Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks

الأصل · EN

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space H, by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators J: H₁ → H₂ and K: H₂ → H₁ which are compatible in a certain sense. With the appropriate definitions of H₁ and J in terms of A and H, we show that (JJ⋆)⁻¹ is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces ℓ²(G) and Hₑ (the energy space).