Let T be a compact operator on an infinite-dimensional Banach space $E$. Let $\sigma(T)$ denote its spectrum. Then $0 \in \sigma(T)$ and every nonzero point $\lambda$ of $\sigma(T)$ is an isolated point.
What is the relation between this result and that concerning Normal compact operators on Hilbert spaces $H$, and does the latter follow from the former (given the extra requirements) in some natural way?
The latter statement complements the first by saying that in addition the eigenspaces corresponding to $\lambda$ are finite dimensional and the compact normal operator can be written as an infinite sum of eigenvalues times mutually orthogonal projections on the eigenspaces, and there is an orthonormal basis of eigenvectors for $H$.