# Isolated points of the spectra of self-adjoint operators on Hilbert spaces

Let $$T$$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $$\mathcal{H}$$.

I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $$T$$ to conclude that the spectrum $$\sigma(T)$$ of $$T$$ cannot have any isolated point except possibly $$\lambda=0$$.

I can prove results of this form for some specific families of operators that arose in my research by analyzing the singularities of the corresponding resolvent operators. However, I kind of have the feeling that it should be possible to give (relatively) explicit sufficient conditions on $$T$$ to get the desired conclusion. Is anybody aware of any result of this form?

PS. I keep the question vague about "what sort of condition" on purpose. Please feel free to mention any observation or result that you think might be relevant. If you happen to be skeptical about the possibility to give any usable mind condition to get a result of the described form, I would also really like to hear it.

• I don't think there can be any such result (other than tautological ones) that applies in a completely general setting. We'd really have to know more about what type of operator you are considering. – Christian Remling Oct 10 at 15:52
• Another remark along these lines is that you want something like stability of the discrete spectrum, which is exactly the part of the spectrum that is most unstable under perturbations. – Christian Remling Oct 10 at 15:53
• @ChristianRemling I am mostly interested in self-adjoint operators on trees of bounded degree (or more generally locally finite). Using known absorption principles, one can give criteria for the non-existence of isolated points. I think that it should be possible to obtain interesting criteria to the non-existence of isolated points other than possibly zero, as well. – Maurizio Moreschi Oct 10 at 16:25
• @ChristianRemling Could you maybe expand a little on the connection with stability? – Maurizio Moreschi Oct 10 at 16:36
• What I had in mind was just the fact that the discrete spectrum can change under (small) rank one perturbations (unlike the essential and ac spectra, which have some stability). But that may not be relevant, if your operators are not perturbations of other operators that can be analyzed. – Christian Remling Oct 10 at 16:46