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I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.

I would like to ask: Are there similar characterizations for singular continuous and absolutely continuous spectrum as well?

In fact, I believe that there should be rather characterizations of sets which cannot support s.c. and a.c. spectrum?- Is that true?

Is it possible to have for example a.c. spectrum on an isolated set of measure zero? Or are there further restrictions on this set so that such a set can no longer support this type of spectrum?

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  • $\begingroup$ No, of course you cannot have ac spectrum of Lebesgue measure zero. The possible ac spectra are exactly the essentially closed (and closed) sets $S$, that is, $|U\cap S|>0$ for every open set $U$ that intersects $S$ at all. $\endgroup$
    – Christian Remling
    Commented Jun 8, 2017 at 16:57
  • $\begingroup$ However, this questions seems too elementary for this site; it might be suited for MSE. $\endgroup$
    – Christian Remling
    Commented Jun 8, 2017 at 16:57
  • $\begingroup$ Finally, exactly the perfect sets (closed, no isolated points) are the possible sc spectra. $\endgroup$
    – Christian Remling
    Commented Jun 8, 2017 at 18:26

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