Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.     

On multivariate projection operators

This paper deals with multivariate Fourier series considering triangular type partial sums. Among others we give the exact order of the corresponding operator norm. Moreover, a generalization of the so-called Faber–Marcinkiewicz–Berman theorem has been proved.     

Two-dimensional algebraic-geometric operators with self-consistent potentials

Landau Institute for Theoretical Physics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 1, pp. 26–40, January–March, 1994.     

Nonstandard extension of not-everywhere-defined positive operators

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 4, pp. 719–727, July–August, 1994.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

Dichotomy of the spectrum of non-self-adjoint operators

Voronezh. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 24–30, May–June, 1991.     

Point spectrum of singular nonselfadjoint differential operators

V. I. Lenin Moldavian State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 78–80, July–September, 1991.     

Potentials of zero radius and Carleman operators

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 4, pp. 828–841, July–August, 1995.     

Compact disjointness preserving operators

Scientific, Technical, and Designing Union Lensistemotekhnika. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 2, pp. 55–57, April–June, 1992.     

Condensing operators

This paper is devoted to a survey of the current state of the theory of measures of noncompactness and condensing operators.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 18, pp. 185–250, 1980.     

Asymptotics of spectral projectors of pseudodifferential operators

Mathematics Institute Computing Center, Academy of Sciences of the Tadzhik Soviet Socialist Republic. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 1, pp. 54–56, January–March, 1992.     

Integral operators determined by quasielliptic equations. II

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 1, pp. 41–65, January–February, 1994.     

The generating elements of certain Volterra operators connected with third- and fourth-order differential operators

Sufficient conditions are established forf (x) to be the generating function for the Volterra operator which is inverse to the Cauchy operator:l [y]=y⁽ⁿ⁾+p₂(x)y^(n–2) + ... +pn(x)y, y(0)=y(0)=...=y^(n–1)(0)=0 (n=3, 4), when the coefficients pi(x) are not analytic.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 715–720, June, 1968.     

On resolvents of integral and partially integral operators

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

Evaluation of integral operators of viscoelasticity

A method of calculating the integral operators of viscoelasticity from arbitrary continuous functions is proposed for the case where the relaxation (creep) function is given in tabulated or graphic form. The problem of constructing the creep function from an experimentally known relaxation function is solved.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 6, pp. 1115–1117, November–December, 1973.     

Integral operators determined by quasielliptic equations. I

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 6, pp. 52–67, November–December, 1993.     

Quasilinear operators and Hammerstein's equation

We describe the class of operators in a Hilbert space H, introduced by A. I. Perov, which can be represented in the form Ax = D(x)x, where D(x) is a self-conjugate operator satisfying the inequalities B_–D(x) B₊ (B_– and B₊ are fixed self-conjugate operators). As an application we obtain new theorems on the solvability of Hammerstein's equation.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 453–464, October, 1972.     

An algebra containing the two-sided convolution operators

We present an intrinsically defined algebra of operators containing the right and left invariant Calderón–Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<∞). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón–Zygmund theory.     

Invertibility of linear differential operators in certain functional spaces

Lipetsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 160–165, May–June, 1991.     

Functorial approach to interpolation of operators of weak type

Samara. Voronezh. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 12–23, May–June, 1991.