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.”     

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.     

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.     

Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.     

微分算子的对称扩张及Friedrichs扩张的辛几何刻画

本文在加权Hilbert空间L2(I,r(x))(I=(a,6),-∞≤a 0)中,利用辛几何,刻画了n阶对称微分算式的最小算子的对称扩张(含自伴扩张)及 Friedrichs扩张,分别获得了其扩张为对称扩张、Friedrichs扩张的充分必要条件．     

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.     

Two-Interval Even Order Differential Operators in Direct Sum Spaces

We characterize the domains of all self-adjoint extensions of the two-interval minimal operator which associated with two general even order linear ordinary differential expressions in terms of real-parameter solutions of the two differential equations. This is for endpoints which are regular or singular and for arbitrary deficiency index.     

Triplet extensions I: Semibounded operators in the scale of Hilbert spaces

The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions can be described by certain boundary conditions, and a natural counterpart of Krein’s resolvent formula is obtained.     

Laplace operator with δ-like potentials

We study the Laplace operator in a punctured domain in a Hilbert space. We obtain an analog of the Green formula and a class of self-adjoint extensions of the Laplacian. We also investigate a certain class of well-posed problems.     

Possibly non-unital operator system structures on a possibly non-unital function system

In this paper, we first give the definition of possibly non-unital function system, which is a characterization of the self-adjoint subspace of the space of continuous functions on a compact Hausdorff space with the induced order and norm structure. Similar to operator system case, we define the unitalization of a possibly non-unital function system. Then we construct two possibly non-unital operator system structures on a given possibly non-unital function system, which are the analogues of minimal and maximal operator spaces on a normed space. These two structures have many interesting relations with the minimal and maximal operator system structures on a given function system.     

On the nonnegative self-adjoint solutions of the operator Riccati equation for infinite dimensional systems

The nonnegative self-adjoint solutions of the operator Riccati equation (ORE) are studied for stabilizable semigroup Hilbert state space systems with bounded sensing and control. Basic properties of the maximal solution of the ORE are investigated: stability of the corresponding closed loop system, structure of the kernel, Hilbert-Schmidt property. Similar properties are obtained for the nonnegative self-adjoint solutions of the ORE. The analysis leads to a complete classification of all nonnegative self-adjoint solutions, which is based on a bijection between these solutions and finite dimensional semigroup invariant subspaces contained in the antistable unobservable subspace.     

On Kre?n's formula

Kre?n's formula provides a parametrization of the generalized resolvents and Štraus extensions of a closed symmetric operator with equal possibly infinite defect numbers in a Hilbert space in terms of Nevanlinna families in a parameter space. The aim of this note is to give a simple complete analytical proof of Kre?n's formula.     

一类两个边界带特征参数的不连续四阶微分算子的自共轭性

本文研究了一类具有特殊转移条件且两个边界条件中带有特征参数的四阶微分算子的自共轭性问题.建立了一个与其相关的新的空间H,将上述问题的研究转化为对此空间中一个线性算子A的研究.     

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.     

自共轭算子束的谱及其应用

首先研究了自共轭算子束L—λV的谱曲线,其中L和V是Hilbert空间H内的自共轭算子.其次研究了谱问题Ly=λVy的特征值.最后,将所得的结论应用到正则和奇异的常微分算子的不定谱问题中.     

On the hard and soft extensions of a nonnegative operator

In terms of spaces of boundary values, i.e., in a form that, in the case of differential operators, leads immediately to the boundary conditions, we construct the hard and soft extensions of a nonnegative operator in Hilbert space, interpreted as perturbations with a change of the domain of definition of a given positivedefinite operator for which these extensions are assumed known. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 7–9.     

Multiplicity of the spectrum of a self-adjoint ordinary differential operator of odd order

A bound is obtained for the multiplicity of the spectrum of the self-adjoint operator generated by a singular ordinary differential operator? of odd order in the Hubert space ?² in terms of solutions of the differential equation?[y]=λy.     

The comparison theorem for Hilbert spaces of entire functions

A generalization of the Sturm comparison theorem is obtained for formally self-adjoint ordinary differential operators of finite order given in canonical form. The result is stated within the vector theory of Hilbert spaces of entire functions when the coefficient space is a finite-dimensional vector space.