Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T01:12:32.654Z Has data issue: false hasContentIssue false
  • English
  • Français

Some algebras of symmetric analytic functions and their spectra

Part of: Measures, integration, derivative, holomorphy Commutative Banach algebras and commutative topological algebras Linear function spaces and their duals

Published online by Cambridge University Press:  20 June 2011

Iryna Chernega ,
Pablo Galindo  and
Andriy Zagorodnyuk
Iryna Chernega
Affiliation:
Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, 3-b, Naukova str., Lviv 79060, Ukraine (icherneha@ukr.net)
Pablo Galindo
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Calle del Doctor Moliner 50, Burjasot, Valencia 46100, Spain (pablo.galindo@uv.es)
Andriy Zagorodnyuk
Affiliation:
Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., Ivano-Frankivsk 76000, Ukraine (andriyzag@yahoo.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

Keywords

Banach spacespolynomialssymmetric polynomialssymmetric and subsymmetric analytic functions of bounded typespectra of algebras

MSC classification

Secondary: 46J15: Banach algebras of differentiable or analytic functions, $H^p$-spaces 46J20: Ideals, maximal ideals, boundaries 46E15: Banach spaces of continuous, differentiable or analytic functions 46E25: Rings and algebras of continuous, differentiable or analytic functions 46G20: Infinite-dimensional holomorphy 46G25: (Spaces of) multilinear mappings, polynomials
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 125 - 142
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Alencar, R., Aron, R., Galindo, P. and Zagorodnyuk, A., Algebras of symmetric holomorphic functions on ℓp, Bull. Lond. Math. Soc. 35 (2003), 55–64.Google Scholar
2.Aron, R. M., Cole, B. J. and Gamelin, T. W., Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51–93.Google Scholar
3.Aron, R. M., Cole, B. J. and Gamelin, T. W., Weak-star continuous analytic funtions, Can. J. Math. 47 (1995), 673–683.CrossRefGoogle Scholar
4.Berglund, J. F., Junghenn, H. D. and Milnes, P., Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts (Canadian Mathematical Society, Toronto, 1989).Google Scholar
5.Dineen, S., Complex analysis in locally convex spaces, Mathematics Studies, Volume 57 (North-Holland, Amsterdam, 1981).Google Scholar
6.Dineen, S., Complex analysis on infinite dimensional spaces, Monographs in Mathematics (Springer, 1999).Google Scholar
7.González, M., Gonzalo, R. and Jaramillo, J., Symmetric polynomials on re-arrangement invariant function spaces, J. Lond. Math. Soc. (2) 59 (1999), 681–697.CrossRefGoogle Scholar
8.Gonzalo, R., Multilinear forms, subsymmetric polynomials and spreading models on Banach spaces, J. Math. Analysis Applic. 202 (1996), 379–397.CrossRefGoogle Scholar
9.Hájek, P., Polynomial algebras on classical Banach spaces, Israel J. Math. 106 (1998), 209–220.Google Scholar
10.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer, 1977).Google Scholar
11.Mujica, J., Complex analysis in Banach spaces (North-Holland, Amsterdam, 1986).Google Scholar
12.Nemirovskii, A. S. and Semenov, S. M., On polynomial approximation of functions on Hilbert space, Math. USSR Sb. 21 (1973), 255–277.CrossRefGoogle Scholar
13.Zagorodnyuk, A., Spectra of algebras of entire functions on Banach spaces, Proc. Am. Math. Soc. 134 (2006), 2559–2569.CrossRefGoogle Scholar
14.Zagorodnyuk, A., Spectra of algebras of analytic functions and polynomials on Banach spaces, Contemp. Math. 435 (2007), 381–394.CrossRefGoogle Scholar