# Authors Index, Vol. 23, 2017

### Transformations of Nevanlinna operator-functions and their fixed points

Methods Funct. Anal. Topology **23** (2017), no. 3, 212-230

We give a new characterization of the class ${\bf N}^0_{\mathfrak M}[-1,1]$ of the operator-valued in the Hilbert space ${\mathfrak M}$ Nevanlinna functions that admit representations as compressed resolvents ($m$-functions) of selfadjoint contractions. We consider the auto\-morphism ${\bf \Gamma}:$ $M(\lambda){\mapsto}M_{{\bf \Gamma}} (\lambda):=\left((\lambda^2-1)M(\lambda)\right)^{-1}$ of the class ${\bf N}^0_{\mathfrak M}[-1,1]$ and construct a realization of $M_{{\bf \Gamma}}(\lambda)$ as a compressed resolvent. The unique fixed point of ${\bf\Gamma}$ is the $m$-function of the block-operator Jacobi matrix related to the Chebyshev polynomials of the first kind. We study a transformation ${\bf\widehat \Gamma}:$ ${\mathcal M}(\lambda)\mapsto {\mathcal M}_{{\bf\widehat \Gamma}}(\lambda) :=-({\mathcal M}(\lambda)+\lambda I_{\mathfrak M})^{-1}$ that maps the set of all Nevanlinna operator-valued functions into its subset. The unique fixed point ${\mathcal M}_0$ of ${\bf\widehat \Gamma}$ admits a realization as the compressed resolvent of the "free" discrete Schrödinger operator ${\bf\widehat J}_0$ in the Hilbert space ${\bf H}_0=\ell^2({\mathbb N}_0)\bigotimes{\mathfrak M}$. We prove that ${\mathcal M}_0$ is the uniform limit on compact sets of the open upper/lower half-plane in the operator norm topology of the iterations $\{{\mathcal M}_{n+1}(\lambda)=-({\mathcal M}_n(\lambda)+\lambda I_{\mathfrak M})^{-1}\}$ of ${\bf\widehat\Gamma}$. We show that the pair $\{{\bf H}_0,{\bf \widehat J}_0\}$ is the inductive limit of the sequence of realizations $\{\widehat{\mathfrak H}_n,\widehat A_n\} $ of $\{{\mathcal M}_n\}$. In the scalar case $({\mathfrak M}={\mathbb C})$, applying the algorithm of I.S. Kac, a realization of iterates $\{{\mathcal M}_n(\lambda)\}$ as $m$-functions of canonical (Hamiltonian) systems is constructed.

### Representation of isometric isomorphisms between monoids of Lipschitz functions

Methods Funct. Anal. Topology **23** (2017), no. 4, 309-319

We prove that each isometric isomorphism between the monoids of all nonnegative $1$-Lipschitz maps defined on invariant metric groups and equipped with the inf-convolution law, is given canonically from an isometric isomorphism between their groups of units.

### On Fourier algebra of a hypergroup constructed from a conditional expectation on a locally compact group

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

Methods Funct. Anal. Topology **23** (2017), no. 1, 37-50

We prove that the Fourier space of a hypergroup constructed from a conditional expectation on a locally compact group has a Banach algebra structure.

### $A$-regular–$A$-singular factorizations of generalized $J$-inner matrix functions

Volodymyr Derkach, Olena Sukhorukova

Methods Funct. Anal. Topology **23** (2017), no. 3, 231-251

Let $J$ be an $m\times m$ signature matrix, i.e., $J=J^*=J^{-1}$. An $m\times m$ mvf (matrix valued function) $W(\lambda)$ that is meromorphic in the unit disk $\mathbb{D}$ is called $J$-inner if $W(\lambda)JW(\lambda)^*\leq J$ for every $\lambda$ from $\mathfrak{h}_W^+$, the domain of holomorphy of $W$, in ${\mathbb{D}}$, and $W(\mu)JW(\mu)^*= J$ for a.e. $\mu\in\mathbb{T}=\partial \mathbb{D}$. A $J$-inner mvf $W(\lambda)$ is called $A$-singular if it is outer and it is called right $A$-regular if it has no non-constant $A$-singular right divisors. As was shown by D. Arov [18] every $J$-inner mvf admits an essentially unique $A$-regular--$A$-singular factorization $W=W^{(1)}W^{(2)}$. In the present paper this factorization result is extended to the class ${\mathcal U}_\kappa^r(J)$ of right generalized $J$-inner mvf's introduced in~\cite{DD09}. The notion and criterion of $A$-regularity for right generalized $J$-inner mvf's are presented. The main result of the paper is that we find a criterion for existence of an $A$-regular--$A$-singular factorization for a rational generalized $J$-inner mvf.

### Myroslav Lvovych Gorbachuk (Obituary)

Methods Funct. Anal. Topology **23** (2017), no. 1, 1-2

### Valentina Ivanivna Gorbachuk (to 80th birthday anniversary)

Methods Funct. Anal. Topology **23** (2017), no. 3, 207-208

### Eduard R. Tsekanovskiĭ (to 80th birthday anniversary)

Methods Funct. Anal. Topology **23** (2017), no. 3, 208-210

### Some remarks on operators of stochastic differentiation in the Lévy white noise analysis

Methods Funct. Anal. Topology **23** (2017), no. 4, 320-345

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 Gaussian white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of so-called normally ordered stochastic equations. During recent years, operators of stochastic differentiation were introduced and studied, in particular, on spaces of regular and nonregular test and generalized functions of the Lévy white noise analysis, in terms of Lytvynov's generalization of the chaotic representation property. But, strictly speaking, the existing theory in the "regular case" is incomplete without one more class of operators of stochastic differentiation, in particular, the mentioned operators are required in calculation of the commutator between the extended stochastic integral and the operator of stochastic differentiation. In the present paper we introduce this class of operators and study their properties. In addition, we establish a relation between the introduced operators and the corresponding operators on the spaces of nonregular test functions. The researches of the paper can be considered as a contribution to a further development of the Lévy white noise analysis.

### Evolution of correlation operators of large particle quantum systems

Methods Funct. Anal. Topology **23** (2017), no. 2, 123-132

The paper deals with the problem of a rigorous description of the evolution of states of large particle quantum systems in terms of correlation operators. A nonperturbative solution to a Cauchy problem of a hierarchy of nonlinear evolution equations for a sequence of marginal correlation operators is constructed. Moreover, in the case where the initial states are specified by a one-particle density operator, the mean field scaling asymptotic behavior of the constructed marginal correlation operators is considered.

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

M. L. Gorbachuk, V. M. Gorbachuk

Methods Funct. Anal. Topology **23** (2017), no. 2, 108-122

For a differential equation of the form $y''(t) - By(t) = 0, \ t \in (0, \infty)$, where $B$ is a weakly positive linear operator in a Banach space $\mathfrak{B}$, the conditions on the operator $B$, under which this equation is uniformly or uniformly exponentially stable are given. As distinguished from earlier works dealing only with continuous at 0 solutions, in this paper no conditions on behavior of a solution near 0 are imposed.

### On universal coordinate transform in kinematic changeable sets

Methods Funct. Anal. Topology **23** (2017), no. 2, 133-154

This work is devoted to a study of abstract coordinate transforms in kinematic changeable sets. Investigations in this direction may be interesting for astrophysics, because there exists a hypothesis that, in a large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws acting in a neighborhood of our solar System.

### The Liouville property for harmonic functions on groups and hypergroups

Methods Funct. Anal. Topology **23** (2017), no. 1, 3-25

A survey is given on the Liouville property of harmonic functions on groups and hypergroups. The discussion of a characterization of that property in terms of the underlying algebraic structures yields interesting open problems.

### Infinitesimal generators of invertible evolution families

Methods Funct. Anal. Topology **23** (2017), no. 1, 26-36

A logarithm representation of operators is introduced as well as a concept of pre-infinitesimal generator. Generators of invertible evolution families are represented by the logarithm representation, and a set of operators represented by the logarithm is shown to be associated with analytic semigroups. Consequently generally-unbounded infinitesimal generators of invertible evolution families are characterized by a convergent power series representation.

### On Barcilon’s formula for Krein’s string

Methods Funct. Anal. Topology **23** (2017), no. 3, 270-276

We find conditions on two sequences of positive numbers that are sufficient for the sequences to be the Neumann and the Dirichlet spectra of a Krein string such that Barcilon’s formula holds true.

### Overdamped modes and optimization of resonances in layered cavities

Illya M. Karabash, Olga M. Logachova, Ievgen V. Verbytskyi

Methods Funct. Anal. Topology **23** (2017), no. 3, 252-260

We study the problem of optimizing the imaginary parts $\mathrm{Im}\, \omega$ of quasi-normal-eigenvalues $\omega$ associated with the equation $y'' = -\omega^2 B y $. It is assumed that the coefficient $B(x)$, which describes the structure of an optical or mechanical resonator, is constrained by the inequalities $0 \le b_1 \le B(x) \le b_2 $. Extremal quasi-normal-eigenvalues belonging to the imaginary line ${\mathrm{i}} {\mathbb R}$ are studied in detail. As an application, we provide examples of $\omega$ with locally minimal $|\mathrm{Im}\, \omega|$ (without additional restrictions on $\mathrm{Re}\, \omega$) and show that a structure generating an optimal quasi-normal-eigenvalue on ${\mathrm{i}} {\mathbb R}$ is not necessarily unique.

### Fixed points of complex systems with attractive interaction

Methods Funct. Anal. Topology **23** (2017), no. 2, 164-176

We study the behavior of complex dynamical systems describing an attractive interaction between two opponents. We use the stochastic interpretation and describe states of systems in terms of probability distributions (measures) and their densities. For the time evolution we derive specific non-linear difference equations which generalize the well-known Lotka-Volterra equations. Our results state the existence of fixed points (equilibrium states) for various kinds of attractive interactions. Besides, we present an explicit description of the limiting distributions and illustrate abstract results by several examples.

### Asymptotic properties of the $p$-adic fractional integration operator

Anatoly N. Kochubei, Daniel S. Soskin

Methods Funct. Anal. Topology **23** (2017), no. 2, 155-163

We study asymptotic properties of the $p$-adic version of a fractional integration operator introduced in the paper by A. N. Kochubei, Radial solutions of non-Archimedean pseudo-differential equations, Pacif. J. Math. 269 (2014), 355-369.

### Sturm-Liouville operators with matrix distributional coefficients

Alexei Konstantinov, Oleksandr Konstantinov

Methods Funct. Anal. Topology **23** (2017), no. 1, 51-59

The paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ with the matrix-valued coefficients $p,q$ such that $$q=Q', \quad p^{-1},\, p^{-1}Q, \,\, Qp^{-1}, \,\, Qp^{-1}Q \in L_1, $$ where the derivative of the function $Q$ is understood in the sense of distributions. Due to a suitable regularization, the corresponding operators are correctly defined as quasi-differentials. Their resolvent convergence is investigated and all self-adjoint, maximal dissipative, and maximal accumulative extensions are described in terms of homogeneous boundary conditions of the canonical form.

### On a function system making a basis in a weight space

V. A. Zolotarev, V. N. Levchuk

Methods Funct. Anal. Topology **23** (2017), no. 3, 261-269

We find necessary and sufficient conditions for systems of functions generated by a second order differential equation to form a basis. The results are applied to show that Mathieu functions make a basis.

### Initial-boundary value problems for two-dimensional parabolic equations in Hörmander spaces

Methods Funct. Anal. Topology **23** (2017), no. 2, 177-191

We investigate a general nonhomogeneous initial-boundary value problem for a two-dimensional parabolic equation in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces.

### Spectral properties and stability of a nonselfadjoint Euler-Bernoulli beam

Methods Funct. Anal. Topology **23** (2017), no. 4, 346-366

In this note we study the spectral properties of an Euler-Bernoulli beam model with damping and elastic forces applying both at the boundaries as well as along the beam. We present results on completeness, minimality, and Riesz basis properties of the system of eigen- and associated vectors arising from the nonselfadjoint spectral problem. Within the semigroup formalism it is shown that the eigenvectors have the property of forming a Riesz basis, which in turn enables us to prove the uniform exponential decay of solutions of the particular system considered.

### On certain spectral features inherent to scalar type spectral operators

Methods Funct. Anal. Topology **23** (2017), no. 1, 60-65

Important spectral features such as the emptiness of the residual spectrum, countability of the point spectrum, provided the space is separable, and a characterization of spectral gap at 0, known to hold for bounded scalar type spectral operators, are shown to naturally transfer to the unbounded case.

### Localization principles for Schrödinger operator with a singular matrix potential

Vladimir Mikhailets, Aleksandr Murach, Viktor Novikov

Methods Funct. Anal. Topology **23** (2017), no. 4, 367-377

We study the spectrum of the one-dimensional Schrödinger operator $H_0$ with a matrix singular distributional potential $q=Q'$ where $Q\in L^{2}_{\mathrm{loc}}(\mathbb{R},\mathbb{C}^{m})$. We obtain generalizations of Ismagilov's localization principles, which give necessary and sufficient conditions for the spectrum of $H_0$ to be bounded below and discrete.

### Weak and vague convergence of spectral shift functions of one-dimensional Schrödinger operators with coupled boundary conditions

Methods Funct. Anal. Topology **23** (2017), no. 4, 378-403

We prove weak and vague convergence results for spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on intervals of the form $(-\ell,\ell)$ to the full-line spectral shift function in the limit $\ell\to \infty$ for a class of coupled boundary conditions. The boundary conditions considered here include periodic boundary conditions as a special case.

### On new inverse spectral problems for weighted graphs

L. P. Nizhnik, V. I. Rabanovich

Methods Funct. Anal. Topology **23** (2017), no. 1, 66-75

In this paper, we consider various new inverse spectral problems (ISP) for metric graphs, using maximal eigen values of the adjacency matrix of the graph and its subgraphs as well as the corresponding eigen vectors or some of their components as spectral data. We give examples of spectral data that uniquely determine the metric on the graph. Effective algorithms for solving the considered ISP are given.

### On well-behaved representations of $\lambda$-deformed CCR

D. P. Proskurin, L. B. Turowska, R. Y. Yakymiv

Methods Funct. Anal. Topology **23** (2017), no. 2, 192-205

We study well-behaved ∗-representations of a λ-deformation of Wick analog of CCR algebra. Homogeneous Wick ideals of degrees two and three are described. Well-behaved irreducible ∗-representations of quotients by these ideals are classified up to unitary equivalence.

### Linear maps preserving the index of operators

Methods Funct. Anal. Topology **23** (2017), no. 3, 277-284

Let $\mathsf{H}$ be an infinite-dimensional separable complex Hilbert space and $\mathcal{B}(\mathsf{H})$ the algebra of all bounded linear operators on $\mathsf{H}.$ In this paper, we prove that if a surjective linear map $ \phi : \mathcal{B}(\mathsf{H}) \longrightarrow \mathcal{B}(\mathsf{H})$ preserves the index of operators, then $\phi$ preserves compact operators in both directions and the induced map $ \varphi : \mathcal{C}( \mathsf{H}) \longrightarrow \mathcal{C}(\mathsf{H}),$ determined by $\varphi(\pi(T)) = \pi( \phi(T)) $ for all $T \in \mathcal{B}(\mathsf{H}),$ is a continuous automorphism multiplied by an invertible element in $\mathcal{C}( \mathsf{H}).$

### On the graph $K_{1,n}$ related configurations of subspaces of a Hilbert space

Methods Funct. Anal. Topology **23** (2017), no. 3, 285-300

We study systems of subspaces $H_1,\dots,H_N$ of a complex Hilbert space H that satisfy the following conditions: for every index $k > 1$, the set $\{\theta_{k,1},\ldots,\theta_{k,m_k}\}$ of angles $\theta_{k,i}\in(0,\pi/2)$ between $H_1$ and $H_k$ is fixed; all other pairs $H_k$, $H_j$ are orthogonal. The main tool in the study is a construction of a system of subspaces of a Hilbert space on the basis of its Gram operator (the G-construction).

### Polarization inequality in complex normed spaces

Methods Funct. Anal. Topology **23** (2017), no. 3, 301-308

We introduce a product in all complex normed vector spaces, which generalizes the inner product of complex inner product spaces. Naturally the question occurs whether the polarization inequality of line (3.1) is fulfilled. We show that the polarization inequality holds for the product from Definition 1.1. This also yields a new proof of the Cauchy-Schwarz inequality in complex inner product spaces, which does not rely on the linearity of the inner product. The proof depends only on the norm in the vector space.

### Tannaka-Krein reconstruction for coactions of finite quantum groupoids

Leonid Vainerman, Jean-Michel Vallin

Methods Funct. Anal. Topology **23** (2017), no. 1, 76-107

We study coactions of finite quantum groupoids on unital $C^*$-algebras and obtain a Tannaka-Krein reconstruction theorem for them.