Spectral properties of self-adjoint subspaces

Abstract

AMS classification: 47A06 47A10 47B15 47B25 39A70 Keywords: Spectrum Hermitian subspace Self-adjoint subspace Singular symmetric linear difference equationThis 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.

Perturbation theory for selfadjoint relations

“…Remark 5. Under the conditions of the previous result, if η − (S) = n < ∞, then, for any ∆ ⊂ρ(S), the spectrum of A in ∆ is discrete and its multiplicity is at most n. This is so because every closed bounded subinterval of ∆ can be covered by a finite number of lacunae ofρ(S) (see [14,Cor. 5.2]).…”

“…We begin this section by stating the following characterization of selfadjoint relations which in its operator version is well known. Another characterization can be found in [14,Thm. 2.5].…”

A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems

“…To proceed it, we use the theory of linear relations and briefly introduce some related results. Let X be a Hilbert space with inner product ·, · , and T be a self-adjoint linear relation in [3]) and σ(T ) = σ(T s ) (see Theorem 2.1 in [17]). We define the product of two linear relations as…”

Absolutely continuous spectrum of fourth order difference equations

“…Assume that H is selfadjoint subspace extension of H 0 and is selfadjoint operator defined on H , then there is a close relationship between the spectral properties of H and those of as stated in Theorem below, see for more details. Theorem If H is a selfadjoint subspace extension of H 0 on and is the selfadjoint operator defined on H , then …”

“…Assume that H is selfadjoint subspace extension of H 0 and H s is selfadjoint operator defined on H , then there is a close relationship between the spectral properties of H and those of H s as stated in Theorem 4.5 below, see [23] for more details. where z ∈ K (z), z = z 0 + iη, is a spectral parameter and defined on 2 w (N).…”