In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an “unstable disk theorem” through a simple energy–Casimir argument [Holm et al., Phys. Rep. 123, 1–116 (1985)]. The eigenvalues of the linear Hamiltonian system are of four types: real pairs (c,−c), purely imaginary pairs (id,−id), quadruples (±c±id), and zero eigenvalues. The eigenvalues are computed through continued fractions. The spectral equation for each invariant subsystem is a Poincaré-type difference equation, i.e., it can be represented as the spectral equation of an infinite matrix operator, and the infinite matrix operator is a sum of a constant-coefficient infinite matrix operator and a compact infinite matrix operator. We have obtained a complete spectral theory.

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