On extensions of the Bessel operator on a finite interval and a half-line

The minimum and maximum operators generated by a Bessel differential expression are studied. The boundary triplets for the maximum operator A _??,max are constructed, and the corresponding Weyl functions are found. All non-negative self-adjoint extensions of the minimum operator A _??,min are described.     

On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval

An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.     

On a spectral problem in the theory of the heat operator

In the theory of the heat operator, we study a spectral problem with squared spectral parameter in the boundary condition. We construct the biorthogonal system and state a nonlocal spectral problem for a complete minimal eigenfunction system.     

Bessel generalized translations and some problems of approximation theory for functions on the half-line

Approximation problems for functions on the half-line [0,+∞) in a weighted L _p -metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of smoothness of arbitrary order which is constructed on the basis of Bessel generalized translation. Equivalence is stated between the modulus of smoothness and the K-functional constructed by the Sobolev space corresponding to the Bessel differential operator. A particular class of entire functions of exponential type is used for approximation. The problems under consideration are studied mostly by means of Fourier-Bessel harmonic analysis.     

A new approach to the convolution operator on a finite interval

The convolution operator on a finite interval defined on a space ofL ₂ functions is studied by relating it to a singular integral operator acting on a space of functions defined on a system of two parallel straight lines in the complex plane . The approach followed in the paper applies both to the case where the Fourier transform of the kernel functions is anL function and to the case where the kernel function is periodic, thus yielding a unified treatment of these two classes of kernel functions. In the non-periodic case it is possible, for a special class of kernel functions, to study the invertibility property of the operator giving an explicit formula for the inverse. An example is presented and generalizations are suggested.     

On the best approximation of the differentiation operator on the half-line

We solve a problem of S. B. Stechkin concerning the best approximation in the metric of C to the operator of k-th order differentiation on certain classes of differentiable functions defined on the half-line, by linear operators whose norms from L₂ into C are bounded. We consider the analogous problem for linear differential operators with constant coefficients. The bibliography contains 10 items.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 573–582, November, 1969.     

Multiparameter spectral theory of a separating operator system

We define the spatial numerical range V[P] of the multiparameter system P(λ), and establish a connection between V[P] and the joint spatial numerical range of a separating operator system.     

Inverse spectral problems for differential systems on a finite interval

Inverse spectral problems are studied for non-selfadjoint systems of ordinary differential equations on a finite interval. We establish properties of the spectral characteristics, and provide a procedure for constructing the solution of the inverse problem of recovering the coefficients of differential systems from the given spectral characteristics.     

New results on the invertibility of the finite interval convolution operator

Conditions for the invertibility and explicit formulas for the inverse of the convolution operator on a finite interval are obtained making use of solutions of corona problems. Using these results, a family of classes of functions is defined for which the study of invertibility can be carried through. An example of one class of this family is presented and a smaller class, for which the calculations are simpler, is more thoroughly studied.Work sponsored by F.C.T. (Portugal) under Project Praxis XXi/2/2.1/MAT/441/94     

On the spectral theory of trees with finite cone type

We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely continuous and to consist of finitely many bands. The main result gives stability of the absolutely continuous spectrum under sufficiently small radially label symmetric perturbations for non-regular trees in this class. In sharp contrast, the absolutely continuous spectrum can be completely destroyed by arbitrary small radially label symmetric perturbations for regular trees in this class.     

Positive solutions of operator equations on half-line

This paper is concerned with an operator equation on half-line. Some theorems for the existence of its positive solutions are obtained by using the Krasnosel'skii-Guo theorem on cone expansion and compression in a special function space.     

On the theory of generalized Abel integral equations on the half-line

The well-known method of reducing a single generalized Abel integral equation to a Cauchy-type integral equation is analyzed in the case of systems.     

Spectral properties of the complex airy operator on the half-line

We prove a theorem on the completeness of the system of root functions of the Schrödinger operator L = ?d ²/dx ² + p(x) on the half-line R₊ with a potential p for which L appears to be maximal sectorial. An application of this theorem to the complex Airy operator L _c = ?d ²/dx ² + cx, c = const, implies the completeness of the system of eigenfunctions of L _c for the case in which |arg c| < 2π/3.We use subtler methods to prove a theorem stating that the system of eigenfunctions of this special operator remains complete under the condition that |arg c| < 5π/6.