On certain spectral properties of a quadratic self-adjoint matrix pencil with dominating main diagonals

Sufficient conditions for the “separation of the spectrum” of a quadratic matrix pencil with self-adjoint coefficients are found. A criterion for the representability of a positive operator in a finite-dimensional space in the form of a matrix with dominating main diagonal in each orthonormal basis is obtained. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 381–390, March, 1997. Translated by A. I. Shtern     

Doubly stochastic operators on a finite-dimensional simplex

Generalizing the concept of a quadratic doubly stochastic operator, we introduce the concept of an arbitrary doubly stochastic operator. We give a necessary condition for double stochasticity. Moreover, we prove an ergodic theorem for doubly stochastic operators. Original Russian Text Copyright ? 2009 Shahidi F. A. __________ Tashkent. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 463–468, March–April, 2009.     

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 the spectra of finite-dimensional quadratic Bose operators

A complete study of the spectrum of a finite-dimensional Bose operator is carried out in the paper. The cases in which the spectrum is discrete or continuous are studied. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 835–854, June, 1997. Translated by A. M. Chebotarev     

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

Spectral functions of self-adjoint and symmetric multiplication operators inL 2(X, μ)-spaces

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution inL ²(X, μ)-space and a convolution-type operator inL ²(ℝ ⁿ )-spaces. Symmetric convolution and multiplication operators are considered inL ²(X, μ) andL ²(ℝ ⁿ )-spaces. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 803–810, June, 2000.     

Singularly perturbed problems with double singularity

A class of singularly perturbed initial and boundary value problems for systems of linear differential equations with singularities of various types is studied. The asymptotics of the solutions of these problems is constructed; in contrast to known results, it involves boundary layers of new types that are dependent not only on the spectrum of the limit operator. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 494–501, October, 1997. Translated by N. K. Kulman     

The leading term of the spectral asymptotics for the Kohn-Laplace operator in a bounded domain

We prove the Weyl asymptotic formula for the number of eigenvalues of the Kohn-Laplace operator on a Heisenberg group and write out the leading term of asymptotics. The method of study is based on estimates of the Green function for the Dirichlet problem for the corresponding parabolic operator and makes use of the classical Hardy-Littlewood Tauberian theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 493–505, October, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00443 and No. 96-01-00503.     

Short-range potential and a model of operator extension theory for resonators with semitransparent boundary

On the basis of operator extension theory, a model of the quantum point is constructed as a resonator with semitransparent boundary. It is proved that the model operator is the limit (in the resolvent sense) of short-range Hamiltonians. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 703–711, May, 1999.     

Invertibility and the fredholm property of difference operators

We obtain conditions for the invertibility and the Fredholm property of the difference operator (Dx)(n)=x(n) -U(n)x(n − 1),n ε ℤ, in the Banach space l _p (ℤ, X),p ε [1, ∞], of vector sequences, whereX is a Banach space andU is a bounded operator function. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 816–827, June, 2000.     

Polynomial approximation of functions with given order of thekth generalized modulus of smoothness

The problem of approximation by algebraic polynomials is considered on function classes characterized by the value of thekth generalized modulus of smoothness defined by the Jacobi generalized shift operator. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 425–436, March, 1998.     

On sufficient conditions for the convergence of successive approximations

We obtain sufficient conditions for the convergence of successive approximations to the solution of operator equations. We show how these conditions can be used in the case of nonlinear integral Hammerstein-type equations. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 854–859, June, 1999.     

Regularized traces of a perturbed laplace-beltrami operator on the unit two-dimensional sphere

The paper contains formulas for regularized traces of the Laplace-Beltrami operator on the unit sphere under perturbation determined by a bounded complex-valued potential and a sufficiently complete justification of these formulas. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 702–705, May, 2000.     

Shift operator generated by trigonometric systems

We study the isometric operator generated by a trigonometric system. This operator is used in problems with directional derivatives and in theory of the Fourier series. We prove that this isometric operator is a pure shift and present application of this fact to the inverse problem of magnetoresistense. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 539–548, April, 2000.     

Measurability of inverses of random operators and existence theorems

Let (Ω,B,μ) be ameasure space andX a separable Hubert space. LetT be a random operator from Ω ×X intoX. In this paper we investigate the measurability ofT ^-1. In our main theorems we show that ifT is a separable random operator withT(w) almost sure invertible and monotone and demicontinuous thenT ^-1is also a random operator. As an application of this we give an existence theorem for random Hammerstein operator equation.     

An infinite-dimensional version of the Borsuk-Ulam theorem

We study the solvability of the equation a(x) = f(x) on a sphere in a Banach space, where a is a closed surjective linear operator and f is an odd a-compact map. We also estimate the topological dimension of the solution set and give applications of the corresponding theorem to some problems in differential equations and other fields of mathematics.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 1–5, 2004Original Russian Text Copyright © by B. D. GelmanSupported by the Russian Foundation for Basic Research, Grant No. 02-01-00189.     

An exact estimate of the boundary behavior of functions from hardy-sobolev classes in the critical case

In the critical case αp=n functions from the Hardy-Sobolev spacesH _α ^p (B ⁿ ) have a limit almost everywhere on the boundary along certain regions of exponential contact with the boundary. It is proved in the paper that the maximal operator associated with these regions is bounded as an operator fromH _α ^p (B ⁿ ) toL ^p (ϖB ⁿ ). Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 527–539, October, 1997. Translated by N. K. Kulman     

Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \mathbb{S}^2

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere . For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 434–448Original Russian Text Copyright © 2005 by V. A. Sadovnichii, Z. Yu. Fazullin.This revised version was published online in April 2005 with a corrected issue number.     

On the strong smoothness of weak solutions of an abstract evolution equation i. differentiability

For the evolution equation y' ^′(t)=Ay(t) with a normal operator A in a Hilbert space, conditions on A are found which are necessary and sufficient for all weak solutions of the equation to be strongly differentiable. Certain effects of smoothness improvement of the weak solutions are analyzed. The strong infinite differentiability of weak solutions of the equation with a symmetric operator is proved.     

Characteristic frequencies of bodies with thin spikes III. Frequency splitting

The eigenvalue problem for the Laplace operator with the Neumann boundary conditions in a domain that has a thin spike of finite length is considered for the case in which the limit value is an eigenvalue both for the main body and the spike. The method of matched asymptotic expansions is used to construct total asymptotics of the eigenvalues of the perturbed problem and obtain closed formulas for the leading asymptotic terms. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 494–502, April, 1997. Translated by V. E. Nazaikinskii