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.     

The index of the pseudo-Riemannian Dirac operator as a transversally elliptic operator

Let (M,r) be a closed, space- and time-orientable, pseudo-Riemannian spin manifold and-let G be a compact group of orientation-preserving isometries on (M,r). If there are no isotropic directions transversal to the orbits of G, then the Dirac operator on (M,r) is transversally elliptic. In this paper we calculate its index.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

On equivariant extensions of a differential operator by the example of the Laplace operator in a circle

We propose a method for investigation of both correctness of the equivariant problem and the spectrum of the corresponding operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 158–169, February, 1999.     

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.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

On a transformation operator

We prove the existence of a transformation operator that takes the solution of the equationy″=λ²ⁿ y to the solution of the equation

Decomposition of a positive operator

Dedicated to Yuri Grigor'evich Reshetnyak on his sixtieth birthday.