# Authors Index, Vol. 21, 2015

### Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them

Methods Funct. Anal. Topology **21** (2015), no. 3, 199-224

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space $\mathfrak H$ with an exit in some larger Hilbert space $\mathfrak H\oplus\mathcal H$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Krein-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.

### Scale-invariant self-adjoint extensions of scale-invariant symmetric operators: continuous versus discrete

Miron B. Bekker, Martin J. Bohner, Mark A. Nudel'man, Hristo Voulov

Methods Funct. Anal. Topology **21** (2015), no. 1, 41-55

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous three articles on the subject. First we study general symmetric and scale-invariant operators on a Hilbert space. We show that if the index of defect of the operator under consideration is $(1,1)$, then the operator either does not admit any scale-invariant self-adjoint extension, or it admits exactly one scale-invariant self-adjoint extension, or it admits exactly two scale-invariant self-adjoint extensions, or all self-adjoint extensions are scale invariant. We then apply these results to the differential operator and the corresponding difference operator under consideration. For the continuous case, we show that the interval of the parameter, for which the differential operator is not semi-bounded, contains an infinite sequence of values for which all self-adjoint extensions are scale-invariant, while for the remaining values of the parameter from that interval, there are no scale-invariant self-adjoint extensions. For the corresponding difference operator, we show that if it is not semi-bounded, then it does not admit any scale-invariant self-adjoint extension. We also show that both differential and difference operators, at value(s) of the parameter that cor espond to the endpoint(s) of the interval(s) of semi-boundedness, have exactly one scale-invariant self-adjoint extension.

### Conservative L-systems and the Livšic function

S. Belyi, K. A. Makarov, E. Tsekanovskiĭ

Methods Funct. Anal. Topology **21** (2015), no. 2, 104-133

We study the connection between the classes of (i) Livsic functions $s(z),$ i.e., the characteristic functions of densely defined symmetric operators $\dot A$ with deficiency indices $(1, 1)$; (ii) the characteristic functions $S(z)$ of a maximal dissipative extension $T$ of $\dot A,$ i.e., the Mobius transform of $s(z)$ determined by the von Neumann parameter $\kappa$ of the extension relative to an appropriate basis in the deficiency subspaces; and (iii) the transfer functions $W_\Theta(z)$ of a conservative L-system $\Theta$ with the main operator $T$. It is shown that under a natural hypothesis {the functions $S(z)$ and $W_\Theta(z)$ are reciprocal to each other. In particular, $W_\Theta(z)=\frac{1}{S(z)}=-\frac{1}{s(z)}$ whenever $\kappa=0$. It is established that the impedance function of a conservative L-system with the main operator $T$ belongs to the Donoghue class if and only if the von Neumann parameter vanishes ($\kappa=0$). Moreover, we introduce the generalized Donoghue class and obtain the criteria for an impedance function to belong to this class. We also obtain the representation of a function from this class via the Weyl-Titchmarsh function. All results are illustrated by a number of examples.

### Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$

C. Boldrighini, A. Marchesiello, C. Saffirio

Methods Funct. Anal. Topology **21** (2015), no. 4, 302-314

In many models of Mathematical Physics, based on the study of a Markov chain $\widehat \eta= \{\eta_{t}\}_{t=0}^{\infty}$ on ${\mathbb Z}^d$, one can prove by perturbative arguments a contraction property of the stochastic operator restricted to a subspace of local functions $\mathcal H_{M}$ endowed with a suitable norm. We show, on the example of a model of random walk in random environment with mutual interaction, that the condition is enough to prove a Central Limit Theorem for sequences $\{f(S^{k}\widehat \eta)\}_{k=0}^{\infty}$, where $S$ is the time shift and $f$ is strictly local in space and belongs to a class of functionals related to the H\"older continuous functions on the torus $T^{1}$.

### On Fourier algebra of a locally compact hypergroup

A. A. Kalyuzhnyi, G. B. Podkolzin, Yu. A. Chapovsky

Methods Funct. Anal. Topology **21** (2015), no. 3, 246-255

We give sufficient conditions for the Fourier and the Fourier-Stieltjes spaces of a locally compact hypergroup to be Banach algebras.

### Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces

Iryna S. Chepurukhina, Aleksandr A. Murach

Methods Funct. Anal. Topology **21** (2015), no. 1, 6-21

We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on two-sided refined scales built on the base of inner product isotropic H\"ormander spaces. The regularity of the distributions forming these spaces are characterized by a real number and an arbitrary function that varies slowly at infinity in the sense of Karamata. For the generalized solutions to the problem, we prove theorems on a priori estimates and local regularity in these scales. As applications, we find new sufficient conditions under which the solutions have continuous classical derivatives of a prescribed order.

### Percolations and phase transitions in a class of random spin systems

Methods Funct. Anal. Topology **21** (2015), no. 3, 225-236

The aim of this paper is to give a review of recent results of Yu. Kondratiev, Yu. Kozitsky, T. Pasurek and myself on the multiplicity of Gibbs states (phase transitions) in infinite spin systems on random configurations, and provide a `pedestrian' route following Georgii–Haggstrom approach to (closely related to phase transitions) percolation problems for a class of random point processes.

### On a class of generalized Stieltjes continued fractions

Vladimir Derkach, Ivan Kovalyov

Methods Funct. Anal. Topology **21** (2015), no. 4, 315-335

With each sequence of real numbers ${\mathbf s}=\{s_j\}_{j=0}^\infty$ two kinds of continued fractions are associated, - the so-called $P-$fraction and a generalized Stieltjes fraction that, in the case when ${\mathbf s}=\{s_j\}_{j=0}^\infty$ is a sequence of moments of a probability measure on $\mathbb R_+$, coincide with the $J-$fraction and the Stieltjes fraction, respectively. A subclass $\mathcal H^{reg}$ of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For ${\mathbf s}\in\mathcal H^{reg}$ the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.

### Volodymyr Vasylyovych Sharko

Methods Funct. Anal. Topology **21** (2015), no. 1, 1-2

### Yurij Makarovych Berezansky (to 90th birthday anniversary)

Methods Funct. Anal. Topology **21** (2015), no. 2, 101–103

### Leonid Pavlovych Nizhnik (to 80th birthday anniversary)

Methods Funct. Anal. Topology **21** (2015), no. 4, 299-301

### Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle

Sergiy Maksymenko, Bohdan Feshchenko

Methods Funct. Anal. Topology **21** (2015), no. 1, 22-40

Let $f:M\to \mathbb{R}$ be a Morse function on a connected compact surface $M$, and $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be respectively the stabilizer and the orbit of $f$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$. In a series of papers the first author described the homotopy types of connected components of $\mathcal{S}(f)$ and $\mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or $\chi(M)<0$. Moreover, in two recent papers the authors considered special classes of smooth functions on $2$-torus $T^2$ and shown that the computations of $\pi_1\mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk and cylinder.

In the present paper we consider another class of Morse functions $f:T^2\to\mathbb{R}$ whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface $Q\subset T^2$, diffeomorphic with a cylinder, such that $\pi_1\mathcal{O}(f)$ is expressed via the fundamental group $\pi_1\mathcal{O}(f|_{Q})$ of the restriction of $f$ to $Q$.

This result holds for a larger class of smooth functions $f:T^2\to \mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple factors.

### Around Ovsyannikov's method

Methods Funct. Anal. Topology **21** (2015), no. 2, 134–150

We study existence, uniqueness, and a limiting behavior of solutions to an abstract linear evolution equation in a scale of Banach spaces. The generator of the equation is a perturbation of the operator which satisfies the classical assumptions of Ovsyannikov's method by a generator of a $C_0$-semigroup acting in each of the spaces of the scale. The results are (slightly modified) abstract version of those considered in [10] for a particular equation. An application to a birth-and-death stochastic dynamics in the continuum is considered.

### On regularity of linear summation methods of Taylor series

Methods Funct. Anal. Topology **21** (2015), no. 1, 56-68

The paper specifies necessary and sufficient conditions for regularity of an infinite matrix of real numbers, which determines some summation method for a class of functions that are analytic on the unit disk and continuous on the closed circle.

### Erratum: F. Gesztesy, S. Hofmann, and R. Nichols, MFAT 19 (2013), no.3, 227-259

Fritz Gesztesy, Steve Hofmann, Roger Nichols

Methods Funct. Anal. Topology **21** (2015), no. 1, 99-99

### Some applications of almost analytic extensions to operator bounds in trace ideals

Methods Funct. Anal. Topology **21** (2015), no. 2, 151–169

Using the Davies-Helffer-Sjostrand functional calculus based on almost analytic extensions, we address the following problem: Given a self-adjoint operator $S$ in $\mathcal H$, and functions $f$ in an appropriate class, for instance, $f \in C_0^{\infty}(\mathbb R)$, how to control the norm $\|f(S)\|_{\mathcal B(\mathcal H)}$ in terms of the norm of the resolvent of $S$, $\|(S - z_0 I_{\mathcal H})^{-1}\|_{\mathcal B(\mathcal H)}$, for some $z_0 \in \mathbb C\backslash\mathbb R$. We are particularly interested in the case where $\mathcal B(\mathcal H)$ is replaced by a trace ideal, $\mathcal B_p(\mathcal H)$, $p \in [1,\infty)$.

### On complex perturbations of infinite band Schrödinger operators

Methods Funct. Anal. Topology **21** (2015), no. 3, 237-245

Let $H_0=-\frac{d^2}{dx^2}+V_0$ be an infinite band Schrödinger operator on $L^2(\mathbb R)$ with a real-valued potential $V_0\in L^\infty(\mathbb R)$. We study its complex perturbation $H=H_0+V$, defined in the form sense, and obtain the Lieb-Thirring type inequ\-alities for the rate of convergence of the discrete spectrum of $H$ to the joint essential spectrum. The assumptions on $V$ vary depending on the sign of $Re V$.

### On the structure of solutions of operator-differential equations on the whole real axis

Methods Funct. Anal. Topology **21** (2015), no. 2, 170–178

We consider differential equations of the form $\left(\frac{d^{2}}{dt^{2}} - B\right)^{m}y(t) = f(t)$, $m \in \mathbb{N}, \ t \in (-\infty, \infty)$, where $B$ is a positive operator in a Banach space $\mathfrak{B}, \ f(t)$ is a bounded continuous vector-valued function on $(-\infty, \infty)$ with values in $\mathfrak{B}$, and describe all their solutions. In the case, where $f(t) \equiv 0$, we prove that every solution of such an equation can be extended to an entire $\mathfrak{B}$-valued function for which the Phragmen-Lindel\"{o}f principle is fulfilled. It is also shown that there always exists a unique bounded on $\mathbb{R}^{1}$ solution, and if $f(t)$ is periodic or almost periodic, then this solution is the same as $f(t)$.

### Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis

Methods Funct. Anal. Topology **21** (2015), no. 4, 336-360

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, and on spaces of regular test and generalized functions of the Levy white noise analysis. In this paper we make the next step: introduce and study operators of stochastic differentiation on spaces of test functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using Lytvynov's generalization of the chaotic representation property. This can be considered as a contribution in a further development of the Levy white noise analysis.

### Fractional contact model in the continuum

Anatoly N. Kochubei, Yuri G. Kondratiev

Methods Funct. Anal. Topology **21** (2015), no. 2, 179–187

We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a behavior of time-dependent correlation functions, essentially different from the one known for the standard contact model.

### Stability of N-extremal measures

Matthias Langer, Harald Woracek

Methods Funct. Anal. Topology **21** (2015), no. 1, 69-75

A positive Borel measure $\mu$ on $\mathbb R$, which possesses all power moments, is N-extremal if the space of all polynomials is dense in $L^2(\mu)$. If, in addition, $\mu$ generates an indeterminate Hamburger moment problem, then it is discrete. It is known that the class of N-extremal measures that generate an indeterminate moment problem is preserved when a finite number of mass points are moved (not ``removed''!). We show that this class is preserved even under change of infinitely many mass points if the perturbations are asymptotically small. Thereby ``asymptotically small'' is understood relative to the distribution of ${\rm supp}\mu$; for example, if ${\rm supp}\mu=\{n^\sigma\log n:\,n\in\mathbb N\}$ with some $\sigma>2$, then shifts of mass points behaving asymptotically like, e.g. $n^{\sigma-2}[\log\log n]^{-2}$ are permitted.

### The projection spectral theorem and Jacobi fields

Methods Funct. Anal. Topology **21** (2015), no. 2, 188–198

We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to L\'evy white noise measures.

### On the Carleman ultradifferentiable vectors of a scalar type spectral operator

Methods Funct. Anal. Topology **21** (2015), no. 4, 361-369

A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space.

### Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation

Methods Funct. Anal. Topology **21** (2015), no. 1, 76-98

Let $H$ be a finite dimensional Hilbert space and let $[H]$ be the set of all li ear operators in $H$. We consider first-order symmetric system $J y'-B(t)y=\Lambda(t) f(t)$ with $[H]$-valued coefficients defined on an interval $[a,b) $ with the regular endpoint $a$. It is assumed that the corresponding minimal relation $T_{\rm min}$ has maximally possible deficiency index $n_+(T_{\rm min})=\dim H$. The main result is a parametrization of all characteristic matrices and pseudospectral (spectral) functions of a given system by means of a Nevanlinna type boundary parameter $\tau$. Similar parametrization for regular systems has earlier been obtained by Langer and Textorius. We also show that the coefficients of the parametrization form the matrix $W(\lambda)$ with the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.

### Spectral and pseudospectral functions of Hamiltonian systems: development of the results by Arov-Dym and Sakhnovich

Methods Funct. Anal. Topology **21** (2015), no. 4, 370-402

The main object of the paper is a Hamiltonian system $J y'-B(t)y=\lambda\Delta(t) y$ defined on an interval $[a,b) $ with the regular endpoint $a$. We define a pseudo\-spectral function of a singular system as a matrix-valued distribution function such that the generalized Fourier transform is a partial isometry with the minimally possible kernel. Moreover, we parameterize all spectral and pseudospectral functions of a given system by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov-Dym and Sakhnovich in this direction.

### Erratum: "On generalized resolvents and characteristic matrices of first-order symmetric systems'', MFAT, Vol. 20, No. 4, 2014, pp. 328–348

Methods Funct. Anal. Topology **21** (2015), no. 1, 100-100

### Inverse spectral problems for Jacobi matrix with finite perturbed parameters

Methods Funct. Anal. Topology **21** (2015), no. 3, 256-265

For Jacobi matrices with finitely perturbed parameters, we get an explicit re\-presentation of the Weyl function, and solve inverse spectral problems, that is, we recover Jacobi matrices from spectral data. For the spectral data, we take the following: the spectral density of the absolutely continuous spectrum, with or without all the eigenvalues; the numerical parameters of the representation of one component of the vector-eigenfunction in terms of Chebyshev polynomials. We prove that these inverse problems have a unique solution, or only a finite number of solutions.

### The multi-dimensional truncated moment problem: maximal masses

Methods Funct. Anal. Topology **21** (2015), no. 3, 266-281

Given a subset $\mathcal K$ of $\mathbb R^d$ and a linear functional $L$ on the polynomials $\mathbb R^d_{2n}[\underline{x}]$ in $d$ variables and of degree at most $2n$ the truncated $\mathcal K$-moment problem asks when there is a positive Borel measure $\mu$ supported by $\mathcal K$ such that $L(p)=\int p\, d\mu$ for $p\in \mathbb R^d_{2n}[\underline{x}]$. For compact sets $\mathcal K$ we investigate the maximal mass of all representing measures at a given point of $\mathcal K$. Various characterizations of this quantity and related properties are developed and a close link to zeros of positive polynomials is established.

### Topological equivalence to a projection

Methods Funct. Anal. Topology **21** (2015), no. 1, 3-5

We present a necessary and sufficient condition for a continuous function on a plane to be topologically equivalent to a projection onto one of the coordinates.

### Tannaka-Krein duality for compact quantum group coactions (survey)

Methods Funct. Anal. Topology **21** (2015), no. 3, 282-298

The last decade saw an appearance of a series of papers containing a very interesting development of the Tannaka-Krein duality for compact quantum group coactions on $C^*$-algebras. The present survey is intended to present the main ideas and constructions underlying this development.