Authors Index, Vol. 18, 2012

On the completeness of general boundary value problems for $2 \times 2$ first-order systems of ordinary differential equations

A. V. Agibalova, M. M. Malamud, L. L. Oridoroga

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 4-18

Let $B={\rm diag} (b_1^{-1}, b_2^{-1}) \not = B^*$ be a $2\times 2$ diagonal matrix with \break $b_1^{-1}b_2 \notin{\Bbb R}$ and let $Q$ be a smooth $2\times 2$ matrix function. Consider the system $$-i B y'+Q(x)y=\lambda y, \; y= {\rm col}(y_1,y_2), \; x\in[0,1],$$ of ordinary differential equations subject to general linear boundary conditions $U_1(y) = U_2(y) = 0.$ We find sufficient conditions on $Q$ and $U_j$ that guaranty completeness of root vector system of the boundary value problem. Moreover, we indicate a condition on $Q$ that leads to a completeness criterion in terms of the linear boundary forms $U_j,\ j\in \{1,2\}.$

Operator-norm approximations of holomorphic one-parameter semigroups of contractions in Hilbert spaces

Yury Arlinskiĭ

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 101-110

We establish the operator-norm convergence of the Iosida and Dunford-Segal approximation formulas for one-parameter semigroups of the class $C_0$, gene ated by maximal sectorial generators in separable Hilbert spaces. Our approach is essentially based on the Crouzeix-Delyon theorem [8] related to the generalization of the von Neumann inequality.

Realizations of stationary stochastic processes: applications of passive system theory

D. Z. Arov, N. A. Rozhenko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 4, 305-331

In the paper, we investigate realizations of a $p$-dimensionalregular weak stationary discrete time stochastic process $y(t)$ asthe output data of a passive linear bi-stable discrete timedynamical system. The state $x(t)$ is assumed to tend to zero as ttends to $-\infty$, and the input data is the $m$-dimensional whitenoise. The results are based on author's development of the Darlington method for passive impedance systems with losses of thescattering channels. Here we establish that considering realizationfor a discrete time process is possible, if the spectral density $\rho(e^{i\mu})$ of the process is a nontangential boundary value ofa matrix valued meromorphic function $\rho(z)$ of rank $m$ withbounded Nevanlinna characteristic in the open unitdisk. A parameterization of all such realizations is given and minimal,optimal minimal, and *-optimal minimal realizations areobtained. The last two coincide with those which are obtained by Kalman filters. This is a further development of the Lindquist-Picci realization theory.

A new metric in the study of shift invariant subspaces of $L^2(\mathbb{R}^n)$

M. S. Balasubramani, V. K. Harish

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 214-219

A new metric on the set of all shift invariant subspaces of $L^2(\mathbb{R}^n)$ is defined and the properties are studied. The limit of a sequence of principal shift invariant subspaces under this metric is principal shift invariant is proved. Also, the uniform convergence of a sequence of local trace functions is characterized in terms of convergence under this new metric.

Sectorial classes of inverse Stieltjes functions and L-systems

S. Belyi, E. Tsekanovskiĭ

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 201-213

We introduce sectorial classes of inverse Stieltjes functions actingon a finite-dimensional Hilbert space as well as scalar classes ofinverse Stieltjes functions based upon their limit behavior at minusinfinity and at zero. It is shown that a function from theseclasses can be realized as the impedance function of a singularL-system and the operator $\tilde A$ in a rigged Hilbert spaceassociated with the realizing system is sectorial. Moreover, it isestablished that the knowledge of the limit values of the scalarimpedance function allows to find an angle of sectoriality of theoperator $\tilde A$ as well as the exact angle of sectoriality of theaccretive main operator $T$ of such a system. The corresponding newformulas connecting the limit values of the impedance function andthe angle of sectoriality of $\tilde A$ are provided. Application ofthese formulas yields that the exact angle of sectoriality ofoperators $\tilde A$ and $T$ is the same if and only if the limitvalue at zero of the corresponding impedance function (along thenegative $x$-axis) is equal to zero. Examples of the realizingL-systems based upon the Schrodinger operator on half-line arepresented.

Linearization of double-infinite Toda lattice by means of inverse spectral problem

Yurij M. Berezansky

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 19-54

The author earlier in [3, 4, 6, 7] proposed some way of integration the Cauchy problem for semi-infinite Toda lattices using the inverse spectral problem for Jacobi matrices. Such a way for double-infinite Toda lattices is more complicated and was proposed in [9]. This article is devoted to a detailed account of the result [3, 4, 6, 7, 9] . It is necessary to note that in the case of double-infinite lattices we cannot give a general solution of the corresponding linear system of differential equations for spectral matrix. Therefore, in this case the corresponding results can only be understood as a procedure of finding the solution of the Toda lattice.

One remark concerning double-infinite Toda lattice

Yurij M. Berezansky

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 4, 332-342

We propose the power moment approach to investigation of double-infinite Toda lattices, which was contained in author's article [6]. As a result, we give the main theorem from [6] in a more effective form.

Maps of 2-manifolds into the plane

A. M. Bondarenko, A. O. Prishlyak

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 220-229

We define a $p$-graph and describe how it changes under isotopy of projections for classification of maps of $2$-manifolds into plane. The problem of graph implementation and maps classification are considered.

Intertwining properties of bounded linear operators on the Bergman space

Namita Das

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 230-242

In this paper we find conditions on $\phi, \psi\in L^{\infty}(\mathbb D)$ that are necessary and sufficient for the existence of bounded linear operators $S,T$ from the Bergman space $L_a^2(\mathbb D)$ into itself such that for all $z\in \mathbb D,$ $ \phi(z)=\langle Sk_z, k_z, \rangle, \psi(z)=\langle Tk_z, k_z \rangle$ and $C_aS=TC_a$ for all $a\in \mathbb D$ where $C_af=f\circ \phi_a$ for all $f\in L_a^2(\mathbb D)$ and $\phi_a(z)=\frac{a-z}{1-\bar a z}, z\in \mathbb D.$ Applications of the results are also discussed.

The complex moment problem in the exponential form with direct and inverse spectral problems for the block Jacobi type correspondence matrices

Mykola E. Dudkin

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 111-139

We present a new generalization of the connection of the classical power moment problem with spectral theory of Jacobi matrices. In the article we propose an analog of Jacobi matrices related to the complex moment problem in the case of exponential form and to the system of orthonormal polynomials with respect to some measure with the compact support on the complex plane. In our case we obtain two matrices that have block three-diagonal structure and acting in the space of $l_2$ type as commuting self-adjoint and unitary operators. With this connection we prove the one-to-one correspondence between the measures defined on a compact set in the complex plane and the couple of block three-diagonal Jacobi type matrices. For simplicity we consider in this article only a bounded self-adjoint operator.

Anatolii Gordeevich Kostyuchenko

Editorial Board

Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 1-3

Trace formulae for graph Laplacians with applications to recovering matching conditions

Yulia Ershova, Alexander V. Kiselev

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 4, 343-359

Graph Laplacians on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of either $\delta$ or $\delta'$ type. In either case, an infinite series of trace formulae which link together two different graph Laplacians provided that their spectra coincide is derived. Applications are given to the problem of reconstructing matching conditions for a graph Laplacian based on its spectrum.

Kawasaki dynamics in the continuum via generating functionals evolution

D. L. Finkelshtein, Yu. G. Kondratiev, M. J. Oliveira

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 55-67

We construct the time evolution of Kawasaki dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, which leads to a local (in time) solution. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.

Schrödinger operators with $(\alpha\delta'+\beta \delta)$-like potentials: norm resolvent convergence and solvable models

Yuriy Golovaty

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 243-255

For real functions $\Phi$ and $\Psi$ that are integrable and compactly supported, we prove the norm resolvent convergence, as $\varepsilon\to0$, of a family $S_\varepsilon$ of one-dimensional Schrödinger operators on the line of the form $$ S_\varepsilon= -\frac{d^2}{d x^2}+\alpha\varepsilon^{-2}\Phi(\varepsilon^{-1}x)+\beta\varepsilon^{-1}\Psi(\varepsilon^{-1}x). $$ The limit results are shape-dependent: without reference to the convergence of potentials in the sense of distributions the limit operator $S_0$ exists and strongly depends on the pair $(\Phi,\Psi)$. A class of nontrivial point interactions which are formally related the pseudo-Hamiltonian $-\frac{d^2}{dx^2}+\alpha\delta'(x)+\beta\delta(x)$ is singled out. The limit behavior, as $\varepsilon\to 0$, of the scattering data for such potentials is also described.

The Dirichlet problem for differential equations in a Banach space

M. L. Gorbachuk, V. I. Gorbachuk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 140-151

In the paper, we consider an abstract differential equation of the form $\left(\frac{\partial^{2}}{\partial t^{2}}- B \right)^{m}y(t) = 0$, where $B$ is a positive operator in a Banach space $\mathfrak{B}$. For solutions of this equation on $(0, \infty)$, it is established the analogue of the Phragmen-Lindelof principle on the basis of which we show that the Dirichlet problem for the above equation is uniquely solvable in the class of vector-valued functions admitting an exponential estimate at infinity. The Dirichlet data may be both usual and generalized with respect to the operator $-B^{1/2}$.The formula for the solution is given, and some applications to partial differential equations are adduced.

Self-adjointness of Schrödinger operators with singular potentials

Rostyslav O. Hryniv, Yaroslav V. Mykytyuk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 152-159

We study one-dimensional Schrödinger operators $S$ with real-valued distributional potentials $q$ in $W^{-1}_{2,\mathrm{loc}}(\mathbb R)$ and prove an extension of the Povzner-Wienholtz theorem on self-adjointness of bounded below $S$ thus providing additional information on its domain. The results are further specified for $q\in W^{-1}_{2,\mathrm{unif}}(\mathbb R)$.

Clark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis

N. A. Kachanovsky

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 160-175

In the classical Gaussian analysis the Clark-Ocone formula can be written in the form $$ F=\mathbf EF+\int\mathbf E_t\partial_t FdW_t, $$ where the function (the random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ denotes the expectation; $\mathbf E_t$ denotes the conditional expectation with respect to the full $\sigma$-algebra that is generated by a Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ is the Hida derivative of $F$; $\int\circ (t)dW_t$ denotes the It\^o stochastic integral with respect to the Wiener process. This formula has applications in the stochastic analysis and in the financial mathematics. In this paper we generalize the Clark-Ocone formula to spaces of test and generalized functions of the so--called Meixner white noise analysis, in which instead of the Gaussian measure one uses the so--called generalized Meixner measure $\mu$ (depending on parameters, $\mu$ can be the Gaussian, Poissonian, Gamma measure etc.). In particular, we study properties of integrands in our (Clark-Ocone type) formulas.

On self-adjontness of 1-D Schrödinger operators with $\delta$-interactions

I. I. Karpenko, D. L. Tyshkevich

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 4, 360-372

In the present work we consider the Schrödinger operator $\mathrm{H_{X,\alpha}}=-\mathrm{\frac{d^2}{dx^2}}+\sum_{n=1}^{\infty}\alpha_n\delta(x-x_n)$ acting in $L^2(\mathbb{R}_+)$. We investigate and complete the conditions of self-adjointness and nontriviality of deficiency indices for $\mathrm{H_{X,\alpha}}$ obtained in [13]. We generalize the conditions found earlier in the special case $d_n:=x_{n}-x_{n-1}=1/n$, $n\in \mathbb{N}$, to a wider class of sequences $\{x_n\}_{n=1}^\infty$. Namely, for $x_n=\frac{1}{n^{\gamma}\ln^\eta n}$ with $\langle\gamma,\eta \rangle\in(1/2,\,1)\!\times\!(-\infty,+\infty)\:\cup\:\{1\}\!\times\!(-\infty,1]$, the description of asymptotic behavior of the sequence $\{\alpha_n\}_{n=1}^{\infty}$ is obtained for $\mathrm{H_{X,\alpha}}$ either to be self-adjoint or to have nontrivial deficiency indices.

Sturm type oscillation theorems for equations with block-triangular matrix coefficients

A. M. Kholkin, F. S. Rofe-Beketov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 176-188

A relation is established between spectral and oscillation properties of the problem on a finite interval and a semi-axis for second order differential equations with block-triangular matrix coefficients.

Controlled fusion frames

Amir Khosravi, Kamran Musazadeh

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 256-265

We use two appropriate bounded invertible operators to define a controlled fusion frame with optimal fusion frame bounds to improve the numerical efficiency of iterative algorithms for inverting the fusion frame operator. We show that controlled fusion frames as a generalization of fusion frames give a generalized way to obtain numerical advantage in the sense of preconditioning to check the fusion frame condition. Also, we consider locally controlled frames for each locally space to obtain new globally controlled frames for our Hilbert space. We develop some well known results in fusion frames to the controlled fusion frames case.

Abstract interpolation problem in generalized Nevanlinna classes

Evgen Neiman

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 266-287

The abstract interpolation problem (AIP) in the Schur class was posed by V. Katznelson, A. Kheifets, and P. Yuditskii in 1987. In the present paper we consider an analog of AIP for the generalized Nevanlinna class $N_κ(L)$ in the nondegenerate case. We associate with the data set of the AIP a symmetric linear relation $\hat A$ acting in a Pontryagin space. The description of all solutions of the AIP is reduced to the problem of description of all $L$-resolvents of this symmetric linear relation $\hat A$. The latter set is parametrized by application of the indefinite version of Kreın’s representation theory for symmetric linear relations in Pontryagin spaces developed by M. G. Kreın and H. Langer in [22] and a formula for the $L$-resolvent matrix obtained by V. Derkach and M. Malamud in [11].

Inverse eigenvalue problems for nonlocal Sturm-Liouville operators on a star graph

L. P. Nizhnik

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 68-78

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

Representations of relations with orthogonality condition and their deformations

V. L. Ostrovskyi, D. P. Proskurin, R. Y. Yakymiv

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 4, 373-386

Irreducible representations of $*$-algebras $A_q$ generated by relations of the form $a_i^*a_i+a_ia_i^*=1$, $i=1,2$, $a_1^*a_2=qa_2a_1^*$, where $q\in (0,1)$ is fixed, are classified up to the unitary equivalence. The case $q=0$ is considered separately. It is shown that the $C^*$-algebras $\mathcal{A}_q^F$ and $\mathcal{A}_0^F$ generated by operators of Fock representations of $A_q$ and $A_0$ are isomorphic for any $q\in (0,1)$. A realisation of the universal $C^*$-algebra $\mathcal{A}_0$ generated by $A_0$ as an algebra of continuous operator-valued functions is given.

On characteristic functions of operators on equilateral graphs

V. Pivovarchik, O. Taystruk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 189-197

Known connection between discrete and continuous Laplacians in case of same symmetric potential on the edges of a quantum graph is used to construct characteristic functions of quantum graphs and to find some parameters of graphs using spectra of boundary value problems.

Eigenvalue asymptotics of perturbed self-adjoint operators

A. A. Shkalikov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 79-89

We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies the "local" subordinate condition $\|B\varphi_k\| \leqslant b\mu_k^{\beta}$ with some $\beta<1$ and $b>0$. Here $\{\varphi_k\}_{k=1}^\infty$ is an orthonormal system of the eigenvectors of the operator $T$ corresponding to the eigenvalues $\{\mu_k\}_{k=1}^\infty$. We introduce the concept of $\alpha$-non-condensing sequence and prove the theorem on the comparison of the eigenvalue-counting functions of the operators $T$ and $T+B$. Namely, it is shown that if $\{\mu_k\}$ is $\alpha-$non-condensing then $$ |n(r,T)- n(r, T+B)| \leqslant C\left[ n(r+ar^\gamma,\, T) - n(r-ar^\gamma,\, T) \right] +C_1 $$ with some constants $C, C_1, a$ and $\gamma = \max(0,\, \beta,\, 2\beta +\alpha -1)\in [0,1)$.

The integration of operator-valued functions with respect to vector-valued measures

Volodymyr Tesko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 3, 288-304

We investigate the $H$-stochastic integral introduced in [24]. It is known that this integral generalizes the classical Ito stochastic integral and the Ito integral on a Fock space. In the present paper we construct and study an extension of the $H$-stochastic integral which will generalize the Hitsuda-Skorokhod integral.

Recovering arbitrary order differential operators on noncompact star-type graphs

V. Yurko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 1, 90-100

We study an inverse spectral problem for arbitrary order ordinary differential equations on noncompact star-type graphs. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are a generalization of the Weyl function for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

The Nevanlinna-type formula for the matrix Hamburger moment problem

S. M. Zagorodnyuk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 4, 387-400

In this paper we obtain a Nevanlinna-type formula for the matrix Hamburger moment problem. We only assume that the problem is solvable and has more than one solution. We express the matrix coefficients of the corresponding linear fractional transformation in terms of the prescribed moments. Necessary and sufficient conditions for the determinacy of the moment problem in terms of the given moments are obtained.

On commuting symmetric operators

S. M. Zagorodnyuk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 18 (2012), no. 2, 198-200

In this paper we present some conditions for a pair of commuting symmetric operators with a joint invariant dense domain in a Hilbert space, to have a commuting self-adjoint extension in the same space. The remarkable Godic-Lucenko theorem allows to get a convenient description of all such extensions.

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