Spaces with an abstract convolution of measures

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.     

Self-adjoint extensions of commuting Hermite operators

It is proved that it is possible to commuting self-adjoint operators two formally commuting Hermite operators, one of which is self-adjoint after closure and the other has equal defect numbers. The operators act in a Hilbert space constructed from the tensor product of two Hilbert spaces by completion with respect to a norm defined by a positive definite kernel which satisfies a certain majorizability condition. The result can be applied to a problem of integral representations and extensions of positive definite kernels.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 695–697, May, 1990.     

Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.     

亏指数为可数无穷的对称微分算子的自伴扩张

应用Ｈｉｌｂｅｒｔ空间内对称算子自伴域一种新的描述方法，得到直和空间内亏指数为可数无穷的对称微分算子自伴扩张的完全解析描述。     

On the nature of the spectrum of self-adjoint extensions of operators admitting separation of variables

We consider the operators: \(L_0 = \overline {M_0 \otimes E'' + E' \otimes Q}\) , acting in the tensor product of the infinite-dimensional Hilbert spaces H′ and H″, where the operator M₀ is symmetric in H′ and Q is self-adjoint in H″. We study the problem concerning the existence of self-adjoint extensions, the spectrum of which possesses certain preassigned properties. In particular, we obtain necessary and sufficient conditions under which the operator L₀ admits self-adjoint extensions with a discrete spectrum.     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

Complex symmetric operators and applications

We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.

     

Self-adjoint extensions of symmetric operators

Let ? denote the Hilbert space of analytic functions on the unit disk which are square summable with respect to the usual area measure. In this paper we consider the formal differential exepressons of order two or greater having the form {fx321-1} and {fx321-2} which give rise to symmetric operators in ?. We show that these operators in ? admit self-adjoint extensions in ?.     

Some Remarks on Krein's Extension Theory

The extension problem of semibounded symmetric operators and symmetric operators with a gap is studied in detail. Using a suitable representation (Krein model) for the inverses of those operators a parameterization of their symmetric and self-adjoint extensions is introduced which improves Krein's famous extension theory. In particular, the parameterization clearly shows which self-adjoint extensions in the gap case correspond to Friedrichs and v. Neumann or Krein extensions in the semibounded case. Moreover, special properties of the extensions as the exactness of the gap are characterized in terms of the parameters.     

Singular point perturbation of an odd operator in ℤ2-Graded space

In the paper we study supersymmetric models for point interaction perturbations of operators of Dirac type and their spectral properties. Such models are considered in the class of odd self-adjoint operators in ℤ₂-graded Pontryagin space. We present in detail the previously considered realization method of strongly singular perturbation by means of their embedding into the theory of self-adjoint extensions. We describe odd self-adjoint extensions of odd symmetric operators with deficiency indices (1,1) in ℤ₂-graded Pontryagin space and squares of such extensions using Krein’s formula for the resolvent. The results obtained are refined in application to singular perturbations of odd self-adjoint differential operators. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 924–940, December, 1999.     

Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints

This paper is concerned with self-adjoint extensions for a linear Hamiltonian system with two singular endpoints. The domain of the closure of the corresponding minimal Hamiltonian operator H₀ is described by properties of its elements at the endpoints of the discussed interval, decompositions of the domains of the corresponding left and right maximal Hamiltonian operators are provided, and expressions of the defect indices of H₀ in terms of those of the left and right minimal operators are given. Based on them, characterizations of all the self-adjoint extensions for a Hamiltonian system are obtained in terms of square integrable solutions. As a consequence, the characterizations of all the self-adjoint extensions are given for systems in several special cases.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

On Some Operators Associated to Superoscillations

In this paper we study the convergence of some sequences of operators associated to the Aharonov and Berry’s superoscillating functions. The main tool to define the sequences of operators is the spectral theorem. In particular we discuss the case of sequences of unbounded self-adjoint operators on a Hilbert space. We apply our results to the case where T is the self-adjoint extension of the momentum operator with unbounded spectrum.     

Generalized random linear operators on a Hilbert space

In this paper, we are concerned with generalized random linear operators on a separable Hilbert space. Generalized random linear bounded operators, generalized random linear normal operators and generalized random linear self-adjoint operators are defined and investigated. The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are obtained.     

Ultracompleteness of hyperspaces of compact sets

A nonnegative linear relation S in a Hilbert space ? is assumed to intertwine in a certain sense two bounded everywhere defined operators B and?C. A?related quotient of the range of S is then provided with a natural inner product and the operators B and C induce two operators on the completion space. This construction is used to show the existence of self-adjoint and nonnegative extensions of the linear relations B ^? S and C ^? S, respectively.     

On self-adjointness of the product of two second-order differential operators

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the productL ₂ L ₁ of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators. Supported by the Royal Society and the National Natural Science Foundation of China and the Regional Science Foundation of Inner Mongolia     

A fundamental approach to the generalized eigenvalue problem for self-adjoint operators

The generalized eigenvalue problem for an arbitrary self-adjoint operator is solved in a Gelfand triple consisting of three Hilbert spaces. The proof is based on a measure theoretical version of the Sobolev lemma, and the multiplicity theory for self-adjoint operators. As an application necessary and sufficient conditions are mentioned such that a self-adjoint operator in L₂(R) has (generalized) eigenfunctions which are tempered distributions.     

Two classes of extensions for generalized Schrödinger operators

Two classes of extensions for generalized Schrödinger operators are considered. One is the Markovian self-adjoint extensions and the other is the extensions in Silverstein's sense. We prove that these classes of extensions are identical. As its application, some properties of drift transformations of Brownian motion are derived.     

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace-Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.