, and therefore also $T$, can be approximated pointwise by bounded operators. In case $\mathcal X=\mathcal Y$, this covers the case of a closed operator with non-empty resolvent set.
Update: In the paper 'Gaussian Measures on a Banach Space', Kuelbs constructs, for any separable Banach space $B$, separable Hilbert spaces $H_1\subset B\subset H_2$, where the inclusions are dense, bounded embeddings. It is claimed by Gill, Basu, Zachary and Steadman in the paper 'Adjoint for Operators in Banach Spaces' that any closed and densely defined operator $T$ on $B$ extends to a closed and densely defined operator on $H_2$ (Theorem 4). This would seem like a large step towards an affirmative answer in the case of a separable Banach space. Unfortunately, I do not understand the proof of Theorem 4, in particular it is not clear to me why (in the notation of the paper) $\left.A'\right|_{H_2'}$ is densely defined.
Edit: A comment in a paper uploaded to arxiv suggests to me that the statement of Theorem 4 was false. So this line of reasoning seems unproductive.
Edit: There is now a counter example to Theorem 4.
Kuelbs, J., Gaussian measures on a Banach space, J. Funct. Anal. 5, 354-367 (1970). ZBL0194.44703.
Gill, Tepper L.; Basu, Sudeshna; Zachary, Woodford W.; Steadman, V., Adjoint for operators in Banach spaces, Proc. Am. Math. Soc. 132, No. 5, 1429-1434 (2004). ZBL1048.46014.