# Vol. 17 (2011), no. 1

### Existence and uniqueness of mild solutions of second order semilinear differential equations in Banach space

Ya. V. Gorbatenko

Methods Funct. Anal. Topology 17 (2011), no. 1, 1-9

We consider the Cauchy problem for second order semilinear differential equations in Banach space. Sufficient conditions of local and global existence and uniqueness of mild solutions are presented.

### G-Frames and operator valued-frames in Hilbert C*-modules

Methods Funct. Anal. Topology 17 (2011), no. 1, 10-19

g-frames and fusion frames in Hilbert C*-modules have been defined by the second author and B.~Khosravi in [15] and operator-valued frames in Hilbert C*-modules have been defined by Kaftal et al in [11]. We show that every operator-valued frame is a g-frame, we also show that in Hilbert C*-modules tensor product of orthonormal basis is an orthonormal basis and tensor product of g-frames is a g-frame, we get some relations between their g-frame operators, and we study tensor product of operator-valued frames in Hilbert C*-modules.

### The infinite direct products of probability measures and structural similarity

Volodymyr Koshmanenko

Methods Funct. Anal. Topology 17 (2011), no. 1, 20-28

We show that any similar structure measure on the segment $[0,1]$ is an image-measure of the appropriate constructed infinite direct product of discrete probability measures.

### A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes

Methods Funct. Anal. Topology 17 (2011), no. 1, 29-46

We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X=\mathbb R^d$, we consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms. This leads us to an equilibrium dynamics of interacting Brownian particles for which a permanental point process is a symmetrizing measure.

### The Faddeev equation and essential spectrum of a Hamiltonian in Fock space

Methods Funct. Anal. Topology 17 (2011), no. 1, 47-57

A Hamiltonian (model operator) $H$ associated to a quantum system describing three particles in interaction, without conservation of the number of particles, is considered. The Faddeev type system of equations for eigenvectors of $H$ is constructed. The essential spectrum of $H$ is described by the spectrum of the channel operator.

### On generalization of the Freudenthal's theorem for compact irreducible standard polyhedric representation for superparacompact complete metrizable spaces

D. K. Musaev

Methods Funct. Anal. Topology 17 (2011), no. 1, 58-64

In this paper for superparacompact complete metrizable spaces, the Freudenthal's theorem for compact irreducible standard polyhedral representation is ge e alized. Furthermore, for superparacompact metric spaces the following is strengthened: 1) the Morita's theorem about universality of the product $Q^\infty\times B(\tau)$ of Hilbert cube $Q^\infty$ to generalized Baire space $B(\tau)$ of the weight $\tau$ in the space of all strongly metrizable spaces of weight $\le \tau$; 2) Nagata's theorem about universality of the product $\Phi^n\times B(\tau)$ of the universal $n$-dimensional compact $\Phi^n$ to $B(\tau)$ in the space of all strongly metrizable spaces $\le\tau$ and dimension $\operatorname{dim}X\le n.$

### Some class of real sequences having indefinite Hankel forms

Methods Funct. Anal. Topology 17 (2011), no. 1, 65-74

In this paper we generalize the results given in [14] about real sequences which are not necessarily positive (i.e, they are not sequences of power moments) but can be mapped, by a difference operator, into a power moment sequence. We prove by elementary methods that the integro-polynomial representation of such sequences remains after dropping the condition on its growth imposed in the mentioned article. Some additional results on the uniqueness of the representation are included.

### Polarization formula for $(p,q)$-polynomials on a complex normed space

Methods Funct. Anal. Topology 17 (2011), no. 1, 75-83

The aim of this paper to give some analogues of polarization formulas and the polarization inequality for $(p,q)$-polynomials between complex normed spaces. Obtained results are useful for investigation of real-differentiable mappings on complex spaces.

### On equiangular configurations of subspaces of a Hilbert space

Methods Funct. Anal. Topology 17 (2011), no. 1, 84-96

In this paper, we find $\tau$, $0<\tau<1$, such that there exists an equiangular $(\Gamma, \tau)$-configuration of one-dimensional subspaces, and describe $(\Gamma, \tau)$-configurations that correspond to unicyclic graphs and to some graphs that have cyclomatic number satisfying $\nu(\Gamma) \geq 2$.