The multi-dimensional truncated moment problem: maximal masses

Given a subset K of <$E{ roman bold R} sup d> and a linear functional L on the polynomials <$D{ roman bold R} sub 2n sup d [x] under> in d variables and of degree at most 2n the truncated K-moment problem asks when there is a positive Borel measure <$Emu> supported by K such that <$EL(p)~=~int~p~d mu> for <$Ep~symbol <174>~{ roman bold R} sub 2n sup d [x] under>. For compact sets K we investigate the maximal mass of all representing measures at a given point of K. Various characterizations of this quantity and related properties are developed and a close link to zeros of positive polynomials is established.

Inverse spectral problems for Jacobi matrix with finite perturbed parameters

For Jacobi matrices with finitely perturbed parameters, we get an explicit representation of the Weyl function, and solve inverse spectral problems, that is, we recover Jacobi matrices from spectral data. For the spectral data, we take the following: the spectral density of the absolutely continuous spectrum, with or without all the eigenvalues; the numerical parameters of the representation of one component of the vector-eigenfunction in terms of Chebyshev polynomials. We prove that these inverse problems have a unique solution, or only a finite number of solutions.

Leonid Pavlovych Nizhnik (to 80-th birthday anniversary)

Tannaka - Krein duality for compact quantum group coactions (survey)

The last decade saw an appearance of a series of papers containing a very interesting development of the Tannaka - Krein duality for compact quantum group coactions on C<^>*-algebras. The present survey is intended to present the main ideas and constructions underlying this development.

On Fourier algebra of a locally compact hypergroup

We give sufficient conditions for the Fourier and the Fourier-Stieltjes spaces of a locally compact hypergroup to be Banach algebras.

Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them

Contractive selfadjoint extensions of a Hermitian contraction B in a Hilbert space n with an exit in some larger Hilbert space <$En~symbol е~H> are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Krem-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.

The projection spectral theorem and Jacobi fields

We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to Levy white noise measures.

On complex perturbations of infinite band Schrodinger operators

Let <$EH sub 0 ~=~--{d sup 2 } over {dx sup 2 } ~+~V sub 0> be an infinite band Schrodinger operator on <$EL sup 2 ( roman bold R)>R) with a real-valued potential <$EV sub 0 ~symbol <174>~L sup inf ( roman bold R)>. We study its complex perturbation H = H0 + V, defined in the form sense, and obtain the Lieb-Thirring type inequalities for the rate of convergence of the discrete spectrum of H to the joint essential spectrum. The assumptions on V vary depending on the sign of Re V.

Percolations and phase transitions in a class of random spin systems

The aim of this paper is to give a review of recent results of Yu. Kondratiev, Yu. Kozitsky, T. Pasurek and myself on the multiplicity of Gibbs states (phase transitions) in infinite spin systems on random configurations, and provide a `pedestrian' route following Georgii-Haggstrom approach to (closely related to phase transitions) percolation problems for a class of random point processes.

On the finiteness of the discrete spectrum of a 3 x 3 operator matrix

An operator matrix H associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of H is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of H is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of H is found.

The investigation of Bogoliubov functionals by operator methods of moment problem

The article is devoted to a study of Bogoliubov functionals by using methods of the operator spectral theory being applied to the classical power moment problem. Some results, similar to corresponding ones for the moment problem, where obtained for such functionals. In particular, the following question was studied: under what conditions a sequence of nonlinear functionals is a sequence of Bogoliubov functionals.

On a generalization of the three spectral inverse problem

We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet - Dirichlet problem (the Sturm - Liouville problem with Dirichlet conditions at both ends) on the whole interval [0, a], parts of spectra of the Dirichlet - Neumann and Dirichlet - Dirichlet problems on [0, a/2] and parts of spectra of the Dirichlet - Newman and Dirichlet - Dirichlet problems on [a/2, a], we find the potential of the Sturm - Liouville equation.

Joint functional calculus in algebra of polynomial tempered distributions

In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol, class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.

Spectral and pseudospectral functions of Hamiltonian systems: development of the results by Arov-Dym and Sakhnovich

The main object of the paper is a Hamiltonian system <$EJy prime ~-~B(t)y~=~lambda DELTA (t)y> defined on an interval [a, b) with the regular endpoint a. We define a pseudospectral function of a singular system as a matrix-valued distribution function such that the generalized Fourier transform is a partial isometry with the minimally possible kernel. Moreover, we parameterize all spectral and pseudospectral functions of a given system by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov-Dym and Sakhnovich in this direction.

On a class of generalized Stieltjes continued fractions

With each sequence of real numbers <$Es~=~left {s sub j right } sub j=0 sup inf> two kinds of continued fractions are associated, - the so-called P-fraction and a generalized Stieltjes fraction that, in the case when <$Es~=~left {s sub j right } sub j=0 sup inf> is a sequence of moments of a probability measure on <$E{ roman bold R} sub +>, coincide with the J-fraction and the Stieltjes fraction, respectively. A subclass H<^>reg of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For <$Es~symbol <174>~H sup reg> the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.

Weak dependence for a class of local functionals of Markov chains on Z^d

In many models of Mathematical Physics, based on the study of a Markov chain <$Eeta hat ~=~left { eta sub t right } sub t=0 sup inf> on <$E{ roman bold Z} sup d>, one can prove by perturbative arguments a contraction property of the stochastic operator restricted to a subspace of local functions Hm endowed with a suitable norm. We show, on the example of a model of random walk in random environment with mutual interaction, that the condition is enough to prove a Central Limit Theorem for sequences <$Eleft { f(S sup k eta hat ) right } sub k=0 sup inf>, where S is the time shift and f is strictly local in space and belongs to a class of functionals related to the Holder continuous functions on the torus T<^>1.

On the Carleman ultradifferentiable vectors of a scalar type spectral operator

A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space.

Operators of stochastic differentiation on spaces of nonregular test functions of Levy white noise analysis

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, and on spaces of regular test and generalized functions of the Levy white noise analysis. In this paper we make the next step: introduce and study operators of stochastic differentiation on spaces of test functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using Lytvynov's generalization of the chaotic representation property. This can be considered as a contribution in a further development of the Levy white noise analysis.

Fractional contact model in the continuum

We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a behavior of time-dependent correlation functions, essentially different from the one known for the standard contact model.

Topological equivalence to a projection

We present a necessary and sufficient condition for a continuous function on a plane to be topologically equivalent to a projection onto one of the coordinates.