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 Vestnik SamU. Estestvenno-Nauchnaya Ser., 2018, Issue 24, Issue 3, Pages 14–22 (Mi vsgu578)

Mathematics

On topological algebras of analytic functionals with a multiplication defined by translations

O. A. Ivanovaa, S. N. Melikhovab

a Institute for Mathematics, Mechanics and Computer Science in the name of I.I. Vorovich, Southern Federal University, 105/42, Bolshaya Sadovaya Street, Rostov-on-Don, 344006, Russian Federation
b Southern Mathematical Institute — the Affiliate of Vladikavkaz Scientific Centre of the Russian Academy of Sciences, 53, Vatutina Street, Vladikavkaz, 362027, Republic of North Ossetia-Alania, Russian Federation

Abstract: We define a multiplication — convolution in the dual of a countable inductive limit $E$ of weighted Fréchet spaces of entire functions of several variables. This algebra is isomorphic to the commutant of the system of partial derivatives in the algebra of all continuous linear operators in $E$. In the constructed algebra of analytic functionals in two pure cases a topology is defined. With this topology the mentioned algebra is topological and it is now topologically isomorphic to the considered commutant with its natural operator topology. It is proved that in this pure situations the present algebra has no zero divisors provided that polynomials are dense in $E$. We show that this condition is essential for the validity of the last statement.

Keywords: weighted space of entire functions, algebra of analytic functionals, topological algebra, communant, convolution operator.

DOI: https://doi.org/10.18287/2541-7525-2018-24-3-14-22

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Document Type: Article
UDC: 517.9

Citation: O. A. Ivanova, S. N. Melikhov, “On topological algebras of analytic functionals with a multiplication defined by translations”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 24:3 (2018), 14–22

Citation in format AMSBIB
\Bibitem{IvaMel18} \by O.~A.~Ivanova, S.~N.~Melikhov \paper On topological algebras of analytic functionals with a multiplication defined by translations \jour Vestnik SamU. Estestvenno-Nauchnaya Ser. \yr 2018 \vol 24 \issue 3 \pages 14--22 \mathnet{http://mi.mathnet.ru/vsgu578} \crossref{https://doi.org/10.18287/2541-7525-2018-24-3-14-22} \elib{http://elibrary.ru/item.asp?id=36731738}