Hankel-Schur multipliers and multipliers of the space H1

In this paper there is given a sufficient condition for a Hankel matrix _F to belong to the space of Schur multipliers of all bounded operators in ² (or, what is the same, to the tensor algebra V²). It is shown that ifw is a nonnegative function on T, such that is a sequence of integers, {F_i}_j1 is a sequence of polynomials,) and, then _FV². It follows from this that under these conditions F is a multiplier of the space H¹, i.e.,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 113–119, 1984     

Spectral properties of one-dimensional disperse crystals

The connection between the spectral characteristics of the one-dimensional Schrödinger operator with periodic potential, asa, and the spectral characteristic of the Schrödinger operatorl(y)=–y+q(x)y with a decreasing potentialq(x) is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 197–211, 1984.     

Estimate of the norm ∥f(A1, A2)-f(B1, B2)∥ for pairs of self-adjoint commuting operators

In this note we announce the estimate for the difference mentioned in the title, where are the lengths of intervals containing the spectra of the operators A₁, B₂ and A₂, B₂; the function f is from the class Lip 1 (with constant 1). The outline of the proof of this estimate is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 175–177, 1984.     

Venkov's reduction theory of positive-quadratic forms

We prove the absolute finiteness of the number of faces (independent of the parameter) of Venkov's reduction domain (Izv. Akad. Nauk SSSR, Ser. Mat.4, 37–52 (1940)) ofn-ary positive quadratic forms. The casen=3 is given special consideration. We study the change of the reduction domain when changes along a line segment in the space of coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 108–116, 1983.     

Transformation of condensers in n-dimensional space

By a condenser in     

Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain by vector-functions of the form , where u is, the solution of the equation Δ^m _u=0 in while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on with the point whose relative neighborhood is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 65, pp. 164–171, 1976.     

Gauge group and gauge transformation in the continual theory of defects

A representation of the algebra (3)=t(3) S0(3, ) by differential Schaefer's operators is proposed, and an external algebra of (3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the (3)-valued field, which enables a group of gauge transformations of the continual theory of defects to be obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 173–184, 1991.I wish to thank V. N. Popov for his interest.     

Two methods for avoiding hereditary completeness

A family of vectors of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families of the same type but already complete in H.. Clearly, in this case cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 183–188, 1976.     

Functions of class II

We consider the class II of operator functions f(z), meromorphic for ¦z¦1, which admit a representation in the form f(z)=f ₂ ^–1 (z)f₁(z), where f₁(z) is an operatorvalued holomorphic function and f₂(z) a scalar-valued bounded holomorphic function, such that the strong limits and coincide a.e. on the unit circle ¦¦=1. We prove that any function of class II can be represented as a block of some J-inner function of class II. We describe all such representations. The results found are applied to the question of realizations of functions of class II as transfer functions of linear systems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta.im. V. A. Steklova AN SSSR, Vol. 135, pp. 5–30, 1984.     

Quasiclassical approximation for models of spin-spin interaction on a one-dimensional lattice

The one-dimensional anisotropic Heisenberg model in an external field is considered. For the corresponding energy operator with parameter, the quasiclassical (h0) spectral series are constructed using the method of the canonical operator. In the integrable case (in the absence of the external field) the results obtained coincide with the exact ones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 63–76, 1984.     

Compact Hopf hypersurfaces of constant mean curvature in complex space forms

We prove that every connected compact Hopf hypersurface of a complex space form, contained in a geodesic ball of radius strictly smaller than the injectivity radius of, having constant mean curvature and with if if < 0 is a geodesic sphere of.Work partially supported by DGICYT Grant No. PB91-0324.     

A contribution to the theory of Maxwell fluids. II

The local classical unique solvability of the periodic initial-boundary problem and of the cauchy problem is proved for the nonstationary quasilinear system describing the motion of Maxwell fluids.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 106–113, 1983.     

Large sample theory for distributions on the hypersphere with rotational symmetries

Summary A distribution on the unit sphere inℝ ^q with a densityf(‖x _v ‖) is considered where is ans(<q) dimensional subspace andx _v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x _v ‖ ², additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.     

Stochastic dilations of uniformly continuous completely positive semigroups

For an arbitrary uniformly continuous completely positive semigroup ( _t :t0) on the space of bounded operators on a Hilbert space, we construct a family (U(t)t0) of unitary operators on a Hilbert space and a conditional expectation from to, such that, for arbitraryt0,. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.     

Convolution of the Fourier coefficients of the Eisenstein-Maass series

The convolution is defined as the sum where for n≠0 and W₀,W₁ are arbitrary smooth functions. Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms. The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function.     

The problem of product of conformal radii of nonoverlapping domains

Let a₁, a₂, a₃, b be distinct points in and let D be the family of all triples of nonoverlapping domains D₁, D₂, D₃ in \ {b} such that a_k∈ D_k, k=1,2,3. For this family we consider the problem on the maximum of the functional I=R₁R₂R₃, where R_k=R(D_k, a_k) is the conformal radius of D_k with respect to a_k. Geometrical properties of the extremal triple of domains are described. We prove that the maximum of I monotonically depends on the position of the point b and find the maximum in some special cases Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 114–128 Translated by N. Yu. Netsvetaev     

Best-possible one-sided approximations in the classes 688-01688-01688-01by trigonometric polynomials in the metric of L1

The quantities are calculated, where is the best approximation from above of the function f by trigonometric polynomials of order ⩽n—1 in the metric of L₁. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 357–370, September, 1977.     

Some finiteness conditions on indecomposable operands

Let S be a monoid such that every left S-operand satisfies the ascending chain condition on orbits. Let X be an indecomposable [0-indecomposable] left S-operand. There is a descending chain of suboperands of X defined in terms of maximal orbits. If this chain terminates after a finite number of terms, the last non-empty [non-zero] term defines a distinguished collection (X) [ ₀(X)] of disjoint [0-disjoint] orbits in X. The assumption that (X) [ ₀(X)] is finite for all appropriate X, together with the additional assumption that every left S-operand satisfies the d.c.c. on orbits, implies that S has a unique minimal left ideal [that the principal left ideals outside of the minimal ideal are linearly ordered]. This research supported in part by the National Science Foundation.     

On a certain nonstationary problem in a dihedral angle. I

An initial-boundary value problem is considered for the heat conduction equation in a dihedral angleD Rⁿ, on one face of which the classical Neumann boundary condition is satisfied, while on the other one the condition where x > 0, h 0, b_j are real constants. The unique solvability of such a problem is proved in weighted Sobolev spaces and estimates of the solution are obtained under natural restrictions on the opening of the dihedral angle.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad Nauk SSSR, Vol. 188, pp. 159–177, 1991.