Asymptotic estimates of wave functions and the existence of wave operators

The existence of wave operators is proved for the case, where the unperturbed operator is the operator of multiplication by a smooth function in momentum space and the perturbation is an arbitrary operator satisfying a fall off condition near infinity or a weighted L_p-estimate in configuration space. Under somewhat more restrictive conditions the invariance principle is also proved.     

Continuous bijective mappings in topological and Banach manifolds

Continuity and monotonicity of the inverse operator is investigated for the case when the original operator is a bijective mapping acting in topological manifolds or in Banach spaces. For finite-dimensional topological manifolds, the inverse operator is always continuous; for Banach spaces, continuity of the inverse operator is guaranteed only if the space is finite-dimensional. For an arbitrary differentiable bijective mapping acting in the Euclidean space, the Jacobian preserves its sign.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 117–124, 1985;     

Application of the internal time operators for the Renyi map

Recently, the internal time operator for the Renyi map has been constructed (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals). It corresponds to a phase space given by the interval [0,1] and to the invariant Lebesgue measure. In this paper, following the idea of (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals), we construct the time operator for a dynamical system with an arbitrary invariant measure μ and an arbitrary phase space X=[a,b] with a and b finite or infinite. We illustrate also the action of such an operator on a fixed initial state.     

On generalized resolvents and spectral functions of differential operators of even order in a space of vector-valued functions

We consider a self-adjoint differential expression of even order in a space of vector-valued functions. We prove that an arbitrary generalized resolvent of the minimal operator L₀ induced by this differential expression is an integral operator and we derive a formula for all of the spectral functions (orthogonal and nonorthogonal) of the operator L₀.     

Polynomial interpolation of operators

Necessary and sufficient conditions for the solvability of the polynomial operator interpolation problem in an arbitrary vector space are obtained (for the existence of a Hermite-type operator polynomial, conditions are obtained in a Hilbert space). Interpolational operator formulas describing the whole set of interpolants in these spaces as well as a subset of those polynomials preserving operator polynomials of the corresponding degree are constructed. In the metric of a measure space of operators, an accuracy estimate is obtained and a theorem on the convergence of interpolational operator processes is proved for polynomial operators. Applications of the operator interpolation to the solution of some problems are described. Bibliography: 134 titles. This paper is a continuation of the work published inObchyslyuval'na ta Prykladna Maternatyka, No. 78 (1994). The numeration of chapters, assertions, and formulas is continued. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp 10–116.     

The Bergman kernel and the green function

Definition of the Bergman space for an arbitrary operator is given. Sufficient conditions for the existence of the Bergman kernel for this space are obtained. For an elliptic operator, the Bergman kernel is represented via the Green function. Bibliography: 12 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 145–166. Translated by S. Yu. Pilyugin.     

Extensions and generalized resolvents of closed dissipative operators

Let B be a closed dissipative operator in a Hilbert space ? with an arbitrary domain of definition. We investigate briefly the problem of describing all the closed (and, in particular, the closed maximal) dissipative extensions B of the operator B. Following this we introduce the concept of a generalized resolvent of a closed dissipative operator with an arbitrary domain of definition, and we study the fundamental properties of generalized resolvents.     

Nonzero fixed points of power-bounded linear operators

This paper provides a variety of sufficient conditions for the existence of a nonzero fixed point of a power-bounded linear operator defined on a real Banach space. In the case of power-bounded positive operators on a Banach lattice, among the conditions we provide are not being strongly stable along with commuting with a compact operator or being quasicompact. These results apply directly to Markov operators. In the case of an arbitrary power-bounded operator on a Hilbert space, being uniformly asymptotically regular and not strongly stable guarantees the existence of a nonzero fixed point.

     

One-loop divergences in theories with an arbitrary nonminimal operator in curved space

We construct an explicit formula for the divergent part of the one-loop effective action for an arbitrary nonminimal operator in curved four-dimensional space. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 351–371, March, 1997.     

The linear factorization of polynomial operator pencils

Sufficient conditions are found for the linear factorization of polynomial operator pencils of arbitrary order in a Banach space. This factorization is generated by the solution of an appropriate operator equation.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 551–559, April, 1973.The author wishes to express his thanks to A. G. Kostyuchenko for the formulation of the problem.     

Generalized fourier transform and its applications

We consider the generalized Fourier transform treated as an operator on the dual of an arbitrary locally convex space. We give a definition of this operator and establish its basic properties. Special attention is paid to cases in which the range of the generalized Fourier transform coincides with a weighted space of entire functions. The results are applied to finding the orders and types of operators in various spaces.     

The use of anti-A priori estimates in the theory of bases in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a one-dimensional Schrödinger operator defined on a finite interval. We show that anti-a priori estimates without a positive power on the right-hand side on the inequality are necessary and sufficient conditions for the unconditional basis property in the space of square integrable functions for an arbitrary choice of associated functions. We suggest a new definition of associated functions of linear operators in a Hilbert space.     

Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces

The following problem arises in thermoacoustic tomography and has intimate connection with PDEs and integral geometry. Reconstruct a function f supported in an n-dimensional ball B given the spherical means of f over all geodesic spheres centered on the boundary of B. We propose a new approach to this problem, which yields explicit reconstruction formulas in arbitrary constant curvature space, including euclidean space ? ⁿ , the n-dimensional sphere, and hyperbolic space. The main idea is analytic continuation of the corresponding operator families. The results are applied to inverse problems for a large class of Euler-Poisson-Darboux equations in constant curvature spaces of arbitrary dimension.     

Jackson inequality in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of entire functions of many variables with mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

Jackson inequality in L2($$ \mathbb{T}^N $$) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of periodic functions of many variables with a mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

Jackson inequality in L2( \mathbbTN \mathbb{T}^N ) with generalized modulus of continuity

The sharp Jackson inequality is proved for the space of periodic functions of many variables with a mean-square norm for an arbitrary modulus of continuity generated by a difference operator with constant coefficients.     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

Approximations by polynomial and trigonometric splines of third order preserving some properties of approximated functions

In this paper with the help of parabolic splines we construct a linear method of approximate recovery of functions by their values on an arbitrary grid. In the method, a spline inherits the properties of monotonicity and convexity from the approximated function, and is sufficiently smooth. In addition, the constructed linear operator as an operator acting from the space of continuous functions to the same space has the norm equal to one. We also obtain similar results for trigonometric splines of third order.