On the problem of Hermite interpolation of operators in a Hilbert space

The problem of Hermite operator interpolation with interpolational conditions containing Gateaux higher-order differentials in arbitrary directions is investigated. A necessary and sufficient condition for solvability of this problem in a Hilbert space is established, and the set of all Hermite operator polynomials and its subset of interpolants preserving operator polynomials of the same degree are described. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 38–48.     

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.     

A comparative analysis of two methods of polynomial operator interpolation in a Hilbert space

A comparative analysis of two methods of polynomial operator interpolation is performed. These methods are shown to be equivalent and, in the case of functional spaces, the efficiency of the algorithm for approximation with respect to the number of arithmetic operations is demonstrated. Translated fromObchyslyuval'na ta Prykladna Maternatyka, No. 78, 1994, pp. 49–54.     

Accurate and truncated three-point variational schemes for the Sturm-Liouville problem with boundary conditions of the second or third kind

On the basis of a variational approach for the eigenvalue problem in the case of a second-order linear differential operator with boundary conditions of the second or third kind, accurate and truncated threepoint schemes of high order of accuracy are constructed. Error estimates of the approximate solution are obtained. The results of the numerical experiment are presented. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 134–144.     

On the asymptotic error estimate for a discrete problem of fourth-order accuracy

The leading term of the error of eigenvalues of a discrete analog of the eigenvalue problem for an elliptic operator with variable coefficients is obtained. A method for refining eigenvalues by evaluating a correction with the help of a discrete problem of second-order accuracy is proposed. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 153–160.     

On the hard and soft extensions of a nonnegative operator

In terms of spaces of boundary values, i.e., in a form that, in the case of differential operators, leads immediately to the boundary conditions, we construct the hard and soft extensions of a nonnegative operator in Hilbert space, interpreted as perturbations with a change of the domain of definition of a given positivedefinite operator for which these extensions are assumed known. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 7–9.     

One-dimensional perturbations of singular unitary operators

Under some natural restrictions, we prove that any one-dimensional perturbation of a singular unitary operator on a Hilbert space is unitarily equivalent to a model operator on a space determined (in a certain way) by two functions from the Hardy space H². Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 118–122. Translated by V. V. Kapustin.     

On the operator representation of the Green's functions of the dynamic theory of elasticity

Starting from the operator notation for the dyadic Green's function for elastic displacement of the nonstationary theory of elasticity, we propose a method of factoring the mutually orthogonal components of the Green's function in the form of tensor products of operators acting respectively on the coordinates of the points of observation and the source. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 78–80.     

Grid approximations of the pseudospectral type with an improved condition number for second-order differential operators

Schemes that are free of the disadvantages of both the finite-difference and finite-element methods and retain the advantages of the saturation-free grid methods are proposed and investigated. The asymptotic behavior of their maximal N th eigenvalue is the same as the behavior of the N the eigenvalue of a differential operator, and it is not difficult to apply these discretizations to nonstationary problems. In contrast to polynomial pseudospectral approximations, the schemes of this paper, as well as of E. B. Karpilovskaya, “Convergence of the collocation method,” Sov. Math. Dokl.,4, No. 2, 1070–1073 (1963) and I. P. Gavrilyuk and L. D. Grekov, On Algorithms for the Realization of Grid Schemes without the Accuracy Staturation for Second-Order Ordinary Differential Equations [in Russian], Deposited at UkrNIINTI 16.08.1991, utilize uniform grids. Bibliography: 27 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 1–27.     

Almost isometric operators: Functional model,invariant subspaces,commutant

A new function model for an arbitrary bounded operator on a Hilbert space is constructed. This model generalizes the model of Sz.-Nagy and Foiaş for contractions and seems to be useful for operators close to an isometry (in a sense). All the model spaces are Hilbert spaces, but instead of dilation a generalization of it is used. The model admits a symmetry with respect to the map z→1/z of the complex plane. In terms of the model the question of lifting the commutant is investigated, a relationship between invariant subspaces of a unitary operator is established, and the characteristic function of the model operator is calculated. Some other problems are solved as well. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 95–116. Translated by V. Vasyunin.     

Some operator-differential equations with increasing spectral multiplicity

We prove a theorem on intermediate derivatives and traces for a class of unbounded nonselfadjoint operators in Hilbert space. The results obtained have application in the qualitative theory of operator-differential equations. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 12–16.     

Interpolation of polynomial operators in a Hilbert space

An interpolant with orthonormal nodes is constructed for a polynomial operator of a given degree defined on an abstract Hilbert space. An estimate for the interpolation accuracy is derived in the metric of the space of operator values, while in the case where nodes are elements of a basis, the pointwise convergence of the interpolational operator process in this metric is proved as the number of nodes increases. Bibliography: 7 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 3–9     

Characteristic functions and their factorizations

A new definition of the characteristic function is introduced for contractions on Hilbert spaces. The relationship with other definitions is established. A factorization formula corresponding to an invariant subspace is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 71–78. Translated by V. V. Kapustin.     

Equipped absolutely continuous subspaces and stationary construction of the wave operators in the non-self-adjoint scattering theory

For a pair of non-self-adjoint operators with characteristic operator-functions having boundary values in the strong operator topology almost everywhere on the real axis, (local) wave operators are constructed. The investigation is based on the local stationary approach arising from the self-adjoint scattering theory and a certain interpretation of the absolutely continuous subspace as an equipped Hilbert space. Sufficient conditions for the existence of the wave operators are obtained and some of their properties are described. Bibliography:28 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 144–171.     

Double operator integrals and their estimates in the uniform norm

In this paper, conditions are considered for the existence of the double operator integral ∫∫ ϕ(λ,μ)dE_λTdF_μ, where E_λ, F_μ are the spectral functions of tow self-adjoint operators A, B on a Hilbert space and T is a bounded operator. In principal, the case where A has finite spectrum is studied. Nonlinear estimates of ‖f(A)T-T f(B)‖ in terms of the norm of ‖AT-TB‖ for f∈ Lip 1 are deduced. Also, a formula for the Fréchet derivative is presented. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 148–173. Translated by S. V. Kislyakov.     

On stable operator polynomials

We attempt to generalize the classical theorem of Wall on the stability of an ordinary numerical polynomial to the operator-valued case. It is shown that such a generalization is admissible if the coefficients of the operator polynomial are uniformly positive commuting operators in Hilbert space. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 144–147.     

On the operator formulation of a class of problems of linear fluid dynamics

We propose a new operator formulation of a rather large class of problems of linear fluid dynamics. We give a result on the unique solvability of an evolution problem. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 118–122.     

Estimates for eigenvalues of a biharmonic operator perturbed by the variation of a domain

Estimates for perturbed eigenvalues and eigenfunctions are established with respect to the distance for which a complex smooth boundary hyperplane with homogeneous Dirichlet conditions defined on it is displaced along the outward normal. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 145–152.     

Ultrametric spaces and related Hilbert spaces

We present simple proofs of the possibility of embedding ultrametric spaces in Hilbert spaces. The main part of the paper deals with ultrametric spaces that we call totally infinite spaces. Related Hilbert spaces, automorphisms of totally infinite spaces, and the corresponding linear operators are considered. Transplated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 223–237, August, 1997. Translated by V. E. Nazaikinskii