Bernstein widths of embeddings of Lebesgue spaces

Let (K, μ) be a measurable space with μ(K)=1. Let Ip,q: L^p (K, μ)→L^q (K, μ) be the embedding operator. The Bernstein widths of I_{p, q} are considered. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 166–169. Translated by S. V. Kislyakov.     

Algebraic complexity of the computation of a pair of bilinear forms

The problem mentioned in the titled reduces to the estimation of the rank of a collection of matrices. The rank of a collection of matrices A₁,...,A_e, denoted rk(A₁,...,A_e), is the least number of such one-dimensional matrices that their linear combinations will represent each matrix of the given collection. For an operator A on ⁿ there exists a space V and a diagonal operator B such that; we denote the minimal dimension of such spaces V by d(V)Theorem. For any matrix A we have the equality rk (E,A.)=n+d(A), where E is the identity matrix.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 159–163, 1974.     

A description of Kac-systems of multiplicative unitary operators

Let V be a multiplicative unitary operator on a separable Hilbert spaceH, then there are two subalgebras ofB( H) denoted byA( V) and ?( V), respectively, which correspond to V. If V satisfiesV ² =I, then we will obtain the necessary and sufficient condition of Baaj and Skandalis’ main theorem, i.e.V has a Kac-system if and only if the linear closed space of the product of the above two algebras is the compact operator space; with this condition the above algebras are also quantum groups.     

Generalized resolvents of closed dissipative operators in a Hilbert space with aF metric

Generalized resolvents of a closedF-dissipative operator in a Hilbert space with aF metric are defined and studied; in particular, a formula of generalized resolvents of such an operator is obtained.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 261–272, August, 1976.In conclusion the author expresses his gratitude to A. V. Shtraus for his scientific guidance.     

Asymptotic resolvent and theorems on generation of semigroups of operators in locally convex spaces

The concept of the asymptotic resolvent of an operator is introduced and used to establish theorems on the generation of semigroups of distributions and operator semigroups of class (ℒ _loc ^p , ℬ) in a locally convex space. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 433–442, September, 1977.     

Weighted approximation by triginometric sums and the Carleman-Golusin-Krylov formula

An explicit formula is given yielding linear combinations of harmonics with negative frequencies approximating a prescribed harmonic with a positive frequency in a weighted space L²(h). The formula is based on the Carleman-Krylov-Goluzin interpolation formula. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 5–14. Translated by V. P. Khavin.     

Analysis of differential operators associated with boundary value problems

We consider boundary value problems in a two-component domain of the Euclidean space R ⁿ obtained by eliminating from R ⁿ the boundary G. Traces on both sides of G are defined without limit passages. In a Hilbert trace space, we introduce orthogonal projections, analogs of the Calderon projections, which are used for constructing operators whose continuous invertibility implies the solvability of the corresponding boundary value problems. For the resolvents we obtain representations similar to the Krein formula. For a symmetric differential operator we show that the constructed resolvents of boundary value problems correspond to closed (not necessarily self-adjoint) extensions of this operator in the sense of von Neumann. Bibliography: 9 titles. Dedicated to N. N. Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 7–48.     

A matrix modification of formulas for the Hermite interpolation of operators in a Hilbert space

Necessary and sufficient conditions for the existence of an Hermite operator polynomial in a Hilbert space are obtained, and Hermite interpolating operator formulas containing vectors and matrices of lower (in comparison with E. G. Kashpur, V. L. Makarov, and V. V. Khlobystov, “On the problem of Hermite interpolation of operators in a Hilbert space”, Vychisl. Prikl. Mat., No. 78, 38–48 (1994)) dimension are derived. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 28–37.     

Generalization of the Mukhamadiev theorem on the invertibility of functional operators in the space of bounded functions

We establish necessary and sufficient conditions for the invertibility of the linear bounded operator d ^m / dt ^m + A in the space of functions bounded on ℝ. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 398–412, March, 2008.     

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.     

Gauge-invariant discrete models of Yang-Mills equations

Two discrete models of Yang-Mill equations are constructed in the space ℝⁿ for some matrix-valued Lie group. A gauge-invariant discrete model is examined. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 742–754, May, 1997. Translated by O. V. Sipacheva     

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Solvability of nonlinear equations in a cone of a banach space

The solvability conditions for the equation Tu+F(u)=0 are found in the case where the operator [T+F′(u)]⁻¹ exists only for u∈K, where K is a cone in a Banach space X. An application concerning the solvability of boundary-value problems for systems of second-order differential equations is provided. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 225–230. Translated by L. Yu. Kolotilina.     

Sarason's transform in a Sobolev space

The lassical Sarason transform is the canonical isomorphism of the model space H²/gqH² (θ is an inner function generated by a single point mass) onto the standard L² space. In the paper the image of the space of smooth functions H ₂ ¹ ={f: f′ ε H²} under this transform is described in explicit terms. Pointwise estimates of the model operator in H ₂ ¹ /θH ₂ ¹ are obtained. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI. Vol. 206, 1993, pp. 33–39. Translated by I. Boricheva.     

Three-dimensional volume of a closed universe as a canonical time parameter

The dynamics of a spatially closed universe is studied in gravitation theory by means of the eigenfunctions and eigenvalues of the 3D projection of the Dirac operator. In a gauge that we call the Ashtekar gauge and construct by using an eigenfunction of this operator, the 3D volume of a spatial section is a canonical parameter, and the energy is a positive-definite functional of the dynamical variables (on the region of the phase space in which V > 0) proportional to the corresponding eigenvalue. There is a discrete set of frames of reference distinguished in this manner.Leningrad State Technical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 541–548, June, 1993.     

Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L ²-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝⁿ, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

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.     

On Toeplitz operators similar to unilateral shifts

Let T be a Toeplitz operator on the Hardy space H² on the unit circle, and let the symbol of T be of the form ϕ/ψ, where ϕ is an inner function, ψ is a finite Blaschke product, and deg ψ ≤ deg ϕ. D. N. Clark proved that such an operator T is similar to an isometry. In this paper, we find necessary and sufficient conditions under which such an operator T is similar to a unilateral shift. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 85–104.     

Solvability of convolution equation in the Gevrey class of functions analytic in a nonconvex region

In the Gevrey space A(q) of functions analytic in a nonconvex bounded region in C^–, we consider a convolution operator L generated by some entire functiona. A necessary and sufficient condition under which the equality L(A(q))=A(q) holds is established.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 35–43, September, 1992.