Let $A$ be an unbounded self-adjoint operator in a Hilbert space $\mathcal{H}$, with domain $\mathrm{dom}\,A$, and consider a growing sequence $(P_n)_{n\in\mathbb{N}}$ of orthogonal projectors in $\mathcal{H}$ which converges strongly to the identity; to fix ideas, we may think of the following case: \begin{equation} P_n=\sum_{j=1}^n\left\langle e_j|\,\cdot\,\right\rangle e_j, \end{equation} with $(e_j)_{j\in\mathbb{N}}$ being an orthonormal basis for $\mathcal{H}$; I'm also going to assume that $e_j\in\mathrm{dom}\,A$ for all $j$. We can therefore construct the operator $A_n=P_nAP_n$, which acts as a symmetric operator on $P_n\mathcal{H}$ and trivially on its orthogonal complement.
By construction, $A_n$ converges strongly to $A$. Does $A_n$ also converge to $A$ in the strong resolvent sense? If not, are there general further assumptions under which resolvent convergence does (or does not) hold?