On estimates for a system of maximal differential polynomials without mixed derivatives

Sufficient conditions for one maximal differential with constant coefficients operator <$E Q (D)> subordinated to a system <$E left { P sub j (D) right } sub 1 sup N> of other operators without mixed derivatives in the spaces <$E L sub p ( OMEGA )> belonged to its linear span are obtained.

On eigenvalues and eigensolutions of the Schrodinger equation on the complement of a set with classical capacity zero

Let <$E GAMMA> be a closed subset of <$E bold roman R sup d> with strictly positive <$E c sub 2>-capacity. We provide a method to construct a large class of selfadjoint operators <$E H sub alpha sup mu> in <$E L sup 2 bold roman (R sup d)> which satisfy <$E H sub alpha sup mu f~=~- DELTA f> for every smooth function with compact support away from <$E GAMMA>. The operators are parametrized via a positive real number <$E alpha> and a measure <$E mu> supported by <$E GAMMA>. We shall show that the negative number <$E - beta> is an eigenvalue of <$E H sub alpha sup mu> if and only if 0 is an eigenvalue of the operator <$E ( - DELTA~+~alpha ) ( - DELTA~+~beta )~-~( alpha~-~beta ) mu> and give a linear bijection between the corresponding eigenspaces. Moreover we shall derive estimates for the number, counting multiplicities, of negative eigenvalues of the operator <$E H sub alpha sup mu>.

On the Cauchy problem for differential equations in a Banach space over the field of P-adic numbers

For the Cauchy problem for an operator differential equation of the form <$E y prime (z)~=~Ay(z)>, where A is a closed linear operator on a Banach space over the field of p-adic numbers, a criterion for well-posedness in the class of locally analytic vector-functions is established. It is shown how the Cauchy - Kovalevskaya theorem for p-adic partial differential equations may be obtained as a particular case from this criterion.

On representation of a scalar operator in the form of a sum of orthoprojections

In this paper we present an explicit construction for a representation, as a sum of orthoprojections, of a positive definite diagonal matrix with an integer trace exceeding its size. The existence of such a representation has been discovered by Fillmore without indicating a precise construction. We also describe scalar operators for which the above representation is unique (up to unitary equivalence). Moreover, we present another proof (see Theorem 2) of a result obtained earlier by Djeldubaev and Kruglyak.

On categorical properties of the functor of order-preserving functionals

For a Tychonoff space X, by O(X) we denote the space of order-preserving functional on <$E C sub b (X)>, where <$E C sub b (X)> is the ring of all continuous bounded functions <$E f~:~X~symbol О~R>. It is shown that the functor O in the category Tych does not preserve preimages and weights of Tychonoff spaces. We proof that the remaining conditions of normality hold with slight modifications.

On a two-dimensional trigonometric moment problem

Positive definite functions defined on a subset of the lattice <$E bold roman Z sup 2> do not admit, generally speaking, positive definite extension on the whole lattice. We give new and elementary proof of the following theorem: Any positive definite function defined on a rectangular of <$E bold roman Z sup 2> with one side from -1 to 1 always admits representation as trigonometric moments of a positive measure on two-dimensional torus, and, therefore, can be extended on the whole lattice.

On covariant maps of matrices

We consider classes of maps, in the matrix space, that commute with the actions of the unitary (general linear) group that acts by conjugation. Functional models for such maps are constructed and some analytic properties are explored in terms of these models.

On connection between <$E bold back 60 +>-decomposability and wildness for algebras generated by idempotents

We introduce a natural semigroup structure on the set of <$E k>-algebras <$E Q sub k (n,~alpha )~=~k~<<~1,~e sub 1 , ... , e sub n~|~e sub 1 sup 2~=~e sub 1 ,..., e sub n sup 2~=~e sub n ,~e sub 1 ...~+~e sub n~=~ alpha~symbol <174>~k~>> > (k is a field), having at least one finite-dimensional representation and, for an algebraically closed field of characteristic 0, obtain a connection between decomposability (in the semigroup) and wildness of such algebras.

Spectral gap inequalities on configuration spaces

In the first part we consider the Laplace operator with Neumann boundary conditions on a configuration space with Poisson measure over a bounded domain. The spectrum of this operator is considered and the structure of its vacuum space is studied. The corresponding spectral gap inequality is proved. The differences between Poincare and spectral gap inequalities are shown, and absence of Poincare inequality is presented. In the second part we study a second order differential operator with grown coefficients on a whole configuration space. The main properties of this operator are considered and <$E roman {Poncar e back 30 up 30 symbol В}> inequality is proved.

Some remarks on singular perturbations of self-adjoint operators

We study, in general, infinite rank singular perturbations of self-adjoint operators. For a given unbounded self-adjoint operator A acting on a separable Hilbert space B and, generally speaking, an unbounded self-adjoint operator G from <$E B sub +2 (A)> into <$E B sub -2 (A)> with the property <$E R(G) Bar~inter~B~=~{0}> we construct a family of regularizations of the formal expression <$E A~+~G> in terms of skew unbounded projections onto <$E B sub +2>, define corresponding self-adjoint in B realizations of this expression and describe their domains and resolvents. This new approach, based on extension theory of symmetric operators with the exit into rigged Hilbert spaces, allows to develop infinite-dimensional version of singular perturbations of self-adjoint operators and in the case of finite rank perturbations to get results close to the corresponding ones obtained by S. Albeverio and P. Kurasov. For a nonnegative self-adjoint operator A we establish for the first time necessary and sufficient conditions on G that guarantee the existence of nonnegative self-adjoint operators among the realizations of <$E A~+~G>, as well as when those nonnegative realizations contain the Krein-von Neumann nonnegative extremal self-adjoint extension. Singular perturbations of the Laplace operator in <$E bold roman R sup 3> by delta potentials are considered.

Von Neumann dimensions of symmetric and antisymmetric tensor products

Let <$E X> be an infinite cover of a compact Riemannian manifold and <$E {roman Har} sup p (X)> be the space of square-integrable harmonic <$Ep>-forms over <$E X>. We define and compute the von Neumann dimension of the symmetric and antisymmetric tensor product <$E {roman Har} sup p (X)~{symbol д} Hat~{roman Har} sup p (X)> and <$E {roman Har} sup p (X)~symbol <185>~{roman Har} sup p (X)> respectively.

Tubular (weakly) <$E bold PI>-complete and (weakly) <$E bold PI>-completeness of continuous mappings, their properties and characterizations by the use of morphisms

In the present paper we give basic properties of new classes of tubular (weakly) (<$E tau ->) <$E PI>-complete and (weakly) (<$E tau ->) -complete mappings, which are mappings that generalize (weakly) (<$E tau ->) <$E PI>-complete and (weakly) (<$E tau ->) superparacompact spaces. We also provide evaluations of these classes with the use of morphisms.

On translation invariant operators in Hardy spaces in tube domains over open cones

We prove that a translation invariant linear bounded operator A from <$E H sup p (T sub GAMMA )> to <$E H sup q (T sub GAMMA )> with <$E p~>>~q> is trivial, <$E A~=~0>. This is an analogue of the well-known theorem by L. Hormander for <$E L sup p (bold roman R sup n )> spaces. Some useful boundary formulae for norm of a function from <$E H sup p (T sub GAMMA )> are also established.

On the Schur algorithm for indefinite moment problem

The paper deals with the Schur algorithm of solution of the indefinite truncated moment problem. The main theorem describes the inductive step which reduces the original problem to a problem in a smaller dimension. The algorithm leads to a factorization of the corresponding solution matrix.

Simultaneous similarity of pairs of convolution Volterra operators to fractional powers of the operator of integration

A criterion for a pair of convolution Volterra operators <$E K sub i> on <$E L sup p [0,~1]> with kernels from the Liouville - Sobolev spaces <$E W sub 1 sup {alpha sub i +1}> (<$E i> = 1, 2) to be simultaneously similar to powers of the operator of integration <$E J> is obtained. Simultaneous similarity means that there exists an automorphism <$E S> on <$E L sup p [0,~1]> such that <$E S sup -1 K sub i S~=~J sup alpha sub i>. The proof of the main result involves a careful analysis of fractional powers of weak positive type convolution Volterra operators.

Schrodinger operators with a number of negative eigenvalues equal to the number of point interactions

We give necessary and sufficient conditions for a one-dimensional Schrodinger operator to have the number of negative eigenvalues equal to the number of point interactions in the cases of <$E delta > and <$E delta prime > interactions.

On a class of generators of a one-parameter <$E bold C sub 0>-semigroup of operators

On a separable Hilbert space, we consider a class of closed, densely defined operators, inverse to which are <$E n>-dimensional perturbations of Volterra dissipative operators. We find conditions such that operators of this class generate a <$E C sub 0>-semigroup.

A description of the anti-Fock representation of a set of *-algebras connected with piecewise linear-fractional unimodal mappings

We give a method for describing the anti-Fock representation of the relation <$E X X sup *~=~f (X sup * X)> for a unimodal mapping f. Examples of a description of the anti-Fock representation for sets of piecewise linear-fractional mappings are given.

Entropy of Bogoliubov automorphisms on II1-representations of the group <$E bold U (inf)>

The authors define analogies of Bogoliubov automorphisms on II1 factor representations of the group <$E U (inf)> and obtain formulas for its CNT-entropy.

Deterministic viscous hydrodynamics via stochastic analysis on groups of diffeomorphisms

The Navier-Stokes equation is derived as an Euler type equation in the "algebra" of the group of diffeomorphisms of flat n-dimensional torus. It is generated by a certain diffusion process on the group, governed by a stochastic analogue of the second Newton's law with a mechanical constraint that guarantees incompressibility. The diffusion term of the process is connected with viscosity coefficient of the fluid. The constraint is given in invariant geometric terms, the Newton's law is formulated in terms of Nelson's mean backward derivatives. The construction is translated into the finite-dimensional language of processes on the torus.