Double unconditional bases associated with a quadratic characteristic parameter problem

A detailed discussion is given of the quadratic characteristic value problem, $\text{x = λPx + (}\text{1}\text{λ}\text{)Qx}$, where P and Q are compact selfadjoint operators in Hilbert space. Theorems are proven showing that unconditional bases of various types may be constructed from the root vectors of $\text{A(λ) = λP + (}\text{1}\text{λ}\text{)Q}$. These theorems are of interest in the stability theory of linear evolution equations, particularly for application to certain problems in the linearized theory of hydrodynamic stability.