# Vol. 12 (2006), no. 2

### Brownian motion and Lévy processes in locally compact groups

David Applebaum

Methods Funct. Anal. Topology 12 (2006), no. 2, 101-112

It is shown that every L\'{e}vy process on a locally compact group $G$ is determined by a sequence of one-dimensional Brownian motions and an independent Poisson random measure. As a consequence, we are able to give a very straightforward proof of sample path continuity for Brownian motion in $G$. We also show that every L\'{e}vy process on $G$ is of pure jump type, when $G$ is totally disconnected.

### Some results on the space of holomorphic functions taking their values in b-spaces

Methods Funct. Anal. Topology 12 (2006), no. 2, 113-123

We define a space of holomorphic functions $O_{1}(U,E/F)$, where $U$ is an open pseudo-convex subset of $\Bbb{C}^{n}$, $E$ is a b-space and $F$ is a bornologically closed subspace of $E$, and we prove that the b-spaces $O_{1}(U,E/F)$ and $O(U,E)/O(U,F)$ are isomorphic.

### Uniform equicontinuity for sequences of homomorphisms into the ring of measurable operators

Methods Funct. Anal. Topology 12 (2006), no. 2, 124-130

We introduce a notion of uniform equicontinuity for sequences of functions with the values in the space of measurable operators. Then we show that all the implications of the classical Banach Principle on the almost everywhere convergence of sequences of linear operators remain valid in a non-commutative setting.

### Generalized zeros and poles of $\mathcal N_\kappa$-functions: on the underlying spectral structure

Methods Funct. Anal. Topology 12 (2006), no. 2, 131-150

Let $q$ be a scalar generalized Nevanlinna function, $q\in\mathcal N_\kappa$. Its gene alized zeros and poles (including their orders) are defined in terms of the function's operator representation. In this paper analytic properties associated with the underlying root subspaces and their geometric structures are investigated in terms of the local behaviour of the function. The main results and various characterizations are expressed by means of (local) moments, asymptotic expansions, and via the basic factorization of $q$. Also an inverse problem for recovering the geometric structure of the root subspace from an appropriate asymptotic expansion is solved.

### On $*$-wildness of a free product of finite-dimensional $C^*$-algebras

Methods Funct. Anal. Topology 12 (2006), no. 2, 151-156

In this paper we study the complexity of representation theory of free products of finite-dimensional $C^*$-algebras.

### A spectral analysis of some indefinite differential operators

A. S. Kostenko

Methods Funct. Anal. Topology 12 (2006), no. 2, 157-169

We investigate the main spectral properties of quasi--Hermitian extensions of the minimal symmetric operator $L_{\rm min}$ generated by the differential expression $-\frac{{\rm sgn}\, x}{|x|^{\alpha}}\frac{d^2}{dx^2} \ (\alpha>-1)$ in $L^2(\mathbb R, |x|^{\alpha})$. We describe their spectra, calculate the resolvents, and obtain a similarity criterion to a normal operator in terms of boundary conditions at zero. As an application of these results we describe the main spectral properties of the operator $\frac{{\rm sgn}\, x}{|x|^\alpha}\left( -\frac{d^2}{dx^2}+c \delta \right), \, \alpha>-1$.

### Continuous frame in Hilbert spaces

Methods Funct. Anal. Topology 12 (2006), no. 2, 170-182

In this paper we introduce a mean of a continuous frame which is a generalization of discrete frames. Since a discrete frame is a special case of these frames, we expect that some of the results that occur in the frame theory will be generalized to these frames. For such a generalization, after giving some basic results and theorems about these frames, we discuss the following: dual to these frames, perturbation of continuous frames and robustness of these frames to an erasure of some elements.

### Strong matrix moment problem of Hamburger

K. K. Simonov

Methods Funct. Anal. Topology 12 (2006), no. 2, 183-196

In this paper we consider the strong matrix moment problem on the real line. We obtain a necessary and sufficient condition for uniqueness and find all the solutions for the completely indeterminate case. We use M.G. Krein’s theory of representations for Hermitian operators and technique of boundary triplets and the corresponding Weyl functions.

### On existence of $*$-representations of certain algebras related to extended Dynkin graphs

Kostyantyn Yusenko

Methods Funct. Anal. Topology 12 (2006), no. 2, 197-204

For $*$-algebras associated with extended Dynkin graphs, we investigate a set of parameters for which there exist representations. We give structure properties of such sets and a complete description for the set related to the graph $\tilde D_4$.