I have a sequence of operators $A_n$ on a separable Hilbert space (not necessarily with the same domain). These operators are unbounded, self-adjoint, and converge in the strong resolvent sense to some operator $A$. Their resolvents are not necessarily compact.
My question is - what can be said about the spectrum of $A$? I'm fine with adding some extra assumptions on $A_n$ if that helps (but not something as restricting as, say, norm resolvent convergence). Some particular questions I care about:
- When can the spectrum of $A$ be obtained as some "limit" of the spectra of $A_n$? I know some results for the case of norm resolvent convergence and for the case of strong convergence, but none for strong resolvent convergence. It is definitely necessary to add some extra assumptions for this, but I am wondering if there are 'natural' assumptions that can be added which are not too restrictive.
- What can be said about the spectral types of $A$. If $A_n$ all have only absolutely continuous spectrum, what about $A$? Same about existence/nonexistence of eigenvalues and singular continuous spectrum. I have seen in a related post that strong resolvent convergence implies strong convergence of the spectral resolutions, but I could not find a source (and also it does not yet answer this question exactly). My hunch is that here as well you need some additional assumptions.
- What about the Lebesgue measure of the spectrum of $A$ in terms of the spectra of $A_n$? Any interesting results?
Here are some related posts here which I found but did not exactly answer my question (maybe they will help others) - post 1, post 2, post 3, post 4.
Thanks in advance.