# Authors Index, Vol. 15, 2009

### Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms

S. Albeverio, Sh. A. Ayupov, A. A. Zaitov, J. E. Ruziev

Methods Funct. Anal. Topology **15** (2009), no. 2, 177-187

In the present paper derivations and $*$-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all $L^0$-linear derivations and $L^{0}$-linear $*$-automorphisms are inner. Moreover, it is proved that each $L^0$-linear automorphism of the algebra of all linear operators on a $bo$-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.

### Conservative discrete time-invariant systems and block operator CMV matrices

Methods Funct. Anal. Topology **15** (2009), no. 3, 201-236

It is well known that an operator-valued function $\Theta$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as a transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $\Theta$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so-called "Schur para-me ers". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are analogs of Cantero--Moral--Vel\'azquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

### On the hyperspace of max-min convex compact sets

Methods Funct. Anal. Topology **15** (2009), no. 4, 322-332

A subset $A$ of $\mathbb R^n$ is said to be max-min convex if, for any $x,y\in A$ and any $t\in \mathbb R$, we have $x\oplus t\otimes y\in A$ (here $\oplus$ stands for the coordinatewise maximum of two elements in $\mathbb R^n$ and $t\otimes (y_1,\dots,y_n)=(\min\{t,y_1\},\dots, \min\{t,y_n\})$). It is proved that the hyperspace of compact max-min convex sets in the Euclidean space $\mathbb R^n$, $n\ge2$, is homeomorphic to the punctured Hilbert cube. This is a counterpart of the result by Nadler, Quinn and Stavrokas proved for the hyperspace of compact convex sets. We also investigate the maps of the hyperspaces of compact max-min convex sets induced by the projection maps of Euclidean spaces. It is proved that this map is a Hilbert cube manifold bundle.

### The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions

Methods Funct. Anal. Topology **15** (2009), no. 2, 101-136

The solution of the Cauchy problem for differential-difference double-infinite Toda lattice by means of inverse spectral problem for semi-infinite block Jacobi matrix is given. Namely, we construct a simple linear system of three differential equations of first order whose solution gives the spectral matrix measure of the aforementioned Jacobi matrix. The solution of the Cauchy problem for the Toda lattice is given by the procedure of orthogonalization w.r.t. this spectral measure, i.e. by the solution of the inverse spectral problem for this Jacobi matrix.

### A haracterization of closure of the set of compactly supported functions in Dirichlet generalized integral metric and its applications

Methods Funct. Anal. Topology **15** (2009), no. 3, 237-250

We obtain conditions under which a function $u(x)$ with finite Dirichlet ge e alized integral over a domain $G$ ($u(x)\in H(G))$ belongs to the closure of the set $C_0^\infty(G)$ in the metrics of this Dirichlet integral (i.e., to the space $H_0(G)$). In the case where $G=R^n \;(n \geq 2)$ using these conditions we construct examples of Dirichlet integrals such that $H(R^n) \neq H_0(R^n)$. For $n=2$ these examples show that in the known Mazia theorem uniform positivity of the Dirichlet integral matrix cannot be replaced with its pointwise positivity. The characterization of the space $H_0(G)$ is also applied to the problem of relative equivalence of the spaces $H(G)$ and $H_0(G)$ concerning the part of the boundary $\Gamma (\Gamma\subseteq \partial G)$. This problem in fact coincides with the problem of possibility to set boundary conditions of corresponding boundary-value problems.

### Aleksandr Yakovlevich Povzner

Methods Funct. Anal. Topology **15** (2009), no. 1, 1-2

### Necessary and sufficient condition for solvability of a partial integral equation

Methods Funct. Anal. Topology **15** (2009), no. 1, 67-73

Let $T_1: L_2(\Omega^2) \to L_2(\Omega^2)$ be a partial integral operator [4,7] with the kernel from $C(\Omega^3)$ where $\Omega=[a,b ]^ u$, $ u \in N$ is fixed. In this paper we investigate solvability of the partial integral equation $f-\varkappa T_1 f=g_0$ in the space $L_2(\Omega^2)$ in the case where $\varkappa$ is a cha ac eristic number. We prove a the theorem that gives a necessary and sufficient condition for solvability of the partial integral equation $f-\varkappa T_1 f=g_0.$

### On the spectrum of a model operator in Fock space

Tulkin H. Rasulov, Mukhiddin I. Muminov, Mahir Hasanov

Methods Funct. Anal. Topology **15** (2009), no. 4, 369-383

A model operator $H$ associated to a system describing four particles in interaction, without conservation of the number of particles, is considered. We describe the essential spectrum of $H$ by the spectrum of the channel operators and prove the Hunziker-van Winter-Zhislin (HWZ) theorem for the operator $H.$ We also give some variational principles for boundaries of the essential spectrum and interior eigenvalues.

### Direct spectral problem for the generalized Jacobi Hermitian matrices

Methods Funct. Anal. Topology **15** (2009), no. 1, 3-14

In this article we will introduce and investigate some generalized Jacobi matrices. These matrices have three-diagonal block structure and they are Hermitian. We will give necessary and sufficient conditions for selfadjointness of the operator which is generated by the matrix of such a type, and consider its generalized eigenvector expansion.

### Inverse spectral problem for some generalized Jacobi Hermitian matrices

Methods Funct. Anal. Topology **15** (2009), no. 4, 333-355

In this article we will investigate an inverse spectral problem for three-diagonal block Jacobi type Hermitian real-valued matrices with "almost" semidiagonal matrices on the side diagonals.

### A class of distal functions on semitopological semigroups

Methods Funct. Anal. Topology **15** (2009), no. 2, 188-194

The norm closure of the algebra generated by the set $\{n\mapsto {\lambda}^{n^k}:$ $\lambda\in{\mathbb {T}}$ and $k\in{\mathbb{N}}\}$ of functions on $({\mathbb {Z}}, +)$ was studied in \cite{S} (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among other things, it is shown that the elements of the involved algebra are distal. In particular, we examine this algebra for $({\mathbb {Z}}, +)$ and (more generally) for the discrete (additive) group of any countable ring. Finally, our results are treated for a bicyclic semigroup.

### Origination of the singular continuous spectrum in the conflict dynamical systems

Methods Funct. Anal. Topology **15** (2009), no. 1, 15-30

We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always" the limiting measures are purely singular continuous. Besides we find the conditions under which the limiting measures belong to one of the spectral type: pure singular continuous, pure point, or pure absolutely continuous.

### Expansion in eigenfunctions of relations generated by pair of operator differential expressions

Methods Funct. Anal. Topology **15** (2009), no. 2, 137-151

For relations generated by a pair of operator symmetric differential expressions, a class of generalized resolvents is found. These resolvents are integro-differential operators. The expansion in eigenfunctions of these relations is obtained.

### Morse functions and flows on nonorientable surfaces

Methods Funct. Anal. Topology **15** (2009), no. 3, 251-258

The present paper deals with the correspondence between Morse functions and flows on nonorientable surfaces. It is proved that for every Morse flow with an indexing of saddle points on a nonorientable surface there is a unique Morse function, up to a fiber equivalence, such that its gradient flow is trajectory equivalent to the initial flow, and the values of the function in the saddle points are ordered according to the indexing. The algorithm for constructing the Morse function from a Morse flow with an indexing is given. Reeb graphs and 3-graphs, which assign Morse functions and the corresponding Morse flows with the number of the saddle points less than $3$ are presented.

### Some properties for Beurling algebras

Methods Funct. Anal. Topology **15** (2009), no. 3, 259-263

Let $G$ be a locally compact group and let $\omega$ be a weight function on $G$. In this paper, among other things, we show that the Beurling algebra $L^1(G,\omega)$ is super-amenable if and only if $G$ is finite and it is biprojective if and only if $G$ is compact.

### Connected components of partition preserving diffeomorphisms

Methods Funct. Anal. Topology **15** (2009), no. 3, 264-279

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial and $\mathcal{S}(f)$ be the group of diffeomorphisms $h$ of $\mathbb{R}^2$ preserving $f$, i.e. $f \circ h =f$. Denote by $\mathcal{S}_{\mathrm{id}}(f)^{r}$, $(0\leq r \leq \infty)$, the identity component of $\mathcal{S}(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $\mathcal{S}_{\mathrm{id}}(f)^{\infty} = \cdots = \mathcal{S}_{\mathrm{id}}(f)^{1}$ for all $f$ and that $\mathcal{S}_{\mathrm{id}}(f)^{1} ot= \mathcal{S}_{\mathrm{id}}(f)^{0}$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.

### Spectral gaps of one-dimensional Schrödinger operators with singular periodic potentials

Vladimir Mikhailets, Volodymyr Molyboga

Methods Funct. Anal. Topology **15** (2009), no. 1, 31-40

The behavior of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schrödinger operators $$ S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q) \right), $$ with real-valued 1-periodic distributional potentials $q(x)\in H_{1\mbox{-}{\operatorname{per}}}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behavior as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\varphi}$, $s>-1$, $\varphi\in \mathrm{SV}$. The case $q(x)\in L_{1\mbox{-}{\operatorname{per}}}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $\varphi\equiv 1$, corresponds to the Marchenko-Ostrovskii Theorem.

### Points of joint continuity of separately continuous mappings

Methods Funct. Anal. Topology **15** (2009), no. 4, 356-360

Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $f:X \times Y \to \mathbb{R}$ be a separately continuous mapping. For each $y \in Y$, we define a game $G(Y, \{ y \})$ between players $O$ and $P$, to show that if in this game either $O$ player has a winning strategy or $X$ is $\alpha$-favorable and $P$ player does not have a winning strategy, then for each countable subset $E$ of $Y$, there exists a dense $G_\delta$ subset $D$ of $X$ such that $f$ is jointly continuous on $D \times E$.

### Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices

Methods Funct. Anal. Topology **15** (2009), no. 3, 280-300

Let $l [y]$ be a formally selfadjoint differential expression of an even order on the interval $[0,b \rangle$, $b\leq \infty$, with operator coefficients, acting in a separable Hilbert space $H$. We introduce the concept of deficiency indices $n_{b\pm}$ of the expression $l$ at the point $b$ and show that in the case $\dim H=\infty$ any values of $n_{b\pm}$ are possible. Moreover the decomposing selfadjoint boundary conditions exist if and only if $n_{b+}=n_{b-}$. Our considerations of differential operators with arbitrary (possibly unequal) deficiency indices are based on the concept of a decomposing $D$-boundary triplet. Such an approach enables to describe extensions of the minimal operator directly in terms of operator boundary conditions at the ends of the interval $[0,b \rangle$. In particular we describe in a compact form selfadjoint decomposing boundary conditions.

Associated to a $D$-triplet is an $m$-function, which can be regarded as a gene
alization of the classical characteristic (Titchmarsh-Weyl) function. Our definition enables to describe all $m$-functions (and, therefore, all spectral functions) directly in terms of boundary conditions at the right end $b$.

### Extension of some Lions-Magenes theorems

Methods Funct. Anal. Topology **15** (2009), no. 2, 152-167

A general form of the Lions-Magenes theorems on solvability of an elliptic boundary-value problem in the spaces of nonregular distributions is proved. We find a general condition on the space of right-hand sides of the elliptic equation under which the operator of the problem is bounded and has a finite index on the corresponding couple of Hilbert spaces. Extensive classes of the spaces satisfying this condition are constructed. They contain the spaces used by Lions and Magenes and many others spaces.

### On the group of foliation isometries

A. Ya. Narmanov, A. S. Sharipov

Methods Funct. Anal. Topology **15** (2009), no. 2, 195-200

The purpose of our paper is to introduce some topology on the group $G_F^{r}(M)$ of all $C^{r}$-isometries of foliated manifold $(M,F)$, which depends on a foliation $F$ and coincides with compact-open topology when $F$ is an $n$-dimensional foliation. If the codimension of $F$ is equal to $n$, convergence in our topology coincides with pointwise convergence, where $ n=\operatorname{dim}M.$ It is proved that the group $G_F^{r}(M)$ is a topological group with compact-open topology, where $r\geq{0}.$ In addition it is showed some properties of F-compact-open topology.

### $ls$-Ponomarev-systems and compact images of locally separable metric spaces

Methods Funct. Anal. Topology **15** (2009), no. 4, 391-400

We introduce the notion of an $ls$-Ponomarev-system $(f, M, X, \{\mathcal{P}_{\lambda,n}\})$, and give necessary and sufficient conditions such that the mapping $f$ is a compact (compact-covering, sequence-covering, pseudo-sequence-covering, sequentially-quotient) mapping from a locally separable metric space $M$ onto a space $X$. As applications of these results, we systematically get characterizations of certain compact images of locally separable metric spaces.

### Inverse eigenvalue problems for nonlocal Sturm-Liouville operators

Methods Funct. Anal. Topology **15** (2009), no. 1, 41-47

We solve the inverse spectral problem for a class of Sturm-Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.

### About $\ast$-representations of polynomial semilinear relations

Methods Funct. Anal. Topology **15** (2009), no. 2, 168-176

In the present paper we study $\ast$-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph $\Gamma$ we construct a polynomial characteristic function such that $\Gamma$ is its graph. Full description of graphs which satisfy polynomial (degree one and two) semilinear relations is obtained. We introduce the $G$-orthoscalarity condition and prove that any semili ear relation with quadratic characteristic function and condition of $G$-orthoscalarity is $\ast$-tame. This class of relations contains, in particular, $\ast$-representations of $U_{q}(so(3)).$

### Compact variation, compact subdifferetiability and indefinite Bochner integral

Methods Funct. Anal. Topology **15** (2009), no. 1, 74-90

The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compact subdifferentiable a.e. mapping is an indefinite Bochner integral.

### Some results on the uniform boundedness theorem in locally convex cones

Asghar Ranjbari, Husain Saiflu

Methods Funct. Anal. Topology **15** (2009), no. 4, 361-368

Walter Roth studied one form of the uniform boundedness theorem in \cite{Rot98}. We investigate some other versions of the uniform boundedness theorem for barreled and upper-barreled locally convex cones. Finally, we show some applications of this theorem.

### Inverse scattering problem on the axis for the triangular $2\times 2$ matrix potential with a virtual level

F. S. Rofe-Beketov, E. I. Zubkova

Methods Funct. Anal. Topology **15** (2009), no. 4, 301-321

The characteristic properties of scattering data for the Schrodinger operator on the axis with a triangular $2\times 2$ matrix potential are obtained under the simple or multiple virtual levels being possibly present. Under a multiple virtual level, a pole for the reflection coefficient at $k=0$ is possible. For this case, the modified Parseval equality is constructed.

### On $n$-tuples of subspaces in linear and unitary spaces

Yu. S. Samoilenko, D. Yu. Yakymenko

Methods Funct. Anal. Topology **15** (2009), no. 1, 48-60

We study a relation between brick $n$-tuples of subspaces of a finite dimensional linear space, and irreducible $n$-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive coefficients, of orthogonal projections onto these subspaces equals the identity operator. We prove that brick systems of one-dimensional subspaces and the systems obtained from them by applying the Coxeter functors (in particular, all brick triples and quadruples of subspaces) can be unitarized. For each brick triple and quadruple of subspaces, we describe sets of characters that admit a unitarization.

### Schrödinger operators with purely discrete spectrum

Methods Funct. Anal. Topology **15** (2009), no. 1, 61-66

We prove $-\Delta +V$ has purely discrete spectrum if $V\geq 0$ and, for all $M$, $|{\{x\mid V(x) < M\}}|<\infty$ and various extensions.

### On *-representations of the perturbation of twisted CCR

Methods Funct. Anal. Topology **15** (2009), no. 4, 384-390

A classification of irreducible *-representations of a certain deformation of twisted canonical commutation relations is given.

### Integral representations for spectral functions of some nonself-adjoint Jacobi matrices

Methods Funct. Anal. Topology **15** (2009), no. 1, 91-100

We study a Jacobi matrix $J$ with complex numbers $a_n,\ n\in\mathbb Z_+,$ in the main diagonal such that $r_0 \leq {\rm Im}\, a_n \leq r_1,\ r_0,r_1\in\mathbb R$. We obtain an integral representation for the (generalized) spectral function of the matrix $J$. The method of our study is similar to Marchenko's method for nonself-adjoint differential operators.