Spectral properties of self-adjoint subspaces

This paper focuses on spectral properties of self-adjoint subspaces in the product space of a Hilbert space. Several classifications of the spectrum of a self-adjoint subspace are introduced, including discrete spectrum, essential spectrum, continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum. Their relationships between the subspace and its operator part are established, respectively. It is shown that all self-adjoint subspace extensions of a closed Hermitian subspace have the same essential spectrum. As a simple application, it is shown that every self-adjoint subspace extension of the corresponding minimal subspace to a singular second-order symmetric linear difference equation has a bounded spectrum under some conditions. This shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators.