A spectral integral representation for decomposable operators

For the operators commuting with a spectral measure, a Bartle type integral representation in the spirit of the one provided by the spectral theorem for normal operators is obtained. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasiclassical spectral asymptotics of one class of integral operators

A class of pseudodifferential operators with a small parameter in the amplitude in considered in . The quasiclassical asymptotics of the spectral characteristics is investigated. The results are applicable to statistical estimation theory.     

On equiconvergence of spectral expansions of self-adjoint integral operators

The paper gives simple sufficient conditions on the kernel of the self-adjoint integral operator considered ensuring the equiconvergence of expansions in eigenfunctions and associated functions with the ordinary trigonometric Fourier series expansions. The presented arguments correct inaccuracies in [1]. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Spectral decomposition of self-adjoint cyclically compact operators and partial integral equations

We prove a spectral decomposition theorem for self-adjoint cyclically compact operators on Hilbert–Kaplansky module over a ring of bounded measurable functions. We apply this result to partial integral equations on the space with mixed norm of measurable functions. We give a condition of solvability of partial integral equations with self-adjoint kernel.     

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K∗ in is less than , whenever is a bounded convex domain and 1<p?2.     

Polysingular integral operators

Summary We study a class of linear singular integral operators of the Cauchy type on the n-torus; an application is given to a boundary value problem for functions of several complex variables.     

Strong integral operators

It is shown that a strong integral (strong B-integral) operator in L₂ is a Hilbert-Schmidt (nuclear) operator.Translated from Matematicheskie Zametki, Vol. 16, No. 6, pp. 907–912, December, 1974.     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

On resolvents of integral and partially integral operators

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 2, pp. 88–91, March–April, 1993.     

On integral convolution operators

We study integral convolutions defined on functions ofn variables in symmetric spaces and obtain new additive estimates for the mean value of the nonincreasing permutation(K ^* f) ^** (t) of the absolute value of the integral convolution on the interval [0,t] for anyt>0. As an example of application of the estimates obtained, we prove the boundedness of the integral convolution operator acting from the intersection of symmetric spaces into the Marcinkiewicz space. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 551–555, October, 1999.