Characterization of compact operators by their spectra

Question

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

2. 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!