A Generalization of Hankel Operators

We introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S.     

Energy dependence of the scattering operator

We prove that the scattering operator S(E) depends continuously on the energy E for a certain class of Schrodinger operators, by an abstract method using trace conditions and the dilation group. We also obtain pointwise bounds on S(E) −1₁as E → ∞, and even as E → 0 in the case of repulsive potentials.     

Eigenvalues of certain differential operators related to automorphic forms

G.Shimura introduced a certain kind of differential operators when he studied the arithmetic properties of Eisenstein series of several variables. The invariable differential operators of the classical domain of types I and II are included in it as its special cases. In this paper the eigenvalues of this kind of differential operators with respect to the eigenfunction (deti( ^tZ−Z))² are calculated. Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

Pseudodifferential Operators on Periodic Graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group \mathbb Zⁿ{{\mathbb {Z}}^n} . The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OPS ⁰ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.     

Generalization of certain subclasses of multivalent functions with negative coefficients defined by using a differential operator

Making use of a differential operator, we introduce and study a certain class SC_n(j,p,q,α,λ) of p-valently analytic functions with negative coefficients. In this paper, we obtain numerous sharp results including (for example ) coefficient estimates, distortion theorem, radii of close-to convexity, starlikeness and convexity and modified Hadamard products of functions belonging to the class SC_n(j,p,q,α,λ). Finally, several applications investigate an integral operator, and certain fractional calculus operators are also considered.     

Poisson structures associated with algebras of differential operators

For differential operators forming an algebra of a certain class that includes algebras of higher derivatives, a Poisson structure is introduced and the first term of the Hochschild spectral sequence is calculated.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 256–271, August, 1995.     

On the uniform modulus of continuity of certain discrete approximation operators

For a certain class of discrete approximation operators B_n^f defined on an interval I and including, e.g., the Bernstein polynomials, we prove that for all f ε C(I), the ordinary moduli of continuity of B_n^f and f satisfy ω(B_n^f; h) cω(f; h), N = 1,2,…, 0 < h < ∞, with a universal constant c > 0. A similar result is shown to hold for a different modulus of continuity which is suitable for functions of polynomial growth on unbounded intervals. Some special operators are discussed in this connection.     

On the Fredholm radius of integral operators of potential theory

The Fredholm radius of the integral operators of the theory of n-dimensional harmonic potentials on surfaces with edges, acting in certain weighted Hölder type spaces, is found. Results for integral operators are derived from theorems on operators in certain auxiliary contact problems.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 109–133, 1990.     

A stationary approach to the existence and completeness of long-range wave operators

The modified wave operators intertwining the Schrödinger operators – and – + V(x), where V(x) is a real long-range potential, are shown to exist and to be complete. The method employed is entirely stationary (time-independent) : One constructs complete stationary wave operators, utilizing a spectral representation (eigenfunction expansion) theory, and then shows that the time-dependent modified wave operators exist and are equal to the stationary ones already constructed.     

Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

We consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H^1+n/p_p( ⁿ) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.     

On the conditional extremum of functionals with discontinuous gradients

We establish the existence condition for points of conditional extremum for functionals in an arbitrary real Banach space, having discontinuous Gâteaux gradients. We point out an application to the eigenfunction problem for nonlinear elliptic operators.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 581–588, October, 1976.     

Ar approximation theorem for semigroups of growth order α and an extension of the Trotter-Lie formula

A semigroup {T(t); t > 0} of linear operators is called of growth order α 0 if its norm behaves like t^−α as t → 0+, essentially. A discrete approximation theorem for convergence of a certain sequence of powers of linear operators towards such a semigroup is proved. Due to the low degree of stability required by this theorem, an extension of the Trotter-Lie product formula can be derived, i.e., a representation of the semigroup generated by , where A and B are the infinitesimal generators of semigroups of growth orders α and β respectively.     

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

The aim of this paper is the study of a new sequence of positive linear approximation operators M_n, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.     

Algebraic integrability for the Schrödinger equation and finite reflection groups

Algebraic integrability of ann-dimensional Schrödinger equation means that it has more thann independent quantum integrals. Forn=1, the problem of describing such equations arose in the theory of finite-gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrödinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero—Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero—Sutherland problem for a special value of the coupling constant.In memory of M. K. PolivanovMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 253–275, February, 1993.     

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.     

A theorem on the extremum of a nonnegative bounded function

Using perturbation formulas for eigenvalues of linear operators, an equation is obtained for the extremal values of certain nonnegative bounded functions. Formulas are obtained for the derivative of regularizing maximal eigenvalues of matrices which depend on a parameter.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 3–7, 1989.     

Hecke operators of the symplectic group of genus two over a real field

We investigate the arithmetic character of the Fourier coefficients of a Hilbert-Siegel modular form which is an eigenfunction of all Hecke operators. We prove the analytic continuability and a functional equation for the corresponding zeta function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 48–58, 1980.     

Stability of Nonlocal Finite-Difference Problems

The article presents a brief review of the stability theory of finite-difference schemes for time-dependent problems of mathematical physics in which the spatial differential operator is constrained by boundary conditions joined on different pieces of the boundary. It is shown for the particular case of the heat-conduction equation that nonlocal boundary conditions may generate both simple finite-difference operators and finite-difference operators with an incomplete eigenfunction system. In both cases a rule is suggested for constructing the energy norm that ensures stability of the finite-difference scheme.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 72–83, 2004.     

A representation of strong Gaussian random operators

An integral representation of Gaussian random operators acting on a real separable Hilbert space is considered. In terms of this representation, boundedness conditions for Gaussian random operators are given. A special class of Gaussian random operators is studied.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 19–23, 1988.     

Spectral Components of Operators with Spectrum on a Curve

Trace class perturbations of normal operators with spectrum on a curve and spectral components of such operators are studied. We establish duality relations for the spectral components of an operator and its adjoint. The generalized Sz.-Nagy–Foia–Naboko functional model introduced in the paper is a basic tool for this theorem. The results have applications in nonself-adjoint scattering theory and to extreme factorizations of J-contraction-valued functions (J-inner-outer and A-regular-singular factorizations).