  It has long been known that the differential operator $D$ represents a
typical examples of unbounded operators in many Banach spaces including the
classical Fock spaces, the Fock--Sobolev spaces, and the generalized Fock
spaces where the weight decays faster than the Gaussian weight. In this note we
identify Fock type spaces where the operator admits some basic spectral
structures including compactness and membership in the Schatten $\mathcal{S}_p$
classes. We also showed that its nontrivial spectrum while acting on such
spaces is precisely the closed unit disk $\D$ in the complex plane.
