Spectral Theory Of Self-adjoint Subspaces

It is well known that under a certain definiteness condition, the minimal operator H0 corresponding to a singular linear continuous Hamiltonian system is a symmetric operators, i.e., a densely defined Hermitian operator, and its ad-joint is equal to the corresponding maximal operator H in the related Hilbert space. If the definiteness condition does not hold, H0 is a non-densely defined operator or multi-valued operator and H is a multi-valued operator. It has been found that for a general singular linear discrete Hamiltonian system and a general difference equation, their minimal operators are non-densely defined in the related Hilbert space and their maximal operators are multi-valued even if the related definiteness condition holds. Therefore, the spectral theory of operators is not applicable in these cases. However, 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 whether its related definiteness condi-tion holds or does not hold. In order to study the spectral theory of linear continuous Hamiltonian systems without definiteness condition, linear discrete Hamiltonian systems, and Hamiltonian systems in a general time scale, it is necessary to first establish spectral theory of subspaces. However, many im-portant issues about spectral theory of subspaces have not been studied or the existing results are not complete..In 1961 [1], R. Arens initiated a study of linear subspace T in X×Y, where X and Y are linear spaces, and, in the case that X=Y is a Hilbert space, he indicated how a self-adjoint subspce could be written as the orthogonal sum of a purely multi-valued part and the graph of a certain self-adjoint operator on a subspace of X. This paper provided a new research subject. E. A. Coddington studied self-adjoint extensions of Hermitian subspaces in the product space X2 in 1973 [3]. He had successfully extended the von Neumann self-adjoint extension theory for symmetric operators to Hermitian subspaces, in which he gave out a sufficient and necessary condition of existence of self-adjoint subspace extension for a Hermitian subspace and a characterization of self-adjoint subspace extensions. In 1976 [6], he study self-adjoint subspace and eigenfunction expansions for ordinary differential subspace, and introduced the concept of spectral family of subspace.Recently, Shi employed Coddington's results to establish the GKN theory for Hermitian subspaces. She gave out some symplectic properties of a Her-mitian subspace and introduced concepts of its defect space [12]. All the self-adjoint subspace extensions of a Hermitian subspace with equal positive and negative defect indices are characterized in terms of GKN-sets. Shi and Sun further employed the GKN theory for subspaces to study self-adjoint exten-sions for second-order symmetric linear difference equations [13]. In this paper, they pointed out that the minimal operator generated by a second-order sym-metric difference equation is a non-densely defined Hermitian operator, and the maximal operator is multi-valued. They introduced minimal, pre-minimal and maximal subspaces for second-order symmetric linear difference equations and studied their properties. In addition, they employ the GKN theory to give out a characterization of all self-adjoint subspace extensions of the mini-mal subspace and a characterization of all self-adjoint operator extensions of the minimal operator. For more results about non-densely defined Hermitian operators or Hermitian subspaces, we refer to [2,4,5,7-11] and some references cited therein.In this paper, we will first employ the existing theory of subspaces to study relationships between a self-adjoint subspace T and its operator part Ts. We will also discuss relationships between operators and their graphs. Thus, we establish relationships between subspaces and operators. Finally, we will employ the various relationships between T and Ts and corresponding theory of operators to study several spectral problems of subspaces.This paper is divided into three chapters.In Chapter 1, we introduce some basic concepts and fundamental theory of subspaces and investigated relationships between the resolvent set, spectrum, point spectrum and boundness from below of subspaces. We give sufficient and necessary conditions of existence of self-adjoint subspace for a Hermitian subspace.In Chapter 2, we study some spectral properties of self-adjoint subspaces. We employ properties of reducing subspaces to study relationships between the resolvent set, spectrum, point spectrum, essential spectrum, and discrete spectrum of self-adjoint subspace T and those sets of its operator part Ts. Be-cause there have been many elegant results about the spectrum of operators, we could employ the relationship between T and Ts and the quite complete theory of operators to study the spectral properties of subspaces. Introduc-ing some special reducing subspaces of self-adjoint subspaces:discontinuous subspace, continuous subspace, singular continuous subspace, absolutely con-tinuous subspace, and singular subspace, we introduce another classification of spectrum of subspaces:continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum. We also establish re-lationships between these reducing subspaces of T and those subspaces of Ts. Finally, we study the spectra of self-adjoint subspace extensions of a closed Hermitian subspaces and show that all self-adjoint subspace extensions of T have the same essential spectrum.In Chapter 3, we study some properties of strong resolvent convergence of self-adjoint subspaces. We introduce concepts of strong resolvent convergence, norm resolvent convergence, spectral exactness, and spectral inclusion. We give necessary and sufficient conditions of strong resolvent convergence, and discuss some properties of strong and norm resolvent convergences. Finally, we give some sufficient conditions of spectral inclusion and spectral exactness.