Vol. 17 (2011), no. 4

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous article. For values of the parameter for which the difference operator is self adjoint, we show that the spectrum of the operator is discrete and simple. When $q$ approaches $1$, the spectrum fills the whole positive or negative semiaxis.

We establish a sufficient condition under which the product of an order bounded almost Dunford-Pettis operator and an order weakly compact operator is Dunford-Pettis. And we derive some consequences.

We discover death-immigration non-equilibrium stochastic dynamics in the continuum also known as the Surgailis process. Explicit expression for the correlation functions is presented. Dynamics of states and their generating functionals are studied. Ergodic properties for the evolutions are considered.

We describe an infinitesimal algebra to a hypergroup constructed from a Lie group and a conditional expectation. We also prove a theorem on a decomposition of the conditional expectation into the product of a counital conditional expectation and the one that arises in the double coset construction.

We prove that the classical Menshov--Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces $L_{2}(X,d\mu;H)$ of vector-valued functions, where $(X,\mu)$ is an arbitrary measure space, and $H$ is a real or complex Hilbert space of an arbitrary dimension.

It is shown that the extremal problem for the one--dimensional Euler--Lagrange variational functional in ${C^1[a;b]}$ under a strengthened Legendre condition can be solved without using the Jacobi equation. In this case, exactly one of the two possible cases requires a restriction to the length of $[a;b]$, defined only by the form of the integrand. The result is extended to the case of compact extremum in ${H^1[a;b]}$.

It is known that a base for a traditional topology, or for a $L$-topology, $\tau$, is a subset ${\mathcal B}$ of $\tau$ with the property that every element $G\in \tau$ can be written as a union of elements of ${\mathcal B}$. In the classical case it is equivalent to say that $G\in \tau$ if and only if for any $x\in G$ we have $B\in {\mathcal B}$ satisfying $x\in B \subseteq G$. This latter property is taken as the foundation for a notion of strong base for a $L$-topology. Characteristic properties of a strong base are given and among other results it is shown that a strong base is a base, but not conversely.

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).