On a singular differential operator

Summary Study of the properties of a singular differential operator in one independent variable. Applications to ordinary and partial differential equations. To Giovanni Sansone on his 70^th birth day. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. 49 (638)-228.     

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

On discrete stochastic bilinear systems stability

Mean square stability conditions for discrete-time bilinear systems operating in a stochastic environment are given in this paper. Only independence and wide sense stationarity are required for the second order disturbance sequences involved, thus dismissing ergodicity and zero-mean assumptions. Stochastic stability conditions are derived by using a deterministic stability result for a class of separable nonlinear dynamical systems evolving in a Banach space.     

On the definition of singular bilinear forms and singular linear operators

We analyze various definitions of the concepts of a singular operator and a singular bilinear form and propose the most suitable ones. We also study the simplest properties of these objects.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1084–1089, August, 1993.     

On the scaling limit of a singular integral operator

The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems. Research supported in part by National Science Foundation grants DMS-02-45371 and DMS-04-05519.     

Order bounded orthosymmetric bilinear operator

It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator b: E×E → F where E and F are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost f-algebras.     

On the discrete spectrum of a perturbed Wiener-Hopf integral operator

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 102–113, January, 1992.     

On boundedness of discrete multilinear singular integral operators

Let m(ξ,η) be a measurable locally bounded function defined in R². Let 1?p₁,q₁,p₂,q₂<∞ such that pi=1 implies qi=∞. Let also 0<p₃,q₃<∞ and 1/p=1/p₁+1/p₂−1/p₃. We prove the following transference result: the operator

     

Additive regularization of singular bilinear forms

Conditions on the additive component of a singular bilinear form are found under which the sum is a regular form. As an application the question of regularization of -shaped forms is considered. The case of tensor products of spaces and forms is considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1199–1204, September, 1990.     

On the basis property of eigenfunctions of a singular second-order differential operator

We consider the first boundary value problem for a singular differential operator of second order on an interval with transmission conditions at an interior point of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L ²(0, 1) and forms a Riesz basis in that space.     

On the discrete spectrum of a perturbed periodic Schrödinger operator

A perturbed periodic Schrödinger operator is considered. Conditions on the perturbation are obtained, under which the discrete spectrum in the gap is finite or, on the contrary, is infinite. In the latter case there are given formulas for the asymptotic behavior of the eigenvalues with respect to the index and conditions under which these formulas hold. The results are formulated in terms of a model problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 157–162, 1991.In conclusion the author expresses his gratitude to his scientific adviser M. Sh. Birman.     

On inverse problem for singular Sturm-Liouville operator from two spectra

We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λ _n } and {λ _n } uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the difference . Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 132–138, January, 2006.     

Weighted Morrey spaces and a singular integral operator

In this paper, we shall introduce a weighted Morrey space and study the several properties of classical operatorsin harmonic analysis on this space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

离散算子的本征投影与本征幂零

本文给出例子，说明离散算子本征投影一致有界推不出其本征幂零一致，有界幂零一致有界也推不出本征幂零的幂一致有界。     

The Bergman projection as a singular integral operator

We show that the Bergman projection operator, associated to one of three classes of domains (all smoothly bounded)-a finite type domain ?²; a decoupled, finite type domain in ?ⁿ; or a convex, finite type domain in wfⁿ-may be viewed as a generalized Calderón-Zygmund operator. As an application of this observation, we show that the Bergman projector on any of these domains preserves the Lebesgue classesL ^p , 1 <p < ∞.     

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.