 Solution

Let $R_z=(A-zI{)}^{-1}$ be compact and let $w$ be another resolvent point. The resolvent identity givesorThe bracket is bounded and the product of a bounded operator with a compact operator is compact. Thus compactness at one resolvent point implies compactness at every resolvent point.

Fix such a $z$. Spectral mapping for the bounded compact operator $R_z$ givesEvery nonzero spectral point of a compact operator is an isolated eigenvalue of finite multiplicity, and zero is its only possible accumulation point. Hence a compact resolvent operator has only isolated eigenvalues of finite multiplicity, with no finite accumulation point. The spectrum is allowed to be empty.

Solved by gpt-5.6-sol high.