# Vol. 13 (2007), no. 4

### A description of characters on the infinite wreath product

Methods Funct. Anal. Topology **13** (2007), no. 4, 301-317

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $\Gamma$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $\Gamma^\infty$ by automorphisms, so one can form a semidirect product $\Gamma^\infty \times \mathfrak{S}_\infty$, known as the wreath product $\Gamma\wr\mathfrak{S}_\infty$ of $\Gamma$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $I\!I_1-$factor-representations of $\Gamma\wr\mathfrak{S}_\infty$ in terms of finite characters of $\Gamma$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\infty\times\mathfrak{S}_\infty$. Also, we discuss certain examples of representations of type $I\!I\!I$, where the modular operator of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the theory of characters of $\mathfrak{S}_\infty$.

### Operator-valued integral of vector-function and bases

Methods Funct. Anal. Topology **13** (2007), no. 4, 318-328

In the present paper we are going to introduce an operator-valued integral of a square modulus weakly integrable mappings the ranges of which are Hilbert spaces, as bounded operators. Then, we shall show that each operator-valued integrable mapping of the index set of an orthonormal basis of a Hilbert space $H$ into $H$ can be written as a multiple of a sum of three orthonormal bases.

### On holomorphic solutions of the heat equation with a Volterra operator coefficient

Sergey Gefter, Anna Vershynina

Methods Funct. Anal. Topology **13** (2007), no. 4, 329-332

Let $A$ be a bounded operator on a Hilbert space and $g$ a vector-valued function, which is holomorphic in a neighborhood of zero. The question about existence of holomorphic solutions of the Cauchy problem $\left\{ \begin{array}{ll} \displaystyle\frac{\partial u}{\partial t}= A\displaystyle\frac{\partial^{2}u}{\partial x^2}\\ u(0,x)=g(x) \\ \end{array} \right.$ is considered in the paper.

### Generalized selfadjoinness of differentiation operator on weight Hilbert spases

Methods Funct. Anal. Topology **13** (2007), no. 4, 333-337

We consider examples of operators that act in some Hilbert rigging from positive Hilbert space into the negative one. For the first derivative operator we investigate a ``generalized'' selfadjointness in the sense of weight Hilbert riggings of the spaces $L^2([0,1])$ and $L^2(\mathbb{R})$. We will show that an example of the operator $i \frac{d}{dt}$ in some rigging scales, which is selfadjoint in usual case and not generalized selfadjoint, can not be constructed.

### On an extended stochastic integral and the Wick calculus on the connected with the generalized Meixner measure Kondratiev-type spaces

Methods Funct. Anal. Topology **13** (2007), no. 4, 338-379

We introduce an extended stochastic integral and construct elements of the Wick calculus on the Kondratiev-type spaces of regular and nonregular gene alized functions, study the interconnection between the extended stochastic integration and the Wick calculus, and consider examples of stochastic equations with Wick-type nonlinearity. Our researches are based on the general approach that covers the Gaussian, Poissonian, Gamma, Pascal and Meixner analyses.

### Another form of separation axioms

S. Athisaya Ponmani, M. Lellis Thivagar

Methods Funct. Anal. Topology **13** (2007), no. 4, 380-385

It is the object of this paper to introduce the $(1, 2)^*$pre-$D_k$ axioms for $k$ = $0$, $1$, $2$.

### Development of the Markov moment problem approach in the optimal control theory

S. Yu. Ignatovich, G. M. Sklyar

Methods Funct. Anal. Topology **13** (2007), no. 4, 386-400

The paper is a survey of the main ideas and results on using of the Markov moment problem method in the optimal control theory. It contains a version of the presentation of the Markov moment approach to the time-optimal control theory, linear and nonlinear.