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In this paper, we study a linearized two-dimensional Euler equation. Thisequation decouples into infinitely many invariant subsystems. Each invariantsubsystem is shown to be a linear Hamiltonian system of infinite dimensions.Another important invariant besides the Hamiltonian for each invariantsubsystem is found, and is utilized to prove an ``unstable disk theorem''through a simple Energy-Casimir argument. The eigenvalues of the linearHamiltonian system are of four types: real pairs ($c,-c$), purely imaginarypairs ($id,-id$), quadruples ($\pm c\pm id$), and zero eigenvalues. Theeigenvalues are computed through continued fractions. The spectral equation foreach invariant subsystem is a Poincar\'{e}-type difference equation, i.e. itcan be represented as the spectral equation of an infinite matrix operator, andthe infinite matrix operator is a sum of a constant-coefficient infinite matrixoperator and a compact infinite matrix operator. We have obtained a completespectral theory

On 2D Euler equations. I. On the energy–Casimir stabilities and the spectra for linearized 2D Euler equations