Self-adjoint Subspace Extensions Research Articles

Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown that every SSE of the minimal subspace has a pure discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral exactness holds, in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, in the limit point and intermediate cases, spectral inclusive holds.

Essential and continuous spectrum of symmetric difference equations

AbstractEssential and continuous spectrum of symmetric difference equations have been investigated. It has been shown that the deficiency indices and the existence of these components of the spectrum are determined by the growth conditions of the coefficients. In particular, the deficiency indices are superimposition of those clusters determined by the coefficient growth. Finally, we have proved the neccessary and sufficient conditions for the existence of essential spectrum of selfadjoint subspace extensions using subspace theory and asymtotic summation.

Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.

Regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations

This paper is concerned with regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations. It is shown that the kth eigenvalue of any given self-adjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular self-adjoint subspace extensions in the case that each endpoint is either regular or in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, it is shown that isolated eigenvalues in every gap of the essential spectrum of any self-adjoint subspace extension are exactly the limits of eigenvalues of suitably chosen induced regular self-adjoint subspace extensions in the case that at least one endpoint is in the limit point case.

Absolutely continuous spectrum of fourth order difference equations

We have studied the absolutely continuous spectrum of a selfadjoint subspace extension generated by a three‐term fourth order difference equation with bounded coefficients using subspace theory. In particular, we have shown that the absolutely continuous spectrum exists outside a certain bounded interval. In addition, we have computed the spectral multiplicity as well as the location of absolutely continuous spectrum of selfadjoint subspace extension under certain asymptotic conditions.

Self-adjoint extensions for discrete linear Hamiltonian systems

This paper is concerned with self-adjoint extensions for a class of discrete linear Hamiltonian systems. By applying the generalized von Neumann theory and the GKN theory for Hermitian subspaces, a complete characterization of all the self-adjoint subspace extensions for the systems are obtained in terms of boundary conditions via linear independent square summable solutions. As a consequence, characterizations of all the self-adjoint subspace extensions are given in the limit point and limit circle cases. In addition, some sufficient conditions for the corresponding minimal subspace to be an operator or a densely defined operator are given, and consequently, characterizations of all the self-adjoint operator extensions for the system are obtained. In particular, even-order formally self-adjoint vector difference equations are discussed as a special case.

Regular approximations of spectra of singular second-order symmetric linear difference equations

This paper is concerned with regular approximations of spectra of singular second-order symmetric linear difference equations. For any given self-adjoint subspace extension of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of regular self-adjoint subspace extensions generated by the same difference expression on smaller finite intervals.

Spectral properties of singular discrete linear Hamiltonian systems

This paper is concerned with spectral properties of singular discrete linear Hamiltonian systems. It is shown that properties of the essential spectrum of each self-adjoint subspace extension (SSE) of the corresponding minimal subspace are independent of the values of the coefficients of the system on any finite subinterval. The analyticity of the Weyl function is studied by employing the Schwarz reflection principle for the system in the limit point case. Based on the above work, several sufficient conditions are obtained for each SSE to have no essential spectrum points in an interval of the real line in the strong limit point case, and then a sufficient condition for the essential spectrum to be bounded from below (above) and a sufficient condition for the pure discrete spectrum are presented, respectively. As a direct consequence, the related spectral properties of singular higher order symmetric vector difference expressions are given.

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.

Self-adjoint extensions for second-order symmetric linear difference equations

In this paper, self-adjoint extensions for second-order symmetric linear difference equations with real coefficients are studied. By applying the Glazman–Krein–Naimark theory for Hermitian subspaces, both self-adjoint subspace extensions and self-adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions, where the two endpoints may be regular or singular.

Self-adjoint subspace extensions satisfying λ-linear boundary conditions

SynopsisLet S be a symmetric subspace in a Hilbert space ℋ2 with finite equal deficiency indices and let S* be its adjoint subspace in ℋ2. We consider those self-adjoint subspace extensions ℋ of S into some larger Hilbert spaces ℋ2 = (ℋ × ℂm)2 which satisfy H⋂({0} × ℂm)2 = {{0,0}}. These extensions H are characterized in terms of inhomogeneous boundary conditions for S*; they are associated with eigenvalue problems for S* depending on λ-linear boundary conditions, which we also characterize.

Boundary conditions for linear manifolds. II algebraic properties and applications

It is shown in [4] (cf. also [3]) that by coordinatizing linear manifolds, the extension theory of certain closed linear manifolds in Banach spaces reduces to matrix analysis. There adjoint spaces, symmetric spaces and selfadjoint spaces are characterized via matrix equations (see [4, Theorem 1.71, also [3, Proposition 2.3, Theorem 3.11). However, such a characterization is theoretical rather than practical because the matrix equations have still to be solved. This is particularly evident if N (the total number of boundary conditions to start with) is large. The aim of this paper is to improve this situation in the case when N ( 00. This is done in a way which is applicable directly to the problem involving multipoint boundary value problems for ordinary linear differential expressions. It is pointed out in Proposition 1 that when the underlying space is a Hilbert space, the numbers of positive and negative eigenvalues of e (see below) are closely related with the deficiency indices of an original symmetric subspace. In Theorem 3 we give convenient formulas for adjoint subspaces. It appears that such formulas have not appeared in the literature even in the case when the subspaces arise from ordinary differential operators. In Theorem 5 we give canonicalized boundary conditions for selfadjoint subspace extensions. This extends Theorem XII, 4.3 1, of Dunford and Schwartz [2] from densely defined symmetric operators in Hilbert space to symmetric linear manifolds in Banach space (Remark 2.3). In Theorem 7 we characterize the symmetricity of linear manifolds via ranks of certain matrices. The idea of such a characterization seems new in the literature. In Section 3 we give an application to ordinary differential operators.

Adjoint subspaces in Banach spaces, with applications to ordinary differential subspaces

Given two subspaces A0 ⊂ A1 ⊂ W=X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those adjoint pairs A, A* satisfying A0 ⊂ A ⊂ A1, A 1 * ⊂ A∗ ⊂ A 0 * ⊂ W+=Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace extensions of symmetric (formally normal) subspaces appear as special cases when Y=X*. These results are then applied to ordinary differential subspaces in W=Lq(ι) ⊕ Lr(ι), 1≦q, r≦∞, where τ is a real interval, and in W=C(\(\bar \iota \)) ⊕ C(\(\bar \iota \)), where\(\bar \iota \) is a compact interval.

Self-adjoint problems for nondensely defined ordinary differential operators and their eigenfunction expansions

The self-adjoint subspace extensions of a possibly nondensely defined symmetric ordinary differential operator in a Hilbert space are described. The operator part of these extensions involve not only the differential operator but boundary-integral terms, and the side conditions which determine the domains of the extensions also involve boundary-integral terms. Corresponding to each self-adjoint subspace extension in a possibly larger Hilbert space an eigenfunction expansion result is obtained. Analogous results for first-order systems of ordinary differential operators are shown to be valid.

Eigenfunction expansions for nondensely defined differential operators

The operator L, is closed, densely defined and symmetric with domain ID(&). Let 5, be a p-dimensional (p is finite) subspace of !+j and let S,, be the restriction of L, to a(&,) = a(L,) n !&l. Then S, is a closed operator but not densely defined on 53. By identifying graph and operator we may speak of the adjoint subspace S = S,,* of S, in a2 = fi x $j and then we have S, C S. We consider self-adjoint subspace extensions H of S,,: S,, C H = H* C S. Such a self-adjoint subspace H gives rise to a self-adjoint operator H, in the smaller Hilbert space @ 0 H(O), where H(0) is the multivalued part of H. The above theory is given by Coddington [3]. The same author characterizes the self-adjoint subspace extensions of nondensely defined operators in terms of abstract boundary conditions, cf. [4], [Sj. Now, in the boundary conditions determining H appear integral terms and in the expression for H, appear both boundary and integral terms. For such operators Coddington [6], [7] has shown the existence of a matrix valued spectral distribution function p in terms of which he gave eigenfunction expansions converging in the Hilbertspace norm. The principal aim of this paper is to derive an expansion theorem in which point-wise convergence is allowed. Coddington’s norm convergence theorem is a straightforward consequence of our theorem. We will prove our results by adaptation of a method used by Niessen [lo], cf. Rellich [ll]. This adaptation is required to take care of the boundary and integral terms

Self-adjoint subspace extensions of nondensely defined symmetric operators

The self-adjoint subspace extensions of a possibly nondensely defined symmetric operator in a Hilbert space are characterized in terms of “generalized boundary conditions.”