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Vol 52, No 1 (2018)

Article

Duhamel Algebras and Applications

Karaev M.T.

Abstract

We introduce Duhamel algebras and study their properties and applications. We prove that a Banach space of analytic functions on the unit disc that satisfy certain conditions is a Duhamel algebra and describe its closed ideals. These results substantially generalize and improve the main results of Wigley’s papers. Some other related questions are also discussed.

Functional Analysis and Its Applications. 2018;52(1):1-8
pages 1-8 views

Results on the Colombeau Products of the Distribution x+r−1/2 with the Distributions xk−1/2 and xk−1/2

Miteva M., Jolevska-Tuneska B., Atanasova-Pacemska T.

Abstract

Results on the products of the distribution x+r−1/2 with the distributions xk−1/2 and xk−1/2 are obtained in the differential algebra G(ℝ) of Colombeau generalized functions, which contains the space D′(ℝ) of Schwartz distributions as a subspace; in this algebra the notion of association is defined, which is a faithful generalization of weak equality in G(ℝ). This enables treating the results in terms of distributions again.

Functional Analysis and Its Applications. 2018;52(1):9-20
pages 9-20 views

On the Distribution of Zero Sets of Holomorphic Functions

Khabibullin B.N., Rozit A.P.

Abstract

Let M be a subharmonic function with Riesz measure νM in a domain D in the n-dimensional complex Euclidean space ℂn, and let f be a nonzero function that is holomorphic in D, vanishes on a set ZD, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of νM near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.

Functional Analysis and Its Applications. 2018;52(1):21-34
pages 21-34 views

Summation of Unordered Arrays

Shchepin E.V.

Abstract

An approach to the summation of unordered number and matrix arrays based on ordering them by absolute value (greedy summation) is proposed. Theorems on products of greedy sums are proved. A relationship between the theory of greedy summation and the theory of generalized Dirichlet series is revealed. The notion of asymptotic Dirichlet series is considered.

Functional Analysis and Its Applications. 2018;52(1):35-44
pages 35-44 views

Brief Communications

On the Pólya–Szégö Inequality for Functionals with Variable Exponent

Bankevich S.V.

Abstract

Analogues of the Pólya–Szégö inequality with variable exponent in the integrand are considered. Necessary and sufficient conditions for the fulfillment of these inequalities are obtained.

Functional Analysis and Its Applications. 2018;52(1):45-48
pages 45-48 views

Restricted Lie (Super)Algebras in Characteristic 3

Bouarroudj S., Krutov A.O., Lebedev A.V., Leites D.A., Shchepochkina I.M.

Abstract

We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic.

Functional Analysis and Its Applications. 2018;52(1):49-52
pages 49-52 views

Invariant Subspaces for Commuting Operators on a Real Banach Space

Lomonosov V.I., Shul’man V.S.

Abstract

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator TA satisfies the condition

\({\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,\)
where ║ · ║e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.

Functional Analysis and Its Applications. 2018;52(1):53-56
pages 53-56 views

On Extrapolation Properties of Schatten–von Neumann Classes

Lykov K.V.

Abstract

For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.

Functional Analysis and Its Applications. 2018;52(1):57-61
pages 57-61 views

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

Pokrovskii A.V.

Abstract

It is shown that, for any compact set K ⊂ ℝn (n ⩾ 2) of positive Lebesgue measure and any bounded domain GK, there exists a function in the Hölder class C1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.

Functional Analysis and Its Applications. 2018;52(1):62-65
pages 62-65 views

Essential Spectrum of Schrödinger Operators on Periodic Graphs

Rabinovich V.S.

Abstract

We give a description of the essential spectra of unbounded operators ℋq on L2(Γ) determined by the Schrödinger operators −d2/dx2 + q(x) on the edges of Γ and general vertex conditions. We introduce a set of limit operators of ℋq such that the essential spectrum of ℋq is the union of the spectra of limit operators. We apply this result to describe the essential spectra of the operators ℋq with periodic potentials perturbed by terms slowly oscillating at infinity.

Functional Analysis and Its Applications. 2018;52(1):66-69
pages 66-69 views

On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Arithmetically Self-Similar Weight of a Generalized Cantor Type

Rastegaev N.V.

Abstract

Spectral asymptotics of the Sturm–Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method for estimating the eigenvalue counting function. This makes it possible to consider a much wider class of self-similar measures.

Functional Analysis and Its Applications. 2018;52(1):70-73
pages 70-73 views

Dichotomy of Iterated Means for Nonlinear Operators

Saburov M.

Abstract

In this paper, we discuss a dichotomy of iterated means of nonlinear operators acting on a compact convex subset of a finite-dimensional real Banach space. As an application, we study the mean ergodicity of nonhomogeneous Markov chains.

Functional Analysis and Its Applications. 2018;52(1):74-76
pages 74-76 views

Monodromization and Difference Equations with Meromorphic Periodic Coefficients

Fedotov A.A.

Abstract

We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.

Functional Analysis and Its Applications. 2018;52(1):77-81
pages 77-81 views