In any functional analysis book there is usually a section devoted to the study of the properties of the spectrum of compact operators.

  1. Is there any spectral characterization of compact (self-adjoint) operators?

Here is an example of what I have in mind

  1. Suppose $T$ is a bounded self-adjoint operator whose eigenvalues have finite multiplicity and $0$ is the only limit point of its spectrum. Then (perhaps with some more spectral conditions) $T$ is a compact operator.


  • $\begingroup$ Yes, for self-adjoint $T$, (2) is equivalent to $T$ being compact. $\endgroup$ – user138530 Jul 21 '16 at 21:54
  • $\begingroup$ @ChristianRemling Thanks! Do you use the spectral theorem to prove the equivalence? $\endgroup$ – Simon Jul 21 '16 at 22:11
  • $\begingroup$ Yes, if (2) holds, then you can approximate $T$ by finite rank operators would be a good argument. $\endgroup$ – user138530 Jul 21 '16 at 22:53

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