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ISSN 1088-6826(e) ISSN 0002-9939(p)

Spectra of algebras of entire functions on Banach spaces

Author: Andriy Zagorodnyuk
Journal: Proc. Amer. Math. Soc. 134 (2006), 2559-2569
MSC (2000): Primary 46J15, 46J20; Secondary 46E15, 46E25, 46G20, 46G25
Posted: February 8, 2006
MathSciNet review: 2213733
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain an explicit description of the spectrum (set of closed maximal ideals) of algebra of analytic functions on a Banach space which are bounded on bounded subsets. We show that the spectrum of admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball $B\subset X$ are indicated.

References

1. R. Alencar, R. Aron, P. Galindo, and A. Zagorodnyuk, Algebras of symmetric holomorphic functions on $\ell _p$ , Bull. Lond. Math. Soc. 35 (2003), 55-64. MR 1934432 (2003m:46082)
2. R.M. Aron and P.D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 3-24. MR 0508947 (80e:46029)
3. R.M. Aron, B.J. Cole, and T.W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991), 51-93. MR 1096902 (92f:46056)
4. R.M. Aron, B.J. Cole, and T.W. Gamelin, Weak-star continuous analytic funtions, Can. J. Math. 47 (1995), 673-683. MR 1346158 (96d:46060)
5. R.M. Aron, P. Galindo, D. Garcia, and M. Maestre, Regularity and algebras of analytic functions in infinite dimensions, Trans. Amer. Math. Soc. 348 (1996), 543-559. MR 1340167 (96g:46032)
6. R.M. Aron and J.B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195-216. MR 0552473 (81c:41078)
7. P. Biström, J.A. Jaramillo, and M. Lindström, Polynomial compactness in Banach spaces, Rocky Mount. J. Math. 28 (1998), 1203-1225. MR 1681664 (2000c:46038)
8. F. Cabello Sánchez, R. García, and I. Villanueva, Extension of multilinear operators on Banach spaces, Extracta Math. 15 (2000), 291-334. MR 1823895 (2002e:46022)
9. D. Carando, D. García and M. Maestre, Homomorphisms and composition operators on algebras of analytic functions of bounded type, Adv. Math. To appear.
10. A.M. Davie and T.W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), 351-356. MR 0947313 (89k:46023)
11. D. Deghoul,Construction de caractères exceptionnels sur une algèbre de Fréchet, C. R. Acad. Sci. Paris. 312 (1991), 579-580. MR 1101036 (92b:46066)
12. V. Dimant and R. Gonzalo, Block diagonal polynomials, Trans. Amer. Math. Soc. 353 (2001), 733-747. MR 1804515 (2002b:46079)
13. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Monographs in Mathematics, Springer, New York, 1999. MR 1705327 (2001a:46043)
14. P. Dixon, Scalar homomorphisms on algebras of infinitely long polynomials with an application to automatic continuity theory, J. London Math. Soc. (2) 19 (1979), 488-496. MR 0540065 (80h:46074)
15. P. Galindo, D. García, M. Maestre, and J. Mujica, Extension of multilinear mappings on Banach spaces, Studia. Math. 108 (1994), 55-76. MR 1259024 (95f:46072)
16. T.W. Gamelin, Analytic functions on Banach spaces, in Complex Potential Theory, Ed. Gauthier and Sabidussi, Kluwer Academic Publishers, Amsterdam, 1994, 187-233. MR 1332962 (96m:46082)
17. D. García, M.L. Lourenço, M. Maestre, and L.A. Moraes, The spectrum of analytic mappings of bounded type, J. Math. Anal. Appl. 245 (2000), 447-470. MR 1758549 (2001f:46069)
18. D. García, M.L. Lourenço, L.A. Moraes, and O.W. Paques, The spectra of some algebras of analytic mappings, Indag. Mathem. 10 (1999), 393-406. MR 1819897 (2002c:46089)
19. R. Gonzalo, Multilinear forms, subsymmetric polynomials, and spreading models on Banach spaces, J. Math. Anal. App. 202 (1996), 379-397. MR 1406236 (98a:46054)
20. E. Michael, Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc., vol. 11, Providence, 1952. MR 0051444 (14:482a)
21. J. Mujica, Complex Analysis in Banach Spaces, North-Holland, Amsterdam, New York, Oxford, 1986. MR 0842435 (88d:46084)
22. J. Mujica, Ideals of holomorphic functions on Tsirelson's space, Archiv der Mathematik 76 (2001), 292-298. MR 1825009 (2002g:46075)

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Additional Information

Andriy Zagorodnyuk
Affiliation: Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, 3 b, Naukova str., Lviv 79060, Ukraine
Address at time of publication: Department of Mathematics and Statistics, McLean Hall, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: zahorodn@math.usask.ca, andriyzag@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08260-8
PII: S 0002-9939(06)08260-8
Keywords: Infinite-dimensional holomorphy, spectra of algebras
Received by editor(s): July 8, 2004
Received by editor(s) in revised form: January 28, 2005, February 1, 2005, and March 15, 2005
Posted: February 8, 2006
Additional Notes: This research was supported in part by NSERC research grant OGP 005616.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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