On the basis property of root vectors of a perturbed self-adjoint operator

We study perturbations of a self-adjoint operator T with discrete spectrum such that the number of its points on any unit-length interval of the real axis is uniformly bounded. We prove that if ‖Bϕ _n ‖ ≤ const, where ϕ _n is an orthonormal system of eigenvectors of the operator T, then the system of root vectors of the perturbed operator T + B forms a basis with parentheses. We also prove that the eigenvalue-counting functions of T and T + B satisfy the relation |n(r, T) − n(r, T + B)| ≤ const.     

Perturbation of a self-adjoint operator by a subordinate symmetric operator

The following variant of Rellich's theorem is proved. Let A,B be operators in a Hilbert space, A=A^*, BB^* and D(B)D(A). We assume that (Bu,u)(Au,u), uD(A) for some> –1. Then the operator A + B with domain of definition D(A) is self-adjoint.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 196–198, 1985.     

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.     

An unusual self-adjoint linear partial differential operator

In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region.

This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic' operator.

The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions.

In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty.

This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty.

Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent.

In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.

     

The spectra of self-adjoint extensions of a symmetric operator

In this article we investigate the nature of the spectra of self-adjoint extensions of a symmetric operator with equal (finite or infinite) deficiency indices. Our results are formulated in terms of abstract boundary conditions.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 429–434, March, 1976.     

Invariant symmetric restrictions of a self-adjoint operator. I

We prove necessary and sufficient conditions of the S-invariance of a subset dense in a separable Hilbert space H. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 623–631, May, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238).     

Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

Invariant symmetric restrictions of a self-adjoint operator. II

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space L ₂(ℝⁿ, dx) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace N ⊂ ℝⁿ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 781–791, June, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238 “Operator”).     

Variational principles for real eigenvalues of self-adjoint operator pencils

Double variational principles are established for eigenvalues of a (norm) continuous self-adjoint operator valued functionL defined on a real interval [, [.L() is not required to be definite for any . Applications are made to linear, quadratic and rational functionsL.This author acknowledges support from NSERC of Canada and the I.W. Killam Foundation.This author was supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 12176-MAT.     

Estimation of the negative spectrum of a lower semibounded self-adjoint operator

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 142–145, December, 1991.     

Repeated eigenvectors and the numerical range of self-adjoint quadratic operator polynomials

LetL() be a self-adjoint quadratic operator polynomial on a Hilbert space with numerical rangeW(L). The main concern of this paper is with properties of eigenvalues on W(L). The investigation requires a careful discussion of repeated eigenvectors of more general operator polynomials. It is shown that, in the self-adjoint quadratic case, non-real eigenvalues on W(L) are semisimple and (in a sense to be defined) they are normal. Also, for any eigenvalue at a point on W(L) where an external cone property is satisfied, the partial multiplicities cannot exceed two.