The Glazman-Krein-Naimark theory for Hermitian subspaces (Q2919625)

Characterizations of self-adjoint extensions are one of the most important problems in the spectral theory of linear Hamiltonian systems. Under certain definiteness conditions, the minimal operator corresponding to a singular continuous Hamiltonian system is a symmetric operator, and its adjoint is the maximal operator in the related Hilbert space. However, for general singular systems, its minimal operator is not necessarily densely defined, and the maximal operator may be multi-valued. So the classical von Neumann self-adjoint extension theory and the Glazman-Krein-Naimark (GKN) theory for symmetric operators are not applicable.NEWLINENEWLINE At the same time, the graph of the minimal operator for a linear Hamiltonian system in a general time scale is a Hermitian subspace in its related product space regardless of whether the definiteness conditions hold or not. \textit{E. A. Coddington} [``Extension theory of formally normal and symmetric subspaces'', Mem. Am. Math. Soc. 134 (1973; Zbl 0265.47023)] extended the von Neumann theory to Hermitian subspaces. The main topic of the present paper is the extension of GKN theory to Hermitian subspaces. One of the main results is a complete characterization of all self-adjoint subspace extensions of a Hermitian subspace with equal positive and negative defect indices in terms of GKN-sets.