I'm reading the chapter about unbounded operators in [Reed,Simon,"Methods in modern mathematical physics", vol. 1]
Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$. If $T_k\to T$ in the norm-resolvent sense, then for any $(a,b)\subset \mathbb R$ with $\{a,b\}\cap \sigma(T)=\emptyset$, the spectral projections $P_{(a,b)}(T_k)\to P_{(a,b)}(T)$ converge in norm. This in particular means that the spectrum of the limiting operator cannot suddenly contract.
While working on a specific case, I constructed a sequence of selfadjoint operators satisfying, for some purely imaginary number $\lambda$ $$\|\Pi_FR(\lambda, T_k)\Pi_F-\Pi_FR(\lambda,T)\Pi_F\|\to 0.$$ $\Pi$ is the orthogonal projection and $F$ is a subspace of $H$. What does this convergence say about the spectrum of $T_k,T$? Maybe about $\Pi_F T\Pi_F$? Is there a weaker version of the previous statement?