# Authors Index, Vol. 20, 2014

### Continuous dual of $c_0(Z,X,\bar\lambda, \bar p)$ and $c(Z,X,\bar\lambda, \bar p)$

Riti Agrawal, J. K. Srivastava

Methods Funct. Anal. Topology **20** (2014), no. 1, 92-100

A bilateral sequence is a function whose domain is the set $Z$ of all integers with natural ordering. In this paper we study the continuous dual of the Banach space of $X$-valued bilateral sequence spaces $c_0(Z,X,\bar\lambda, \bar p)$ and $c(Z,X,\bar\lambda, \bar p)$.

### Arens algebras of measurable operators for Maharam traces

Methods Funct. Anal. Topology **20** (2014), no. 2, 175-185

We study order and topological properties of the non-commutative Arens algebra associated with arbitrary Maharam trace.

### Parameter-elliptic problems and interpolation with a function parameter

Anna V. Anop, Aleksandr A. Murach

Methods Funct. Anal. Topology **20** (2014), no. 2, 103-116

Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to a Hilbert Sobolev scale. The latter are the Hörmander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at $+\infty$. We prove that the operator corresponding to this problem sets isomorphisms between appropriate Hörmander spaces provided the absolute value of the parameter is large enough. For solutions to the problem, we establish two-sided estimates, in which the constants are independent of the parameter.

### A simplicity criterion for symmetric operator on a graph

E. N. Ashurova, A. N. Kandagura, I. I. Karpenko

Methods Funct. Anal. Topology **20** (2014), no. 2, 117-123

In the present paper we show that the topology of the underlying graph determines the domain and deficiency indices of a certain associated minimal symmetric operator. We obtaine a criterion of simplicity for the minimal operator associated with the graph.

### An exponential representation for some integrals with respect to Lebesgue-Poisson measure

Methods Funct. Anal. Topology **20** (2014), no. 2, 186-192

We prove a theorem that allows to simplify some combinatorial calculations. An example of application of this theorem in statistical mechanics is given.

### On large coupling convergence within trace ideals

Methods Funct. Anal. Topology **20** (2014), no. 1, 3-9

Let $\mathcal E$ and $\mathcal P$ be nonnegative quadratic forms such that $\mathcal E + b \mathcal P$ is closed and densely defined for every nonnegative real number $b$. Let $H_b$ be the selfadjoint operator associated with $\mathcal E + b\mathcal P.$ By Kato's monotone convergence theorem, there exists an operator $L$ such that $(H_b+1)^{-1}$ converges to $L$ strongly, as $b$ tends to infinity. We give a condition which is sufficient in order that the operators $(H_b+1)^{-1}$ converge w.r.t. the trace norm with convergence rate $O(1/b)$. As an application we discuss trace norm resolvent convergence of Schrodinger operators with point interactions.

### Continuity of operator-valued functions in the $*$-algebra of locally measurable operators

Methods Funct. Anal. Topology **20** (2014), no. 2, 124-133

In the present paper we establish sufficient conditions for a complex-valued function $f$ defined on $\mathbb{R}$ which guarantee continuity of an operator-function $T\mapsto f(T)$ w.r.t. the topology of local measure convergence in the $*$-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated to a von Neumann algebra $\mathcal{M}$.

### On the discrete spectrum of a linear operator pencil arising in transport theory

Methods Funct. Anal. Topology **20** (2014), no. 1, 10-16

We study the problem of the finiteness of the discrete spectrum for linear operator pencils occurring in one-velocity transport theory. The results are obtained using direct methods of perturbation theory for linear operators. The proposed approach allowed to give a relatively quick proofs of the main results improving related results obtained previously by K. M. Case and C. G. Lekkerkerker.

### Schatten class operators on the Bergman space over bounded symmetric domain

Methods Funct. Anal. Topology **20** (2014), no. 3, 193-212

Let $\Omega$ be a bounded symmetric domain in $\mathbb{C}^{n}$ with Bergman kernel $K(z, w)$. Let $dV_{\lambda}(z)=K(z, z)\frac{dV(z)}{C_{\lambda}}$, where $C_{\lambda}=\displaystyle\int_{\Omega}K(z, z)^{\lambda}dV(z)$, $\lambda\in\mathbb{R}$, $dV(z)$ is the volume measure of $\Omega$ normalized so that $K(z, 0)=K(0, w)=1$. In this paper we have shown that if the Toeplitz operator $T_{\phi}$ defined on $L_{a}^{2}(\Omega, \frac{dV}{C_{0}})$ belongs to the Schatten $p$-class, $1\leq p<\infty$, then $\widetilde{\phi}\in L^{p}(\Omega, d\eta)$, where $d\eta(z)=K(z, z)\frac{dV(z)}{C_{0}}$ and $\widetilde{\phi}$ is the Berezin transform of $\phi$. Further if $\phi\in L^{p}(\Omega, d\eta_{\lambda})$, then $\widetilde{\phi_{\lambda}}\in L^{p}(\Omega, d\eta_{\lambda})$ and $T_{\phi}^{\lambda}$ belongs to Schatten $p$-class. Here $d\eta_{\lambda}=K(z, z)\frac{dV(z)}{C_{\lambda}}$, the function $\widetilde{\phi_{\lambda}}$ is the Berezin transform of $\phi$ in $L_{a}^{2}(\Omega, dV_{\lambda})$ and $T_{\phi}^{\lambda}$ is the Toeplitz operator defined on $L_{a}^{2}(\Omega, dV_{\lambda})$. We also find conditions on bounded linear operator $C$ defined from $L_{a}^{2}(\Omega, dV_{\lambda})$ into itself such that $C$ belongs to the Schatten $p$-class by comparing it with positive Toeplitz operators defined on $L_{a}^{2}(\Omega, dV_{\lambda})$. Applications of these results are obtained and we also present Schatten class characterization of little Hankel operators defined on $L_{a}^{2}(\Omega, dV_{\lambda})$.

### Nonzero capacity sets and dense subspaces in scales of Sobolev spaces

Mykola E. Dudkin, Volodymyr D. Koshmanenko

Methods Funct. Anal. Topology **20** (2014), no. 3, 213-218

We show that for a compact set $K\subset{\mathbb R}^n$ of nonzero $\alpha$-capacity, $C_\alpha(K)>0$, $\alpha\geq 1$, the subspace $\overset{\circ}{W}{^{\alpha,2}}(\Omega)$, $\Omega={\mathbb R}^n\setminus K$ in ${W}{^{\alpha,2}}({\mathbb R}^n)$ is dense in $W^{m,2}({\mathbb R}^n)$, $m\leq\alpha-1$, iff the $m$-capacity of $K$ is zero, $C_{m}(K)=0$.

### Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem

Mykola E. Dudkin, Valentyna I. Kozak

Methods Funct. Anal. Topology **20** (2014), no. 3, 219-251

We 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 Jacobi matrices related to some system of orthonormal polynomials with respect to the measure on the real plane. In our case we obtained two matrices that have a block three-diagonal structure and are symmetric operators acting in the space of $l_2$ type. With this connection we prove the one-to-one correspondence between such measures defined on the real plane and two block three-diagonal Jacobi type symmetric matrices. For the simplicity we investigate in this article only bounded symmetric operators. From the point of view of the two dimensional moment problem this restriction means that the measure in the moment representation (or the measure, connected with orthonormal polynomials) has compact support.

### Volodymyr Dmytrovych Koshmanenko (to his 70th birthday)

Methods Funct. Anal. Topology **20** (2014), no. 1, 1-2

### Yury Stefanovich Samoilenko (to 70th birthday anniversary)

Methods Funct. Anal. Topology **20** (2014), no. 2, 101-102

### Damir Zyamovich Arov (to 80th birthday anniversary)

Methods Funct. Anal. Topology **20** (2014), no. 4, 297-298

### Vadym Movsesovich Adamyan (to 75th birthday anniversary)

Methods Funct. Anal. Topology **20** (2014), no. 4, 299-300

### Trace formulae for Schrödinger operators on metric graphs with applications to recovering matching conditions

Yulia Ershova, Alexander V. Kiselev

Methods Funct. Anal. Topology **20** (2014), no. 2, 134-148

The paper is a continuation of the study started in [8]. Schrödinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of $\delta$ type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of $L_1$ and $C^M$ edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

### Delta-type solutions for a system of induction equations with discontinuous velocity field

A. I. Esina, A. I. Shafarevich

Methods Funct. Anal. Topology **20** (2014), no. 1, 17-33

We study asymptotic solutions of a Cauchy problem for induction equations describing magnetic field in a well conducting fluid. We assume that the coefficient (the velocity field of the fluid) changes rapidly in a small vicinity of a two-dimensional surface. We prove that the weak limit of the solution has delta-type singularity on this surface; in the case of a perfectly conducting fluid, we describe several regularizations of the problem with discontinuous coefficients which allow to define generalized solutions.

### On the a.c. spectrum of the 1D discrete Dirac operator

Sylvain Golénia, Tristan Haugomat

Methods Funct. Anal. Topology **20** (2014), no. 3, 252-273

In this paper, under some integrability condition, we prove that an electrical perturbation of the discrete Dirac operator has purely absolutely continuous spectrum for the one dimensional case. We reduce the problem to a non-self-adjoint Laplacian-like operator by using a spin up/down decomposition and rely on a transfer matrices technique.

### On behavior at infinity of solutions of parabolic differential equations in a Banach space

M. L. Gorbachuk, V. I. Gorbachuk

Methods Funct. Anal. Topology **20** (2014), no. 3, 274-283

For a differential equation of the form $y'(t) + Ay(t) = 0, \ t \in (0, \infty)$, where $A$ is the generating operator of a $C_{0}$-semigroup of linear operators on a Banach space $\mathfrak{B}$, we give conditions on the operator $A$, under which this equation is uniformly (uniformly exponentially) stable, that is, every its weak solution defined on the open semiaxis $(0, \infty)$ tends (tends exponentially) to 0 as $t \to \infty$. As distinguished from the previous works dealing only with solutions continuous at 0, in this paper no conditions on the behavior of a solution near 0 are imposed. In the case where the equation is parabolic, there always exist weak solutions which have singularities of any order. The criterions below not only generalize, but make more precise a number of earlier results in this direction.

### Schrödinger operators with non-symmetric zero-range potentials

Methods Funct. Anal. Topology **20** (2014), no. 1, 34-49

Non-self-adjoint Schrödinger operators $A_{\mathbf{T}}$ which correspond to non-symmetric zero-range potentials are investigated. For a given $A_{\mathbf{T}}$, a description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of $A_{\mathbf{T}}$ as a self-adjoint operator in a Krein space is studied, the problem of similarity of $A_{\mathbf{T}}$ to a self-adjoint operator in a Hilbert space is solved.

### Eigenfunction expansions associated with an operator differential equation non-linearly depending on a spectral parameter

Methods Funct. Anal. Topology **20** (2014), no. 1, 68-91

For an operator differential equation that depends on a spectral parameter in the Nevanlinna manner we obtain expansions in eigenfunctions.

### Existence theorems of the ω-limit states for conflict dynamical systems

Methods Funct. Anal. Topology **20** (2014), no. 4, 379-390

We introduce a notion of the conflict dynamical system in terms of probability measures, study the behavior of trajectories of such systems, and prove the existence theorems of the ω-limit states.

### Darboux transformation of generalized Jacobi matrices

Methods Funct. Anal. Topology **20** (2014), no. 4, 301-320

Let $\mathfrak{J}$ be a monic generalized Jacobi matrix, i.e. a three-diagonal block matrix of special form, introduced by M.~Derevyagin and V.~Derkach in 2004. We find conditions for a monic generalized Jacobi matrix $\mathfrak{J}$ to admit a factorization $\mathfrak{J}=\mathfrak{LU}$ with $\mathfrak{L}$ and $\mathfrak{U}$ being lower and upper triangular two-diagonal block matrices of special form. In this case the Darboux transformation of $\mathfrak{J}$ defined by $\mathfrak{J}^{(p)}=\mathfrak{UL}$ is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and the second kind, corresponding to the Darboux transformation $\mathfrak{J}^{(p)}$ are found.

### Spectral gaps of the Hill-Schrödinger operators with distributional potentials

Vladimir Mikhailets, Volodymyr Molyboga

Methods Funct. Anal. Topology **20** (2014), no. 4, 321-327

The paper studies the Hill-Schrödinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences that arise as lengths of spectral gaps of these operators. The space $H^\omega$ coincides with the H\"{o}rmander space $H^{\omega}_2\left(\mathbb{T}, \mathbb{R}\right)$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ belongs to Avakumovich's class $\mathrm{OR}$. In particular, if the functions $\omega$ are power, then these spaces coincide with the Sobolev spaces. The functions $\omega$ may be nonmonotonic.

### On generalized rezolvents and characteristic matrices of first-order symmetric systems

Methods Funct. Anal. Topology **20** (2014), no. 4, 328-348

We study general (not necessarily Hamiltonian) first-ordersymmetric system $J y'-B(t)y=\Delta(t) f(t)$ on an interval $\mathcal I=[a,b)$ with the regular endpoint $a$ and singular endpoint $b$. It isassumed that the deficiency indices $n_\pm(T_{\min})$ of thecorresponding minimal relation $T_{\min}$ in $L_\Delta^2(\mathcal I)$ satisfy$n_-(T_{\min})\leq n_+(T_{\min})$. We describe all generalized resolvents$y=R(\lambda)f, \; f\in L_\Delta^2(\mathcal I),$ of $T_{\min}$ in terms of boundary problemswith $\lambda$-depending boundary conditions imposed on regular andsingular boundary values of a function $y$ at the endpoints $a$and $b$ respectively. We also parametrize all characteristicmatrices $\Omega(\lambda)$ of the system immediately in terms of boundaryconditions. Such a parametrization is given both by the blockrepresentation of $\Omega(\lambda)$ and by the formula similar to thewell-known Krein formula for resolvents. These results develop the Straus' results on generalized resolvents and characteristicmatrices of differential operators.

### Hypercyclic composition operators on Hilbert spaces of analytic functions

Z. H. Mozhyrovska, A. V. Zagorodnyuk

Methods Funct. Anal. Topology **20** (2014), no. 3, 284-291

In the paper we consider composition operators on Hilbert spaces of analytic functions of infinitely many variables. In particular, we establish some conditions under which composition operators are hypercyclic and construct some examples of Hilbert spaces of analytic functions which do not admit hypercyclic operators of composition with linear operators.

### On the accelerants of non-self-adjoint Dirac operators

Methods Funct. Anal. Topology **20** (2014), no. 4, 349-364

We prove that there is a homeomorphism between the space of accelerants and the space of potentials of non-self-adjoint Dirac operators on a finite interval.

### Comment on 'A uniform boundedness theorem for locally convex cones' [W. Roth, Proc. Amer. Math. Soc. 126 (1998), 1973-1982]

Davod Saeedi, Ismail Nikoufar, Husain Saiflu

Methods Funct. Anal. Topology **20** (2014), no. 3, 292-295

In page 1975 of [W. Roth, *A uniform boundedness theorem for locally convex cones*, Proc. Amer. Math. Soc. **126** (1998), no.7, 1973-1982] we can see: In a locally convex vector space $E$ a barrel is defined to be an absolutely convex closed and absorbing subset $A$ of $E$. The set $U = \{(a,b)\in E^2,\ a-b\in A\}$ then is seen to be a barrel in the sense of Roth's definition. With a counterexample, we show that it is not enough for $U$ to be a barrel in the sense of Roth's definition. Then we correct this error with providing its converse and an application.

### Spectral analysis of metric graphs with infinite rays

Methods Funct. Anal. Topology **20** (2014), no. 4, 391-396

We conduct a detailed analysis for finite metric graphs that have a semi-infinite chain (a ray) attached to each vertex. We show that the adjacency matrix of such a graph gives rise to a selfadjoint operator that is unitary equivalent to a direct sum of a finite number of simplest Jacobi matrices. This permitted to describe spectrums of such operators and to explicitly construct an eigenvector decomposition.

### On a criterion of mutual adjointness for extensions of some nondensely defined operators

Methods Funct. Anal. Topology **20** (2014), no. 1, 50-58

In the paper the role of initial object is played by a pair of closed linear densely defined operators $L_0$ and $M_0$, where $L_0 \subset M_0^{\ast}:= L,$ acting in Hilbert space. A criterion of mutual adjointness for some classes of the extensions of finite-dimensional (non densely defined) restrictions of $L_0$ and $M_0$ are established. The main results are based on the theory of linear relations in Hilbert spaces and are formulated in the terms of abstract boundary operators.

### Some remarks on Hilbert representations of posets

Methods Funct. Anal. Topology **20** (2014), no. 2, 149–163

For a certain class of finite posets, we prove that all their irreducible orthoscalar representations are finite-dimensional and describe those, for which there exist essential (non-degenerate) irreducible orthoscalar representations.

### Spectral problem for a graph of symmetric Stieltjes strings

Methods Funct. Anal. Topology **20** (2014), no. 2, 164-174

A spectral problem generated by the Stieltjes string recurrence relations with a finite number of point masses on a connected graph is considered with Neumann conditions at pendant vertices and continuity and Kirchhoff conditions at interior vertices. The strings on the edges are supposed to be the same and symmetric with respect to the midpoint of the string. The characteristic function of such a problem is expressed via characteristic functions of two spectral problems on an edge: one with Dirichlet conditions at the both ends and the other one with the Neumann condition at one end and the Dirichlet condition at the other end. This permits to find values of the point masses and the lengths of the subintervals into which the masses divide the string from knowing the spectrum of the problem on the graph and the length of an edge. If the number of vertices is less than five then the spectrum uniquely determines the form of the graph.

### On the common point spectrum of pairs of self-adjoint extensions

Methods Funct. Anal. Topology **20** (2014), no. 1, 59-67

Given two different self-adjoint extensions of the same symmetric operator, we analyse the intersection of their point spectra. Some simple examples are provided.

### Factorization formulas for some classes of generalized $J$-inner matrix valued functions

Methods Funct. Anal. Topology **20** (2014), no. 4, 365-378

The class $\mathcal{U}_\kappa(j_{pq})$ of generalized $j_{pq}$-inner matrix valued functions (mvf's) %and its subclass $\mathcal{U}^r_\kappa(j_{pq})$ was introduced in [2]. For a mvf $W$ from a subclass $\mathcal{U}^r_\kappa(j_{pq})$ of $\mathcal{U}_\kappa(j_{pq})$ the notion of the right associated pair was introduced in [13] and some factorization formulas were found. In the present paper we introduce a dual subclass $\mathcal{U}^\ell_\kappa(j_{pq})$ and for every mvf $W\in \mathcal{U}^\ell_\kappa(j_{pq})$ a left associated pair $\{\beta_1,\beta_2\}$ is defined and factorization formulas for $W$ in terms of $\beta_1,\beta_2$ are found. The notion of a singular generalized $j_{pq}$-inner mvf $W$ is introduced and a characterization of singularity of $W$ is given in terms of associated pair.