Abstract:
We study the cluster value theorem for Hb(X), the Fréchet algebra of holomorphic functions bounded on bounded sets of X. We also describe the (size of) fibers of the spectrum of Hb(X). Our results are rather complete whenever X has an unconditional shrinking basis and for X = ℓ1. As a byproduct, we obtain results on the spectrum of the algebra of all uniformly continuous holomorphic functions on the ball of ℓ1. © 2015 American Mathematical Society.
Registro:
Documento: |
Artículo
|
Título: | Cluster values of holomorphic functions of bounded type |
Autor: | Aron, R.M.; Carando, D.; Lassalle, S.; Maestre, M. |
Filiación: | Department of Mathematical Sciences, Kent State University, Kent, OH 44242, United States Departamento de Matemática, Pab I, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, Buenos Aires, VIC B1644BID, Argentina Departmento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, Burjasot, Valencia 6100, Spain
|
Palabras clave: | Analytic functions of bounded type; Banach space; Cluster value; Fiber; Spectrum |
Año: | 2016
|
Volumen: | 368
|
Número: | 4
|
Página de inicio: | 2355
|
Página de fin: | 2363
|
DOI: |
http://dx.doi.org/10.1090/tran/6407 |
Título revista: | Transactions of the American Mathematical Society
|
Título revista abreviado: | Trans. Am. Math. Soc.
|
ISSN: | 00029947
|
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029947_v368_n4_p2355_Aron |
Referencias:
- Acosta, M.A., Lourenço, M.L., Shilov boundary for holomorphic functions on some classical Banach spaces (2007) Studia Math, 179 (1), pp. 27-39. , MR2291721 (2008c:46061)
- Aron, R.M., Berner, P.D., A Hahn-Banach extension theorem for analytic mappings (English, with French summary) (1978) Bull. Soc. Math. France, 106 (1), pp. 3-24. , MR508947 (80e:46029)
- Aron, R.M., Cole, B.J., Gamelin, T.W., Spectra of algebras of analytic functions on a Banach space (1991) J. Reine Angew. Math, 415, pp. 51-93. , MR1096902 (92f:46056)
- Aron, R.M., Carando, D., Gamelin, T.W., Lassalle, S., Maestre, M., Cluster values of analytic functions on a Banach space (2012) Math. Ann, 353 (2), pp. 293-303. , MR2915537
- Aron, R.M., Diestel, J., Rajappa, A.K., (1985) Weakly Continuous Functions on Banach Spaces Containing L1, 1166, pp. 1-3. , Banach spaces (Columbia, Mo., 1984), Springer, Berlin, MR827751 (87g:46022)
- Aron, R.M., Galindo, P., García, D., Maestre, M., Regularity and algebras of analytic functions in infinite dimensions (1996) Trans. Amer. Math. Soc, 348 (2), pp. 543-559. , MR1340167 (96g:46032)
- Boyd, C., Ryan, R.A., Bounded weak continuity of homogeneous polynomials at the origin, Arch (1998) Math. (Basel), 71 (3), pp. 211-218. , MR1637369 (99k:46076)
- Carando, D., García, D., Maestre, M., Sevilla-Peris, P., On the spectra of algebras of analytic functions, Topics in complex analysis and operator theory, Contemp. Math (2012) Amer. Math. Soc, 561, pp. 165-198. , Providence, RI, MR2905544 (2012k:46001)
- Carleson, L., Interpolations by bounded analytic functions and the corona problem (1962) Ann. of Math, 2 (76), pp. 547-559. , MR0141789 (25 #5186)
- Collingwood, E.F., Lohwater, A.J., The theory of cluster sets (1966) Cambridge Tracts in Mathematics and Mathematical Physics, 56. , Cambridge University Press, Cambridge, MR0231999 (38 #325)
- Davie, A.M., Gamelin, T.W., A theorem on polynomial-star approximation (1989) Proc. Amer. Math. Soc, 106 (2), pp. 351-356. , MR947313 (89k:46023)
- Diestel, J., Sequences and series in Banach spaces (1984) Graduate Texts in Mathematics, 92. , Springer-Verlag, New York, MR737004 (85i:46020)
- Dineen, S., Complex analysis on infinite-dimensional spaces (1999) Springer Monographs in Mathematics, , Springer-Verlag London, Ltd., London, MR1705327 (2001a:46043)
- Jeff, D., Farmer, Fibers over the sphere of a uniformly convex Banach space, Michigan (1998) Math. J, 45 (2), pp. 211-226. , MR1637638 (99h:46063)
- Theodore, W., Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N (1969) J., , MR0410387 (53 #14137)
- Gamelin, T.W., Analytic functions on Banach spaces, Complex potential theory (Montreal, PQ, 1993), NATO Adv (1994) Sci. Inst. Ser. C Math. Phys. Sci, 439, pp. 187-233. , Kluwer Acad. Publ., Dordrecht, MR1332962 (96m:46082)
- Johnson, W.B., Ortega Castillo, S., The cluster value problem in spaces of continuous functions (2015) Proc. Amer. Math. Soc, 143 (4), pp. 1559-1568. , MR3314069
- Johnson, W.B., Ortega Castillo, S., The Cluster Value Problem for Banach Spaces, , preprint
- Llavona, J.G., Moraes, L.A., The Aron-Berner extension for polynomials defined in the dual of a Banach space (2004) Publ. Res. Inst. Math. Sci, 40 (1), pp. 221-230. , MR2030074 (2004k:46069)
- Mujica, J., Complex analysis in Banach spaces, Holomorphic functions and domains of holomorphy in finite and infinite dimensions, Notas de Matemáica [Mathematical Notes], 107 (1986) North-Holland Mathematics Studies, 120. , North-Holland Publishing Co., Amsterdam, MR842435 (88d:46084)
- Munkres, J.M., (2000) Topology, , Second Edition. Prentice Hall, Upper Saddle River
- Schark, I.J., Maximal ideals in an algebra of bounded analytic functions (1961) J. Math. Mech, 10, pp. 735-746. , (“I. J. Schark” is a pseudonym for the group: Irving Kaplansky, John Wermer, Shizuo Kakutani, R. Creighton Buck, Halsey Royden, Andrew Gleason, Richard Arens and Kenneth Hoffman., MR0125442 (23 #A2744)
Citas:
---------- APA ----------
Aron, R.M., Carando, D., Lassalle, S. & Maestre, M.
(2016)
. Cluster values of holomorphic functions of bounded type. Transactions of the American Mathematical Society, 368(4), 2355-2363.
http://dx.doi.org/10.1090/tran/6407---------- CHICAGO ----------
Aron, R.M., Carando, D., Lassalle, S., Maestre, M.
"Cluster values of holomorphic functions of bounded type"
. Transactions of the American Mathematical Society 368, no. 4
(2016) : 2355-2363.
http://dx.doi.org/10.1090/tran/6407---------- MLA ----------
Aron, R.M., Carando, D., Lassalle, S., Maestre, M.
"Cluster values of holomorphic functions of bounded type"
. Transactions of the American Mathematical Society, vol. 368, no. 4, 2016, pp. 2355-2363.
http://dx.doi.org/10.1090/tran/6407---------- VANCOUVER ----------
Aron, R.M., Carando, D., Lassalle, S., Maestre, M. Cluster values of holomorphic functions of bounded type. Trans. Am. Math. Soc. 2016;368(4):2355-2363.
http://dx.doi.org/10.1090/tran/6407