# Characterization of compact operators by their spectra

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.

Thanks!

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