# Vol. 23 (2017), no. 3

### 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

### 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.

### $A$-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.

### 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.

### 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.

### 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.

### 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.