# Vol. 16 (2010), no. 3

### Boundary problems for the wave equation with the Lévy Laplacian in Shilov's class

S. Albeverio, Ya. I. Belopolskaya, M. N. Feller

Methods Funct. Anal. Topology **16** (2010), no. 3, 197-202

We present solutions to some boundary value and initial-boundary value problems for the "wave" equation with the infinite dimensional L\'evy Laplacian $\Delta _L$ $$\frac{\partial^2 U(t,x)}{\partial t^2}=\Delta_LU(t,x)$$ in the Shilov class of functions.

### The strong Hamburger moment problem and related direct and inverse spectral problems for block Jacobi-Laurent matrices

Yurij M. Berezansky, Mykola E. Dudkin

Methods Funct. Anal. Topology **16** (2010), no. 3, 203-241

In this article we propose an approach to the strong Hamburger moment problem based on the theory of generalized eigenvectors expansion for a selfadjoint operator. Such an approach to another type of moment problems was given in our works earlier, but for strong Hamburger moment problem it is new. We get a sufficiently complete account of the theory of such a problem, including the spectral theory of block Jacobi-Laurent matrices.

### On the $H$-ring structure of infinite Grassmannians

Methods Funct. Anal. Topology **16** (2010), no. 3, 242-258

The $H$-ring structure of certain infinite dimensional Grassmannians is discussed using various algebraic and analytical methods but avoiding cellular arguments. These methods allow us to treat these Grassmannians in a greater generality.

### On mixing and completely mixing properties of positive $L^1$-contractions of finite real W* -algebras

Methods Funct. Anal. Topology **16** (2010), no. 3, 259-263

We consider a non-commutative real analogue of Akcoglu and Sucheston's result about the mixing properties of positive L$^1$-contractions of the L$^1$-space associated with a measure space with probability measure. This result generalizes an analogous result obtained for the L$^1$-space associated with a finite (complex) W$^*$-algebras.

### Unitarization of Schur representations of a poset corresponding to $\widetilde{E_8}$

Methods Funct. Anal. Topology **16** (2010), no. 3, 264-270

We prove that every Schur representation of a poset corresponding to $\widetilde{E_8}$ can be unitarized with some character.

### A description of all solutions of the matrix Hamburger moment problem in a general case

Methods Funct. Anal. Topology **16** (2010), no. 3, 271-288

We describe all solutions of the matrix Hamburger moment problem in a general case (no conditions besides solvability are assumed). We use the fundamental results of A. V. Shtraus on the generalized resolvents of symmetric operators. All solutions of the truncated matrix Hamburger moment problem with an odd number of given moments are described in an "almost nondegenerate" case. Some conditions of solvability for the scalar truncated Hamburger moment problem with an even number of given moments are given.