# Authors Index,

### Higher powers of q-deformed white noise

Methods Funct. Anal. Topology **12** (2006), no. 3, 205-219

We introduce the renormalized powers of $q$-deformed white noise, for any $q$ in the open interval $(-1,1)$, and we extend to them the no--go theorem recently proved by Accardi--Boukas--Franz in the Boson case. The surprising fact is that the lower bound ( ef{basicineq}), which defines the obstruction to the positivity of the sesquilinear form, uniquely determined by the renormalized commutation relations, is independent of $q$ in the half-open interval $(-1,1]$, thus including the Boson case. The exceptional value $q=-1$, corresponding to the Fermion case, is dealt with in the last section of the paper where we prove that the argument used to prove the no--go theorem for $q \ne 0$ does not extend to this case.

### Lévy-Dirichlet forms. II

S. Albeverio, Ya. Belopolskaya, M. Feller

Methods Funct. Anal. Topology **12** (2006), no. 4, 302-314

A Dirichlet form associated with the infinite dimensional symmetrized Levy-Laplace operator is constructed. It is shown that there exists a natural connection between this form and a Markov process. This correspondence is similar to that studied in a previous paper by the same authors for the non-symmetric Levy Laplacian.

### $*$-wildness of some classes of $C^*$-algebras

Sergio Albeverio, Kate Jushenko, Daniil Proskurin, Yurii Samoilenko

Methods Funct. Anal. Topology **12** (2006), no. 4, 315-325

We consider the complexity of the representation theory of free products of $C^*$-algebras. Necessary and sufficient conditions for the free product of finite-dimensional $C^*$-algebras to be $*$-wild is presented. As a corollary we get criteria for $*$-wildness of free products of finite groups. It is proved that the free product of a non-commutative nuclear $C^*$-algebra and the algebra of continuous functions on the one-dimensional sphere is $*$-wild. This result is applied to estimate the complexity of the representation theory of certain $C^*$-algebras generated by isometries and partial isometries.

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

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

B. Aqzzouz, M. T. Belghiti, M. H. Elalj, R. Nouira

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.

### Properties of the spectrum of type $\pi_{+}$ and type $\pi_{-}$ of self-adjoint operators in Krein spaces

Jussi Behrndt, Friedrich Philipp, Carsten Trunk

Methods Funct. Anal. Topology **12** (2006), no. 4, 326-340

We investigate spectral points of type $\pi_{+}$ and type $\pi_{-}$ for self-adjoint operators in Krein spaces. In particular a sharp lower bound for the codimension of the linear manifold $H_0$ occuring in the definition of spectral points of type $\pi_+$ and type $\pi_-$ is determined. Furthermore, we describe the structure of the spectrum in a small neighbourhood of such points and we construct a finite dimensional perturbation which turns a real spectral point of type $\pi_{+}$ (type $\pi_{-}$) into a point of positive (resp.\ negative) type. As an application we study a singular Sturm-Liouville operator with an indefinite weight.

### The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices

Yurij M. Berezansky, Mykola E. Dudkin

Methods Funct. Anal. Topology **12** (2006), no. 1, 1-31

We continue to generalize the connection between the classical power moment problem and the spectral theory of selfadjoint Jacobi matrices. In this article we propose an analog of the Jacobi matrix related to the complex moment problem and to a system of polynomials orthogonal with respect to some probability measure on the complex plane. Such a matrix has a block three-diagonal structure and gives rise to a normal operator acting on a space of l2 type. Using this connection we prove existence of a one-to-one correspondence between probability measures defined on the complex plane and block three-diagonal Jacobi type normal matrices. For simplicity, we investigate in this article only bounded normal operators. From the point of view of the complex moment problem, this restriction means that the measure in the moment representation (or the measure, connected with the orthonormal polynomials) has compact support.

### 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.

### Permutations in tensor products of factors, and $L^{2}$ cohomology of configuration spaces

Alexei Daletskii, Alexander Kalyuzhnyi

Methods Funct. Anal. Topology **12** (2006), no. 4, 341-352

We prove that the natural action of permutations in a tensor product of type $\mathrm{II}$ factors is free, and compute the von Neumann trace of the projection onto the space of symmetric and antisymmetric elements respectively. We apply this result to computation of von Neumann dimensions of the spaces of square-integrable harmonic forms ($L^{2}$-Betti numbers) of $N$-point configurations in Riemannian manifolds with infinite discrete groups of isometries.

### Continuous frame in Hilbert spaces

A. Rahimi, A. Najati, Y. N. Dehghan

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.

### Borg-type theorems for generalized Jacobi matrices and trace formulas

Methods Funct. Anal. Topology **12** (2006), no. 3, 220-233

The paper deals with two types of inverse spectral problems for the class of generalized Jacobi matrices introduced in [9]. Following the scheme proposed in [5], we deduce analogs of the Hochstadt--Lieberman theorem and the Borg theorem. Properties of a Weyl function of the generalized Jacobi matrix are systematically used to prove the uniqueness theorems. Trace formulas for the generalized Jacobi matrix are also derived.

### To the memory of Yury L'vovich Daletskii

Methods Funct. Anal. Topology **12** (2006), no. 4, 301-301

### Quantum of Banach algebras

Methods Funct. Anal. Topology **12** (2006), no. 1, 32-37

A variety of Banach algebras is a non-empty class of Banach algebras, for which there exists a family of laws such that its elements satisfy all of the laws. Each variety has a unique core (see [3]) which is generated by it. Each Banach algebra is not a core but, in this paper, we show that for each Banach algebra there exists a cardinal number (quantum of that Banach algebra) which shows the elevation of that Banach algebra for bearing a core. The class of all cores has interesting properties. Also, in this paper, we shall show that each core of a variety is generated by essential elements and each algebraic law of essential elements permeates to all of the elements of all of the Banach algebras belonging to that variety, which shows the existence of considerable structures in the cores.

### On completeness of the set of root vectors for unbounded operators

Myroslav L. Gorbachuk, Valentyna I. Gorbachuk

Methods Funct. Anal. Topology **12** (2006), no. 4, 353-362

For a closed linear operator $A$ in a Banach space, the notion of a vector accessible in the resolvent sense at infinity is introduced. It is shown that the set of such vectors coincides with the space of exponential type entire vectors of this operator and the linear span of root vectors if, in addition, the resolvent of $A$ is meromorphic. In the latter case, the completeness criteria for the set of root vectors are given in terms of behavior of the resolvent at infinity.

### 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

Ekaterina Jushenko, Konstantin Sukretniy

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 generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise

Methods Funct. Anal. Topology **12** (2006), no. 4, 363-383

We introduce and study a generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise. Properties of this derivative are quite analogous to the properties of the stochastic derivative in the Gaussian analysis. As an example we calculate the generalized stochastic derivative of the solution of some stochastic equation with Wick-type nonlinearity.

### A spectral analysis of some indefinite differential operators

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 $\frac{{\rm sgn}\, x}{|x|^\alpha}\left( -\frac{d^2}{dx^2}+c \delta \right), \, \alpha>-1$.

### $\nabla$-Fredgolm operators in Banach-Kantorovich spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 234-242

The paper is devoted to studying $\nabla$-Fredholm operators in Banach--Kantorovich spaces over a ring of measurable functions. We show that a bounded linear operator acting in Banach--Kantorovich space is $\nabla$-Fredholm if and only if it can be represented as a sum of an invertible operator and a cyclically compact operator.

### Finite rank self-adjoint perturbations

Methods Funct. Anal. Topology **12** (2006), no. 3, 243-253

Finite rank perturbations of a semi-bounded self-adjoint operator $A$ are studied. Different types of finite rank perturbations (regular, singular, mixed singular) are described from a unique point of view and by the same formula with the help of quasi-boundary value spaces. As an application, a Schr\"{o}dinger operator with nonlocal point interactions is considered.

### Cyclical elements of operators which are left-inverses to multiplication by an independent variable

Methods Funct. Anal. Topology **12** (2006), no. 4, 384-388

We study properties of operators which are left-inverses to the operator of multiplication by an independent variable in the space $\mathcal H (G)$ of functions that are analytic in an arbitrary domain $G$. This space is endowed with topology of compact convergence. A description of cyclic elements for such operators is obtained. The obtained statements generalize known results in this direction.

### A note on one decomposition of Banach spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 254-257

For a scalar type spectral operator $A$ in complex Banach space $X$, the decomposition of $X$ into the direct sum \begin{equation*} X=\ker A\oplus \overline{R(A)}, \end{equation*} where $\ker A$ is the kernel of $A$ and $\overline{R(A)}$ is the closure of its range $R(A)$ is established.

### Nevanlinna type families of linear relations and the dilation theorem

Methods Funct. Anal. Topology **12** (2006), no. 1, 38-56

Let H1 be a subspace in a Hilbert space H0 and let $\widetilde C(H_0,H_1)$ be the set of all closed linear relations from $H_0$ to $H_1$. We introduce a Nevanlinna type class $\widetilde R_+ (H_0,H_1)$ of holomorphic functions with values in $\widetilde C(H_0,H_1)$ and investigate its properties. In particular we prove the existence of a dilation for every function $\tau_+(\cdot)\in \widetilde R_+ (H_0,H_1)$. In what follows these results will be used for the derivation of the Krein type formula for generalized resolvents of a symmetric operator with arbitrary (not necessarily equal) deficiency indices.

### Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers

Methods Funct. Anal. Topology **12** (2006), no. 3, 258-280

Let $H$ be a Hilbert space and let $A$ be a symmetric operator in $H$ with arbitrary (not necessarily equal) deficiency indices $n_\pm (A)$. We introduce a new concept of a $D$-boundary triplet for $A^*$, which may be considered as a natural generalization of the known concept of a boundary triplet (boundary value space) for an operator with equal deficiency indices. With a $D$-triplet for $A^*$ we associate two Weyl functions $M_+(\cdot)$ and $M_-(\cdot)$. It is proved that the functions $M_\pm(\cdot)$ posses a number of properties similar to those of the known Weyl functions ($Q$-functions) for the case $n_+(A)=n_-(A)$. We show that every $D$-triplet for $A^*$ gives rise to Krein type formulas for generalized resolvents of the operator $A$ with arbitrary deficiency indices. The resolvent formulas describe the set of all generalized resolvents by means of two pairs of operator functions which belongs to the Nevanlinna type class $\bar R(H_0,H_1)$. This class has been earlier introduced by the author.

### Systems of $n$ subspaces and representations of $*$-algebras generated by projections

Yu. P. Moskaleva, Yu. S. Samoĭlenko

Methods Funct. Anal. Topology **12** (2006), no. 1, 57-73

In the present work a relationship between systems of n subspaces and representations of *-algebras generated by projections is investigated. It is proved that irreducible nonequivalent *-representations of *-algebras P4,com generate all nonisomorphic transitive quadruples of subspaces of a finite dimensional space.

### Two-weighted inequality for parabolic sublinear operators in Lebesgue spaces

Methods Funct. Anal. Topology **12** (2006), no. 1, 74-81

In this paper, the author establishes the boundedness in weighted $L_p$ spaces on $\mathbb R^{n+1}$ with a parabolic metric for a large class of sublinear operators generated by parabolic Calderon-Zygmund kernels. The conditions of these theorems are satisfied by many important operators in analysis. Sufficient conditions on weighted functions $\omega$ and $\omega_1$ are given so that certain parabolic sublinear operator is bounded from the weighted Lebesgue spaces $L_{p,\omega}(\mathbb R^{n+1})$ into $L_{p,\omega_1}(\mathbb R^{n+1})$.

### A locally convex quotient cone

Asghar Ranjbari, Husain Saiflu

Methods Funct. Anal. Topology **12** (2006), no. 3, 281-285

We define a quotient locally convex cone and verify some topological properties of it. We show that the extra conditions are necessary.

### On similarity of convolution Volterra operators in Sobolev spaces

Methods Funct. Anal. Topology **12** (2006), no. 3, 286-295

Necessary and sufficient conditions for a convolution Volterra operator to be similar in a Sobolev space to the operator $J^\alpha$ are obtained. A criterion of similarity is obtained as well.

### Lagrangian pairs in Hilbert spaces

Methods Funct. Anal. Topology **12** (2006), no. 1, 82-100

Weakly Lagrangian pairs and Lagrangian pairs in a pair of Hilbert spaces $(H_1, H_2)$ are defined. The weakly Lagrangian pair and Lagrangian pair extensions in $(H_1, H_2)$ of a given weakly Lagrangian pair in $(H_1, H_2)$ are characterized and those extensions which are operators are identified. A description of all Lagrangian pair extensions in a larger pair of Hilbert spaces $(\tilde H_1, \tilde H_2)$ of a given weakly Lagrangian pair in $(H_1, H_2)$ is also given.

### On enveloping $C^*$-algebra of one affine Temperley-Lieb algebra

Methods Funct. Anal. Topology **12** (2006), no. 3, 296-300

$C^*$-algebras generated by orthogonal projections satisfying relations of Temperley-Lieb type are constructed.

### About Kronrod-Reeb graph of a function on a manifold

Methods Funct. Anal. Topology **12** (2006), no. 4, 389-396

We study Kronrod-Reeb graphs of functions with isolated critical points on smooth manifolds. We prove that any finite graph, which satisfies the condition $\Im$ is a Kronrod-Reeb graph for some such function on some manifold. In this connection, monotone functions on graphs are investigated.

### Strong matrix moment problem of Hamburger

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

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 $\tilde D_4$.