Spectral Properties of Ordinary Differential Operators with Involution

Abstract—Let P and Q be ordinary differential operators of order n and m generated by s boundary conditions (where s = max{n, m}) on a bounded interval [a, b]. We study operators of the form L = JP + Q, where J is an involution operator in the space L₂[a, b]. Three cases are considered, namely, n > m, n < m, and n = m, for which the concepts of regular, almost regular, and normal boundary conditions are defined. Theorems on an unconditional basis property and the completeness of the root functions of the operator L depending on the type of boundary conditions from the chosen classes are announced.