A characterization of positive self-adjoint extensions and its application to ordinary differential operators

Abstract

A new characterization of the positive self-adjoint extensions of symmetric operators, T 0 T_0 , is presented, which is based on the Friedrichs extension of T 0 , T_0, a direct sum decomposition of domain of the adjoint T 0 ∗ T_0^{*} and the boundary mapping of T 0 ∗ T_0^{*} . In applying this result to ordinary differential equations, we characterize all positive self-adjoint extensions of symmetric regular differential operators of order 2 n 2n in terms of boundary conditions.