Compact resolvent

A closed densely defined operator has compact resolvent when $(A-zI{)}^{-1}$ is compact at one, equivalently every, resolvent point. Its spectrum consists only of isolated eigenvalues of finite multiplicity, with possible accumulation only at infinity.

The imaginary Airy operator $-d^2/d{x}^2+ix$ on $L^2(\mathbb{R})$ has compact resolvent but empty spectrum. Real translations shift it by a purely imaginary scalar, forcing vertical translation invariance of its spectrum.