Criteria for operators to have infinitely many eigenvalues

Question

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.

For non-normal operators this no longer has to be true. There exist even examples of compact operators without eigenvalues such as weighted shifts and the Volterra operator $Tf(t) = \int_0^t f(s) \ ds$ on $L^2(0,1),$ which when applied to a polynomial basis can also be interpreted as a shift. So a perhaps bold guess would be that we want to avoid some generalized infinite Jordan blocks.

I would therefore like to understand: Do there exist other criteria not assuming normality that imply the existence of infinitely many eigenvalues for compact operators on Hilbert spaces?

Answer 1

The only criterion I know is based on a Theorem in the second book of Dunford and Schwartz, see Theorem X1.6.29 and following. If the resolvent of an Hilbert-Schimdt operator satisfies some decay estimates on some rays dividing the complex plane, then the span of the generalized eigenfuntions is dense. The theorem generalizes to operators having trace-class powers and to closed operators with trace-class powers of the resolvent.