Email this article 
ON TOPOLOGICAL ALGEBRAS OF ANALYTIC FUNCTIONALS WITH A MULTIPLICATION DEFINED BY TRANSLATIONS
- Authors: Ivanova O.A.1, Melikhov S.N.1
-
Affiliations:
- Southern Federal University
- Issue: Vol 24, No 3 (2018)
- Pages: 14-22
- Section: Articles
- URL: https://journals.ssau.ru/est/article/view/6448
- DOI: https://doi.org/10.18287/2541-7525-2018-24-3-14-22
- ID: 6448
Cite item
Full Text
Abstract
We define a multiplication — convolution in the dual of a countable inductive limit E of weighted Fr´echet 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.
About the authors
O. A. Ivanova
Southern Federal University
Author for correspondence.
Email: morenov@ssau.ru
S. N. Melikhov
Southern Federal University
Email: morenov@ssau.ru
References
- Helemskii A.Ya. Gomologiia v banakhovykh i topologicheskikh algebrakh . M.: Izd-vo MGU, 1986, 288 p. .
- Helemskii A.Ya. Banakhovy i polinormirovannye algebry: obshchaia teoriia, predstavleniia, gomologii . M.: Nauka, 1989, 466 p. .
- Tkachenko V.A. Ob operatorakh, kommutiruiushchikh s obobshchennym integrirovaniem v prostranstvakh analiticheskikh funktsionalov . Matem. zametki , 1979, Vol. 25, no 2, pp. 141–146. Available at: http://mi.mathnet.ru/mz8303 .
- Tkachenko V.A. Spektral’nye razlozheniia v prostranstvakh analiticheskikh funktsionalov . Izv. AN SSSR. Ser. matem. , 1980, Vol. 14, no. 3, pp. 597–651. DOI: http://dx.doi.org/10.1070/IM1980v014n03ABEH001146
- .
- Tkachenko V.A. Spektral’naia teoriia v prostranstvakh analiticheskikh funktsionalov dlia operatorov, porozhdaemykh umnozheniem na nezavisimuiu peremennuiu . Matem. sb. , 1981, Vol. 40, no 3, pp. 387–427. DOI: http://dx.doi.org/10.1070/SM1981v040n03ABEH001833 .
- Gurevich D.I. Operatory obobshchennogo sdviga s pravym infinitezimal’nym operatorom Shturma-–Liuvillia . Matem. zametki Mathematical Notes, 1979, Vol. 25, no. 3, pp. 208–215. DOI: https://doi.org/10.1007/BF01159513 .
- Litvinov G.L. O preobrazovanii Laplasa na gruppakh Li . Funkts. analiz i ego pril. , 1972, Vol. 6, no 1, pp. 76–77. DOI: https://doi.org/10.1007/BF01075519
- .
- Rashevskii P.K. Assotsiativnaia sverkhobolochka algebry Li, ee reguliarnoe predstavlenie i idealy . Tr. MMO , 1966, Vol. 15, pp. 3–54. .
- Ivanova O.A, Melikhov S.N., Melikhov Yu.N. O kommutante operatorov differentsirovaniia i sdviga v vesovykh prostranstvakh tselykh funktsii . Ufimskii matem. zhurn. , 2017, Vol. 9, no 3, pp. 37–47. DOI: https://doi.org/10.13108/2017-9-3-37.
- Edwards R.E. Funktsional’nyi analiz. Teoriia i prilozheniia . M.: Mir, 1969, 1072 p. .
- Schaefer H. Topologicheskie vektornye prostranstva . M.: Mir, 1971, 360 p. .
- Meise R., Vogt D. Introduction to Functional Analysis. Oxford: Clarendon, 1997, 448 p. .
- Ivanova O.A., Melikhov S.N. Ob operatorakh, perestanovochnykh s operatorom tipa Pomm’e v vesovykh prostranstvakh tselykh funktsii . Algebra i analiz , 2017, Vol. 28, no 2, pp. 209–224. DOI: https://doi.org/10.1090/spmj/1447 .
- Ivanova O.A., Melikhov S.N. On the completeness of orbits of a Pommiez operator in weighted (LF)-spaces of entire functions. Complex Analysis and Operator Theory, 2017, Vol. 11, pp. 1407–1424. DOI: https://doi.org/10.1007/s11785-016-0617-5
- .
- Korobeinik Yu.F., Morzhakov V.V. Obshchii vid izomorfizmov, perestanovochnykh s operatorom differentsirovaniia, v prostranstvakh tselykh funktsii medlennogo rosta . Matem. sb. , 1973, Vol. 20, no 4, pp. 493–505. DOI: http://dx.doi.org/10.1070/SM1973v020n04ABEH001886
- .
- Gorodentsev A.L. Algebra. Uchebnik dlia studentov-matematikov. Ch. 1 . M.: Izd-vo MTsNMO, 2013, 486 p. .
- Lelong P., Gruman L. Tselye funktsii mnogikh kompleksnykh peremennykh . M.: Mir, 1989, 352 p. .
- H¨ormander L. Analiz lineinykh differentsial’nykh operatorov s chastnymi proizvodnymi. T. 1. Teoriia raspredelenii i analiz Fur’e . M.: Mir, 1986, 464 p. .
- Martineau A. Equations differentielles d’ordre infini. Bull. Soc. Math. France, 1967, Vol. 95, pp. 109–154 .
Supplementary files
