×

zbMATH — the first resource for mathematics

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces. (English) Zbl 1174.46008
Summary: We show that a Banach space  \(E\) has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on  \(E\) can be uniformly approximated on compact sets by homogeneous polynomials which belong to the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established for the compact approximation property.
For an erratum to this paper see ibid. 60, No. 3, 887–887 (2010; Zbl 1224.46033).

MSC:
46B28 Spaces of operators; tensor products; approximation properties
46G25 (Spaces of) multilinear mappings, polynomials
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] R. M. Aron, G. Galindo: Weakly compact multilinear mappings. Proc. Edinb. Math. Soc. 40 (1997), 181–192. · Zbl 0901.46038
[2] R. M. Aron, C. Hervés, and M. Valdivia: Weakly continuous mappings on Banach spaces. J. Funct. Anal. 52 (1983), 189–204. · Zbl 0517.46019
[3] K. Astala, H.-O. Tylli: Seminorms related to weak compactness and to Tauberian operators. Math. Proc. Camb. Philos. Soc. 107 (1990), 367–375. · Zbl 0709.47009
[4] G. Botelho: Weakly compact and absolutely summing polynomials. J. Math. Anal. Appl. 265 (2002), 458–462. · Zbl 1036.46034
[5] G. Botelho: Ideals of polynomials generated by weakly compact operators. Note Mat. 25 (2005/2006), 69–102. · Zbl 1223.46047
[6] G. Botelho, D. M. Pellegrino: Two new properties of ideals of polynomials and applications. Indag. Math. 16 (2005), 157–169. · Zbl 1089.46027
[7] C. Boyd: Montel and reexive preduals of the space of holomorphic functions. Stud. Math. 107 (1993), 305–315. · Zbl 0811.46029
[8] H. A. Braunss, H. Junek: Factorization of injective ideals by interpolation. J. Math. Anal. Appl. 297 (2004), 740–750. · Zbl 1053.47059
[9] P. G. Casazza: Approximation properties. In: Handbook of the Geometry of Banach Spaces, Vol. I (W. Johnson, J. Lindenstrauss, eds.). North-Holland, Amsterdam, 2001, pp. 271–316.
[10] E. Çalışkan: Aproximação de funções holomorfas em espaços de dimensão in-nita. PhD. Thesis. Universidade Estadual de Campinas, São Paulo, 2003.
[11] E. Çalışkan: Bounded holomorphic mappings and the compact approximation property. Port. Math. 61 (2004), 25–33. · Zbl 1059.46025
[12] E. Çalışkan: Approximation of holomorphic mappings on infinite dimensional spaces. Rev. Mat. Complut. 17 (2004), 411–434. · Zbl 1061.46040
[13] A. M. Davie: The approximation problem for Banach spaces. Bull. London Math. Soc. 5 (1973), 261–266. · Zbl 0267.46013
[14] W. Davis, T. Figiel, W. Johnson, and A. Pełczyński: Factoring weakly compact operators. J. Funct. Anal. 17 (1974), 311–327. · Zbl 0306.46020
[15] S. Dineen: Complex Analysis on In-nite Dimensional Spaces. Springer Monographs in Math. Springer-Verlag, Berlin, 1999. · Zbl 1034.46504
[16] T. Figiel: Factorization of compact opertors and applications to the approximation problem. Stud. Math. 45 (1973), 191–210. · Zbl 0257.47017
[17] N. Grønbæk, G. A. Willis: Approximate identities in Banach algebras of compact operators. Can. Math. Bull. 36 (1993), 45–53. · Zbl 0794.46017
[18] S. Heinrich: Closed operator ideals and interpolation. J. Funct. Anal. 35 (1980), 397–411. · Zbl 0439.47029
[19] Å. Lima, O. Nygaard, and E. Oja: Isometric factorization of weakly compact operators and the approximation property. Isr. J. Math. 119 (2000), 325–348. · Zbl 0983.46024
[20] J. Lindenstrauss: Weakly compact sets–their topological properties and the Banach spaces they generate. In: Symposium on Infinite Dimensional Topology. Ann. Math. Stud. (R. D. Anderson, eds.). Princeton Univ. Press, Princeton, 1972, pp. 235–273. · Zbl 0232.46019
[21] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I. Sequence Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0362.46013
[22] J. Mujica: Complex Analysis in Banach Spaces. North-Holland Math. Stud. North-Holland, Amsterdam, 1986. · Zbl 0586.46040
[23] J. Mujica: Linearization of bounded holomorphic mappings on Banach spaces. Trans. Am. Math. Soc. 324 (1991), 867–887. · Zbl 0747.46038
[24] J. Mujica: Reflexive spaces of homogeneous polynomials. Bull. Pol. Acad. Sci. Math. 49 (2001), 211–223. · Zbl 1068.46027
[25] J. Mujica, M. Valdivia: Holomorphic germs on Tsirelson’s space. Proc. Am. Math. Soc. 123 (1995), 1379–1384. · Zbl 0823.46047
[26] A. Pietsch: Operator Ideals. North Holland, Amsterdam, 1980.
[27] A. Pietsch: Ideals of multilinear functionals. In: Proceedings of the Second International Conference on Operator Algebras, Ideals and Applications in Theoretical Physics. Teubner, Leipzig, 1983, pp. 185–199. · Zbl 0561.47037
[28] R. Ryan: Applications of topological tensor products to infinite dimensional holomorphy. PhD. Thesis. Trinity College, Dublin, 1980.
[29] G. Willis: The compact approximation property does not imply the approximation property. Stud. Math. 103 (1992), 99–108. · Zbl 0814.46017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.