Alexander Abanin

Southern Federal University | sfedu · Faculty of Mathematics, Mechanics and Computer Sciences

During last two decades investigations of many known specialists are devoted to the topological and dynamical properties of the classical operators on weighted Banach spaces of holomorphic functions having uniform estimates with respect to a given radial weight. At present complete results are established for only the problem of a characterization...

We state conditions under which some classical operators acting from abstract quasi-Banach spaces of functions holomorphic in a plain domain into a weighted space of the same functions with sup-norm are compact. It is obtained abstract criteria for the compactness of a linear operator on an arbitrary quasi-Banach space which are stated in terms of...

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces....

We consider the topological structure problem for the space of composition operators as well as the space of weighted composition operators on weighted Banach spaces with sup-norm. For the first space, we prove that the set of all composition operators that differ from the given one by a compact operator is path connected; however, in general, it i...

We review recent results in the theory of classical operators (embedding, differentiation, and integration) in weighted Banach spaces of holomorphic functions with uniform estimates. We formulate and analyze results based on associated and essential weights.

We study topological properties of weighted (LB)-spaces of holomorphic functions on open sets with o-growth condition. We show that such spaces are semi-Montel if and only if they are (DFS). We also establish that, for a wide family of open sets, some important topological properties of such holomorphic spaces are always equivalent and necessary fo...

The spaces dual to spaces of holomorphic functions of given growth on Carathéodory domains are described by using the Cauchy transform of functionals. A pseudoanalytic extension of such transforms to the whole plane is constructed, which makes it possible to remove convexity constrains and consider spaces determined by weights of general form, rath...

Recently we have described the topological structure of sampling sets for the Hörmander algebra A-∞(BN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{-\infty }({\ma...

С помощью преобразования Коши функционалов дается описание сопряженных с пространствами голоморфных функций заданного роста в областях Каратеодори. За счет конструктивного псевдоаналитического продолжения таких преобразований во всю плоскость удается избавиться от требований на определенную выпуклость и одномерную зависимость весов от функции расст...

We develop an approach to describe invariant subspaces of the integration operator on various scales of weighted spaces of holomorphic functions on the unit disk and the complex plane. It allows us to solve the problem for wide classes of Bergman, Bloch, Dirichlet, and Fock spaces, while all previous known results concern spaces defined by some wei...

We obtain a complete description of families of continuous and compact composition operators on Hilbert spaces of entire functions. These operators were introduced by K. Chan and J. Shapiro for studying some dynamic properties of translation operators. In contrast to recent papers devoted to the same problems, we make no additional assumptions on t...

Motivated by some recent results on the boundedness and dynamical properties of the differentiation and integration operators in weighted Banach spaces of holomorphic functions, we study conditions on weights that guarantee the compactness of these two operators in the corresponding weighted spaces.

Let Omega be a Caratheodory domain in the complex plane C, A(-infinity)(Omega) the space of functions that are holomorphic in Omega with polynomial growth near the boundary partial derivative Omega, and A(infinity)(C Omega) the space of holomorphic functions in the interior of C Omega := (C) over bar\Omega, vanishing at infinity and being in C-infi...

We show that some previous results concerning the boundedness of differentiation and integration operators on weighted spaces given by radial weights in the unit disk or the complex plane might fail without some natural additional conditions. In view of this we develop a new elementary approach which is essentially different from the previous one a...

We study algebraic equalities and their topological consequences in weighted Banach, Fréchet, or (LB) spaces of holomorphic-like functions on a locally compact and σ -compact Hausdorff space X . Our main results are the following: (1) The algebraic equality VA(X)=V0A(X)VA(X)=V0A(X) for (LB)-spaces with O - and o -growth conditions given by a weight...

We consider the convolution operator in spaces of holomorphic functions, defined in convex subdomains of the complex plane, with polynomial growth at a boundary. We prove that if this operator is surjective on the class of all bounded convex domains, then it always has a linear continuous right inverse one.

Under study are sufficient sets in Fréchet spaces of entire functions with uniform weighted estimates. We obtain general results on the a priori overflow of these sets and introduce the concept of their minimality. We also establish necessary and sufficient conditions for a sequence of points on the complex plane to be a minimal sufficient set for...

We obtain, in terms of associated weights, natural criteria for compact embedding of weighted Banach spaces of holomorphic functions on a wide class of domains in the complex plane. Our study is based on a complete characterization of finite-dimensional weighted spaces and canonical weights for them. In particular, we show that for a domain whose c...

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ n and have a polynomial growth near its boundary. Acharacterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the op...

In this paper we consider a problem of extension of solutions to homogeneous convolution equations defined by operators acting from a space A(-infinity)(D + K) of holomorphic functions with polynomial growth near the boundary of D + K into another space of such a type A(-infinity)(D) (D and K being a bounded convex domain and a convex compact set i...

In this article, we give a description, via the Cauchy–Fantappiè transformation of analytic functionals, of the mutual dualities between the dual Fréchet–Schwartz (FS)-space A (Ω) of holomorphic functions in a bounded lineally convex domain Ω of ℂ (n ≥ 2) with polynomial growth near the boundary ∂Ω, and the (FS)-space of holomorphic functions in th...

In this paper we describe, via the Laplace transformation of analytic functionals, a pre-dual to the function algebra A −∞(D) (D being either a bounded C 2-smooth convex domain in $${\mathbb{C}^N (N > 1)}$$ , or a bounded convex domain in $${\mathbb{C}}$$) as a space of entire functions with certain growth. A possibility of representation of functi...

In this Note we present the following results: (i) a description, via the Cauchy–Fantappiè transformation of analytic functionals, of the mutual dualities between the (DFS)-space A−∞(Ω) of holomorphic functions in a bounded lineally convex domain Ω of Cn(n⩾2) with polynomial growth near the boundary ∂Ω, and the (FS)-space A∞(Ω˜) of holomorphic func...

We study a minimal property of absolutely representing systems of exponentials EΛ=(eλkz) in the space A−∞(Ω) of holomorphic functions in a bounded convex domain Ω of the complex plane C, with polynomial growth near the boundary. A relationship between the absolutely representing property and absolutely nontrivial expansions of zero in A−∞(Ω) is als...

In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-space A − ∞ ( D ) A^{-\infty }(D) ( D D being either a bounded C 2 C^2 -smooth convex domain in C N \mathbb {C}^N , with N > 1 N>1 , or a bounded convex domain in C \mathbb {C} ) as an (FS)-space A D − ∞ A^...

In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-space A(-infinity)(D) (D being either a bounded C-2-smooth convex domain in C-N, with N > 1, or a bounded convex domain in C) as an (FS)-space A(D)(-infinity) of entire functions satisfying a certain growth...

In this paper we consider a problem of existence of Schauder basis of special form for the kernels (null spaces) of convolution operators acting from the space A −∞ (D +K) of holomorphic functions with polynomial growth near the boundary of D + K, into the space A −∞ (D) of such a type (D and K being a bounded convex domain and a convex compact set...

The goal of this Note is to prove criteria for surjectivity of convolution operators acting from A−∞(Ω+K) into A−∞(Ω) (Ω and K being a bounded convex domain and a convex compact set in Cn(n>1), respectively). This is obtained in a connection with the division problem. The explicit representation of solutions of the corresponding convolution equatio...

As it is known, Roumieu–Komatsu theory of ultradistributions is strictly larger than Beurling–Björck one and that the latter theory is established by the class of all subadditive weight functions. In its own turn, Roumieu–Komatsu theory is equivalent to Braun–Meise–Taylor one which is given by the class of all weight functions. We prove that the cl...

We prove a generalization of the well-known Hörmander theorem on continuation of holomorphic functions with growth conditions from complex planes in ℂp into the whole ℂp. We apply this result to construct special families of entire functions playing an important role in convolution equations, interpolation and extension of infinitely differentiable...

The goal of this Note is to prove that the Laplace transformation of analytic functionals establishes the mutual duality between the spaces A−∞(D)A−∞(D) and AD−∞ (D being a bounded convex domain in CNCN) and that functions from AD−∞ can be represented in a form of Dirichlet series with frequencies from D. To cite this article: A.V. Abanin, L.H. Kho...

We develop a generalization of Beurling's approach to the construction of ultradistribution theory in which Fourier transformation is a basic tool. We establish a structure theorem on the representation of ultradistributions and a theorem of Paley-Wiener-Schwartz type. We illustrate the key role of extending the weights determining the spaces from...

Using the Fourier-Laplace transform for functionals, we describe the duals of some spaces of the infinitely differentiable functions given on convex compact sets or convex domains in ℝN and such that the growth of their derivatives is determined by weight sequences of a general form.

The development of the Berling-Bjorck's approach is presented, containing as individual cases the ultradistribution spaces from the corresponding published works. The developed analysis allows establishing the analogues of classic distribution theory results. Generalization of Paley-Wiener-Schwartz theorem is obtained in the refined formed revealin...

Let E be a ring of entire functions on \mathbbC\user1N \mathbb{C}^\user1{N} with the operation of pointwise multiplication, and let f 1,...,f m be a set of nonzero elements in E. The ideal E(f 1,...,f m ) in E with generators f 1,...,f m is said to be generating if E(f 1,...,f m ) = E. The generating ideals in rings of entire functions on...

For weighted rings of entire functions given by systems of weight functions depending on moduli of variables we obtain a characterization of finitely generated ideals coinciding with the whole ring. This characterization is given in terms of zero sets of generators.

We study conditions on a domainG in the extended complex plane [^(\mathbbC)]\widehat{\mathbb{C}} and a sequence L = { lk }\Lambda = \left\{ {\lambda _k } \right\} of points in the complement Gc = [^(\mathbbC)]\GG^c = \widehat{\mathbb{C}}\backslash G ofG under which every functionf(z) holomorphic inG and vanishing atz=∞ (if ∞∈G) admits a represent...

В статье дается описа ние общего вида абсол ютно сходящегося в локаль но выпуклом пространствеH разлож ения нуля по системе ℰΛ:={e(λk)}