# Authors Index,

### Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators

Miron B. Bekker, Martin J. Bohner, Alexander P. Ugolʹnikov, Hristo Voulov

Methods Funct. Anal. Topology **24** (2018), no. 1, 1-15

On a Hilbert space $\frak H$, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scale-invariant self-adjoint extension in $ \frak H$. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.

### Lie derivations on the algebras of locally measurable operators

Vladimir Chilin, Ilkhom Juraev

Methods Funct. Anal. Topology **24** (2018), no. 1, 16-26

We prove that every Lie derivation on a solid $\ast$-subalgebra in an algebra of locally measurable operators is equal to a sum of an associative derivation and a center-valued trace.

### On the numerical range with respect to a family of projections

Waed Dada, Joachim Kerner, Nazife Erkurşun-Özcan

Methods Funct. Anal. Topology **24** (2018), no. 4, 297-304

In this note we introduce the concept of a numerical range of a bounded linear operator on a Hilbert space with respect to a family of projections. We give a precise definition and elaborate on its connection to the classical numerical range as well as to generalizations thereof such as the quadratic numerical range, block numerical range, and product numerical range. In general, the importance of this new notion lies within its unifying aspect.

### On extensions of linear functionals with applications to non-symmetrically singular perturbations

Mykola Dudkin, Tetiana Vdovenko

Methods Funct. Anal. Topology **24** (2018), no. 3, 193-206

The article is devoted to extensions of linear functionals, generated by scalar products, in a scale of Hilbert spaces. Such extensions are used to consider non-symmetrically singular rank one perturbations of ${\mathcal H}_{-2}$-class. For comparison, we give main definitions and descriptions of singular non-symmetric perturbations of ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$-classes.

### Myroslav Lvovych Gorbachuk (to his 80th birthday)

Methods Funct. Anal. Topology **24** (2018), no. 2, 91-97

### Volodymyr Dmytrovych Koshmanenko (to 75th birthday anniversary)

Methods Funct. Anal. Topology **24** (2018), no. 3, 187-189

### Yuii Stefanovych Samoilenko (to 75th birthday anniversary)

Methods Funct. Anal. Topology **24** (2018), no. 3, 190-192

### Ergoregions between two ergospheres

Gregory Eskin, Michael A. Hall

Methods Funct. Anal. Topology **24** (2018), no. 2, 98-106

For a stationary spacetime metric, black holes are spatial regions out of which disturbances do not propagate. In our previous work an existence and regularity theorem was proven for black holes in two space dimensions, in the case where the boundary of the ergoregion is a simple closed curve surrounding a singularity. In this paper we study the case of an annular ergoregion, whose boundary has two components.

### Complex powers of abstract pseudodifferential operators

Methods Funct. Anal. Topology **24** (2018), no. 4, 305-338

Under suitable assumptions, we show that the abstract pseudodifferen\-tial operators introduced by Connes and Moscovici possess complex powers that belong to this class of operators. We analyse several spectral functions obtained via the (super)trace including the zeta function and the heat trace. We present examples showing that the analysis is explicit and tractable.

### On similarity of unbounded perturbations of selfadjoint operators

Methods Funct. Anal. Topology **24** (2018), no. 1, 27-33

We consider a linear unbounded operator $A$ in a separable Hilbert space with the following property: there is an invertible selfadjoint operator $S$ with a discrete spectrum such that $\|(A-S)S^{-\nu}\|<\infty$ for a $\nu\in [0,1]$. Besides, all eigenvalues of $S$ are assumed to be different. Under certain assumptions it is shown that $A$ is similar to a normal operator and a sharp bound for the condition number is suggested. Applications of that bound to spectrum perturbations and operator functions are also discussed. As an illustrative example we consider a non-selfadjoint differential operator.

### Self-consistent translational motion of reference frames and sign-definiteness of time in universal kinematics

Methods Funct. Anal. Topology **24** (2018), no. 2, 107-119

Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists a hypothesis that, in the 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 present paper is devoted to investigation of self-consistent translational motion of reference frames in abstract universal kinematics. In the case of self-consistent translational motion we can give a clear and unambiguous definition of displacement as well as the average and the instantaneous speed of the reference frame. Hence the uniform rectilinear motion is a particular case of self-consistent translational motion. So, the investigation of self-consistently translational motion is technically necessary for definition of classes of inertially-related reference frames (being in the state of uniform rectilinear mutual motion) in universal kinematics. In the paper we investigate the correlations between self-consistent translational motion and definiteness of time direction for reference frames in universal kinematics.

### A probabilistic proof of the Vitali Covering Lemma

E. Gwaltney, P. Hagelstein, D. Herden

Methods Funct. Anal. Topology **24** (2018), no. 1, 34-40

The classical Vitali Covering Lemma on $\mathbb{R}$ states that there exists a constant $c > 0$ such that, given a finite collection of intervals $\{I_j\}$ in $\mathbb{R}$, there exists a disjoint subcollection $\{\tilde{I}_j\} \subseteq \{I_j\}$ such that $|\cup \tilde{I}_j| \geq c |\cup I_j|$. We provide a new proof of this covering lemma using probabilistic techniques and Padovan numbers.

### Quasi-invariance of completely random measures

Habeebat O. Ibraheem, Eugene Lytvynov

Methods Funct. Anal. Topology **24** (2018), no. 3, 207-239

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on $X$ that is concentrated on $\mathbb K(X)$. We consider the multiplicative (current) group $C_0(X\to\mathbb R_+)$ consisting of functions on $X$ that take values in $\mathbb R_+=(0,\infty)$ and are equal to 1 outside a compact set. Each element $\theta\in C_0(X\to\mathbb R_+)$ maps $\mathbb K(X)$ onto itself; more precisely, $\theta$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i \theta(s_i)s_i\delta_{x_i}$. Thus, elements of $C_0(X\to\mathbb R_+)$ transform the weights of discrete Radon measures. We study conditions under which the measure $\mu$ is quasi-invariant under the action of the current group $C_0(X\to\mathbb R_+)$ and consider several classes of examples. We further assume that $X=\mathbb R^d$ and consider the group of local diffeomorphisms $\operatorname{Diff}_0(X)$. Elements of this group also map $\mathbb K(X)$ onto itself. More precisely, a diffeomorphism $\varphi\in \operatorname{Diff}_0(X)$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i s_i\delta_{\varphi(x_i)}$. Thus, diffeomorphisms from $\operatorname{Diff}_0(X)$ transform the atoms of discrete Radon measures. We study quasi-invariance of $\mu$ under the action of $\operatorname{Diff}_0(X)$. We finally consider the semidirect product $\mathfrak G:=\operatorname{Diff}_0(X)\times C_0(X\to \mathbb R_+)$ and study conditions of quasi-invariance and partial quasi-invariance of $\mu$ under the action of $\mathfrak G$.

### Automorphisms generated by umbral calculus on a nuclear space of entire test functions

Ferdinand Jamil, Yuri Kondratiev, Sheila Menchavez, Ludwig Streit

Methods Funct. Anal. Topology **24** (2018), no. 4, 339-348

In this paper we show that Sheffer operators, mapping monomials to certain Sheffer polynomial sequences, such as falling and rising factorials, Charlier, and Hermite polynomials extend to continuous automorphisms on the space of entire functions of first order growth and minimal type.

### Elliptic problems with boundary operators of higher orders in Hörmander–Roitberg spaces

Tetiana Kasirenko, Aleksandr Murach

Methods Funct. Anal. Topology **24** (2018), no. 2, 120-142

We investigate elliptic boundary-value problems for which the maximum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an arbitrary problem of this kind is bounded and Fredholm between appropriate Hilbert spaces which form certain two-sided scales and are built on the base of isotropic Hörmander spaces. The differentiation order for these spaces is given by an arbitrary real number and positive function which varies slowly at infinity in the sense of Karamata. We establish a local a priori estimate for the generalized solutions to the problem and investigate their local regularity (up to the boundary) on these scales. As an application, we find sufficient conditions under which the solutions have continuous classical derivatives of a given order.

### Fractional kinetics in a spatial ecology model

José Luís da Silva, Yuri Kondratiev, Pasha Tkachov

Methods Funct. Anal. Topology **24** (2018), no. 3, 275-287

In this paper we study the effect of subordination to the solution of a model of spatial ecology in terms of the evolution density. The asymptotic behavior of the subordinated solution for different rates of spatial propagation is studied. The difference between subordinated solutions to non-linear equations with classical time derivative and solutions to non-linear equation with fractional time derivative is discussed.

### Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces

Methods Funct. Anal. Topology **24** (2018), no. 1, 41-52

There is still a big gap between knowing that a Riemann surface of genus $g$ has $g$ holomorphic differential forms and being able to find them explicitly. The aim of this paper is to show how to construct holomorphic differential forms on compact Riemann surfaces. As known, the dimension of the space $H^1(\mathcal{D}, \mathbb{C})$ of holomorphic differentials of a compact Riemann surface $\mathcal{D}$ is equal to its genus, $\dim H^1(\mathcal{D}, \mathbb{C})=g(\mathcal{D})=g$. When the Riemann surface is concretely described, we show that one can usually present a basis of holomorphic differentials explicitly. We apply the method to the case of relatively complicated Riemann surfaces.

### On the mean ergodicity of weak solutions of an abstract evolution equation

Methods Funct. Anal. Topology **24** (2018), no. 1, 53-70

Found are conditions of rather general nature sufficient for the existence of the limit at infinity of the Cesàro means $$ \frac{1}{t} \int_0^ty(s)\,ds $$ for every bounded weak solution $y(\cdot)$ of the abstract evolution equation $$ y'(t)=Ay(t),\ t\ge 0, $$ with a closed linear operator $A$ in a Banach space $X$.

### On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator

Methods Funct. Anal. Topology **24** (2018), no. 4, 349-369

Found are conditions on a scalar type spectral operator $A$ in a complex Banach space necessary and sufficient for all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\quad t\ge 0, \end{equation*} to be strongly Gevrey ultradifferentiable of order $\beta\ge 1$, in particular analytic or entire, on $[0,\infty)$. Certain inherent smoothness improvement effects are analyzed.

### One-dimensional parameter-dependent boundary-value problems in Hölder spaces

Hanna Masliuk, Vitalii Soldatov

Methods Funct. Anal. Topology **24** (2018), no. 2, 143-151

We study the most general class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the complex H\"older space $C^{n+r,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0<\alpha\leq1$. We prove a constructive criterion under which the solution to an arbitrary parameter-dependent problem from this class is continuous in $C^{n+r,\alpha}$ with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of this solution to the solution of the corresponding nonperturbed problem.

### Schrödinger operators with measure-valued potentials: semiboundedness and spectrum

Vladimir Mikhailets, Volodymyr Molyboga

Methods Funct. Anal. Topology **24** (2018), no. 3, 240-254

We study 1-D Schrödinger operators in the Hilbert space $L^{2}(\mathbb{R})$ a with real-valued Radon measure $q'(x)$, $q\in \mathrm{BV}_{loc}(\mathbb{R})$ as potentials. New sufficient conditions for minimal operators to be bounded below and selfadjoint are found. For such operators, a criterion for discreteness of the spectrum is proved, which generalizes Molchanov's, Brinck's, and the Albeverio-Kostenko-Malamud criteria. The quadratic forms corresponding to the investigated operators are described.

### Categories of unbounded operators

Methods Funct. Anal. Topology **24** (2018), no. 1, 71-81

In this article we introduce the concept of an $LK^\ast$-algebroid, which is defined axiomatically. The main example of an $LK^\ast$-algebroid is the category of all subspaces of a Hilbert space and closed (not necessarily bounded) linear operators. We prove that for any $LK^\ast$-algebroid there is a faithful functor that respects its structure and maps it into this main example.

### Symmetric extensions of symmetric linear relations (operators) preserving the multivalued part

Methods Funct. Anal. Topology **24** (2018), no. 2, 152-177

Let $\mathfrak H$ be a Hilbert space and let $A$ be a symmetric linear relation (in particular, a nondensely defined operator) in $\mathfrak H$. By using the concept of a boundary triplet for $A^*$ we characterize symmetric extensions $\widetilde A\supset A$ preserving the multivalued part of $A$. Such a characterization is given in terms of an abstract boundary parameter and the Weyl function of the boundary triplet. Application of these results to the Hamiltonian system $Jy'-B(t)y=\lambda\Delta(t) y$ enabled us to describe its matrix solutions generating the generalized Fourier transform with the nonempty set of respective spectral functions.

### Lacunary $\mathcal{I}$-convergent and lacunary $\mathcal{I}$-bounded sequence spaces defined by a Musielak-Orlicz function over $n$-normed spaces

Methods Funct. Anal. Topology **24** (2018), no. 4, 370-380

In the present paper we defined $\mathcal{I}$-convergent and $\mathcal{I}$-bounded sequence spaces defined by a Musielak-Orlicz function $\mathcal{M} = (M_k)$ over $n$-normed spaces. We also make an effort to study some topological properties and prove some inclusion relation between these spaces.

### On the inverse eigenvalue problems for a Jacobi matrix with mixed given data

Methods Funct. Anal. Topology **24** (2018), no. 2, 178-186

We give necessary and sufficient conditions for existence and uniqueness of a solution to inverse eigenvalues problems for Jacobi matrix with given mixed initial data. We also propose effective algorithms for solving these problems.

### On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex

Methods Funct. Anal. Topology **24** (2018), no. 3, 255-264

The Perron--Frobenius theorem states that a linear stochastic operator associated with a positive square stochastic matrix has a unique fixed point in the simplex and it is strongly ergodic to that fixed point. However, in general, the similar result for quadratic stochastic operators associated with positive cubic stochastic matrices does not hold true. Namely, it may have more than one fixed point in the simplex. Moreover, the uniqueness of fixed points does not imply the strong ergodicity of quadratic stochastic operators. In this paper, for some classes of positive cubic stochastic matrices, we provide a uniqueness criterion for fixed points of quadratic stochastic operators acting on a 2D simplex. Some supporting examples are also presented.

### On the spectrum of multiplication operators

Methods Funct. Anal. Topology **24** (2018), no. 3, 265-274

We study relations between spectra of two operators that are connected to each other through some intertwining conditions. As an application, we obtain new results on the spectra of multiplication operators on $B(\mathcal H)$ relating it to the spectra of the restriction of the operators to the ideal $\mathcal C_2$ of Hilbert-Schmidt operators. We also solve one of the problems, posed in [6], about the positivity of the spectrum of multiplication operators with positive operator coefficients when the coefficients on one side commute. Using the Wiener-Pitt phenomena we show that the spectrum of a multiplication operator with normal coefficients satisfying the Haagerup condition might be strictly larger than the spectrum of its restriction to $\mathcal C_2$.

### On new points of the discrete spectrum under singular perturbations

Methods Funct. Anal. Topology **24** (2018), no. 3, 288-296

We study the emergence problem of new points in the discrete spectrum under singular perturbations of a positive operator. We start with the sequential approach to construction of additional eigenvalues for perturbed operators, which was produced by V. Koshmanenko on the base of rigged Hilbert spaces methods. Two new observations are established. We show that one can construct a point of the discrete spectrum of any finite multiplicity in a single step. And that the method of rigged Hilbert spaces admits an application to the modified construction of a new point of the discrete spectrum under super-singular perturbations.

### Strong convergence in topological spaces

Mehmet Ünver, Şeyhmus Yardimci

Methods Funct. Anal. Topology **24** (2018), no. 1, 82-90

Study of summability theory in an arbitrary topological space is not always an easy issue as many of the convergence methods need linear structure in the space. The concept of statistical convergence is one of the exceptional concepts of summability theory that can be considered in a topological space. There is a strong relationship between this convergence method and strong convergence which is another interesting concept of summability theory. However, dependence of the strong convergence to the metric, studying similar relationship directly in arbitrary Hausdorff spaces is not possible. In this paper we introduce a convergence method which extends the notion of strong convergence to topological spaces. This new definition not only helps us to investigate a similar relationship in a topological space but also leads to study a new type of convergence in topological spaces. We also give a characterization of statistical convergence.

### Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions

T. V. Vasylyshyn, A. V. Zagorodnyuk

Methods Funct. Anal. Topology **24** (2018), no. 4, 381-398

Vector spaces of all homogeneous continuous polynomials on infinite dimensional Banach spaces are infinite dimensional. But spaces of homogeneous continuous polynomials with some additional natural properties can be finite dimensional. The so-called symmetry of polynomials on some classes of Banach spaces is one of such properties. In this paper we consider continuous symmetric $3$-homogeneous polynomials on the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ and on the Cartesian square of this space. We construct Hamel bases of spaces of such polynomials and prove formulas for representing of polynomials as linear combinations of base polynomials. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on $L_\infty$ and on its Cartesian square. In particular, in order to describe appropriate topologies on the spectrum (the set of complex valued homomorphisms) of a given algebra of analytic functions, it is useful to have representations for polynomials, obtained in this paper.