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In their recent paper, Marcoux, Radjavi and Popov show that for $T\in B(\mathcal{H})$, total discontinuity of $\sigma(T)$ implies the existence of almost invariant subspace for $T$. So a natural question is: what class of operators have totally disconnected spectra?

I know that every nonempty compact subset of $\mathbb{C}$ is the spectrum of a diagonal operator, but such kind of result seems artificial.

I am looking for something more 'natural'. A nice class I can think of is compact operators, their spectra are totally disconnected.

Thanks!

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I couldn't find "totally disconnected" in that paper. Are you sure you referenced the right one? –  Robert Israel Jul 22 '12 at 6:55
    
In particular, what if the spectrum is a single point? That is totally disconnected. –  Robert Israel Jul 22 '12 at 7:03
    
@RobertIsrael Oh, I made a mistake. I am thinking about a slightly different problem. But the theorem I referred to is Lemma 2.7, which states that if $\sigma(T)$ contains infinitely many connected components, then $T$ has an almost invariant half-space. –  Hui Yu Jul 22 '12 at 13:57
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