V. A. Tesko

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.

We investigate the $H$-stochastic integral introduced in [24]. It is known that this integral generalizes the classical Ito stochastic integral and the Ito integral on a Fock space. In the present paper we construct and study an extension of the $H$-stochastic integral which will generalize the Hitsuda-Skorokhod integral.

Let $\ast_P$ be a product on $l_{fin}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{\infty}$ of real polynomials on $\mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $\ast_P$-positive functionals on $l_{fin}.$ If $(P_n)_{n=0}^{\infty}$ is a family of the Newton polynomials $P_n(x)=\prod_{i=0}^{n-1}(x-i)$ then the corresponding product $\star=\ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a "Fock space". We get an explicit expression for the product $\star$ and establish a connection between $\star$-positive functionals on $l_{fin}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).

In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space.