# Vol. 17 (2011), no. 2

### On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, G. Torbin

MFAT **17** (2011), no. 2, 97-111

97-111

We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

### Complement on order weakly compact operators

Belmesnaoui Aqzzouz, Jawad H'mishane

MFAT **17** (2011), no. 2, 112-117

112-117

We generalize a result on the duality property for order weakly compact operators and use it to establish some characterizations of positive operators.

### Boundary problems and initial-boundary value problems for one class of nonlinear parabolic equations with Lévy Laplacian

MFAT **17** (2011), no. 2, 118-125

118-125

We develop a method to construct a solution to a boundary problem and an initial-boundary value problem in a fundamental domain of a Hilbert space for a class of nonlinear parabolic equations not containing explicitly the unknown function, $$\frac{\partial U(t,x)}{\partial t}=f(t,\Delta_LU(t,x)),$$ where $\Delta _L$ is the infinite dimensional Lévy Laplacian.

### Multi-dimensional Schrödinger operators with point interactions

MFAT **17** (2011), no. 2, 126-143

126-143

We study two- and three-dimensional matrix Schrödinger operators with $m\in \Bbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by the other authors in this field. For instance, we parametrize all self-adjoint extensions of the initial minimal symmetric Schrödinger operator by abstract boundary conditions and characterize their spectra. Particularly, we find a sufficient condition in terms of distances and intensities for the self-adjoint extension $H_{\alpha,X}^{(3)}$ to have $m'$ negative eigenvalues, i.e., $\kappa_-(H_{\alpha,X}^{(3)})=m'\le m$. We also give an explicit description of self-adjoint nonnegative extensions.

### Unconditional bases of de Branges spaces and interpolation problems corresponding to them

MFAT **17** (2011), no. 2, 144-149

144-149

In this paper the unconditional bases of de Branges spaces are constructed from the values of reproducing kernels. Appropriate problems of interpolation by entire functions are also considered. The paper is a continuation of papers [2, 3].

### Notes on Wick calculus on parametrized test functions spaces of Meixner white noise

MFAT **17** (2011), no. 2, 150-167

150-167

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we construct elements of a Wick calculus on parametrized Kondratiev-type spaces of test functions; consider the interconnection between the extended stochastic integration and the Wick calculus; and give an example of a stochastic equation with a Wick-type nonlinearity. The main results consist in studying properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces of test functions. These results are necessary, in particular, in order to describe properties of solutions of stochastic equations with Wick type nonlinearities in the "Meixner analysis".

### On $C^*$-algebra generated by Fock representation of Wick algebra with braided coefficients

MFAT **17** (2011), no. 2, 168-173

168-173

We consider $C^*$-algebras $\mathcal{W}(T)$ generated by operators of Fock representations of Wick $*$-algebras with a braided coefficient operator $T$. It is shown that for any braided $T$ with $||T||<1$ one has the inclusion $\mathcal{W}(0)\subset\mathcal{W}(T)$. Conditions for existence of an isomorphism $\mathcal{W}(T)\simeq\mathcal{W}(0)$ are discussed.

### On $(\bigwedge, \mu)$-closed sets in generalized topological spaces

MFAT **17** (2011), no. 2, 174-179

174-179

In this paper, we introduce and study $(\bigwedge ,\mu )$-open sets and $(\bigwedge ,\mu )$-closed sets via $\mu $-open and $\mu $-closed sets in generalized topological spaces. Moreover, we introduce some generalized separation axioms in generalized topological spaces.

### Factor representations of infinite semi-direct products

MFAT **17** (2011), no. 2, 180-192

180-192

In this article, we propose a new method to study unitary representations of inductive limits of locally compact groups. For the group of infinite upper triangular matrices, we construct a family of type III factorial representations. These results are complements to previous results of A. V. Kosyak, and Albeverio and Kosyak [1, 5].