Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

Çalışkan, Erhan

Czechoslovak Mathematical Journal, Tome 57 (2007) no. 2, pp. 763-776

Abstract (VO)

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 are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.

Détail

Classification : 46B28, 46G20, 46G25, 47B10, 47L20
Keywords: compact approximation property; weakly compact approximation property; ideals of homogeneous polynomials

Bibliographie

[1] R. M.  Aron, G. Galindo: Weakly compact multilinear mappings. Proc. Edinb. Math. Soc. 40 (1997), 181–192. | DOI | MR

[2] R. M.  Aron, C.  Hervés, and M.  Valdivia: Weakly continuous mappings on Banach spaces. J.  Funct. Anal. 52 (1983), 189–204. | DOI | MR

[3] K.  Astala, H.-O.  Tylli: Seminorms related to weak compactness and to Tauberian operators. Math. Proc. Camb. Philos. Soc. 107 (1990), 367–375. | DOI | MR

[4] G.  Botelho: Weakly compact and absolutely summing polynomials. J. Math. Anal. Appl. 265 (2002), 458–462. | DOI | MR | Zbl

[5] G.  Botelho: Ideals of polynomials generated by weakly compact operators. Note Mat. 25 (2005/2006), 69–102. | MR

[6] G.  Botelho, D. M.  Pellegrino: Two new properties of ideals of polynomials and applications. Indag. Math. 16 (2005), 157–169. | DOI | MR

[7] C.  Boyd: Montel and reflexive preduals of the space of holomorphic functions. Stud. Math. 107 (1993), 305–315. | DOI | MR

[8] H. A.  Braunss, H.  Junek: Factorization of injective ideals by interpolation. J.  Math. Anal. Appl. 297 (2004), 740–750. | DOI | MR

[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. | MR | Zbl

[10] E.  Çalışkan: Aproximação de funções holomorfas em espaços de dimensão infinita. 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. | MR

[12] E. Çalışkan: Approximation of holomorphic mappings on infinite dimensional spaces. Rev. Mat. Complut. 17 (2004), 411–434. | MR

[13] A. M.  Davie: The approximation problem for Banach spaces. Bull. London Math. Soc. 5 (1973), 261–266. | DOI | MR | Zbl

[14] W.  Davis, T.  Figiel, W.  Johnson, and A. Pełczyński: Factoring weakly compact operators. J.  Funct. Anal. 17 (1974), 311–327. | DOI | MR

[15] S.  Dineen: Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Math. Springer-Verlag, Berlin, 1999. | MR

[16] T.  Figiel: Factorization of compact opertors and applications to the approximation problem. Stud. Math. 45 (1973), 191–210. | DOI | MR

[17] N.  Grønbæk, G. A.  Willis: Approximate identities in Banach algebras of compact operators. Can. Math. Bull. 36 (1993), 45–53. | DOI | MR

[18] S.  Heinrich: Closed operator ideals and interpolation. J.  Funct. Anal. 35 (1980), 397–411. | DOI | MR | Zbl

[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. | DOI | MR

[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. | MR | Zbl

[21] J.  Lindenstrauss, L.  Tzafriri: Classical Banach Spaces  I. Sequence Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1977. | MR

[22] J.  Mujica: Complex Analysis in Banach Spaces. North-Holland Math. Stud. North-Holland, Amsterdam, 1986. | MR

[23] J.  Mujica: Linearization of bounded holomorphic mappings on Banach spaces. Trans. Am. Math. Soc. 324 (1991), 867–887. | DOI | MR | Zbl

[24] J.  Mujica: Reflexive spaces of homogeneous polynomials. Bull. Pol. Acad. Sci. Math. 49 (2001), 211–223. | MR | Zbl

[25] J.  Mujica, M.  Valdivia: Holomorphic germs on Tsirelson’s space. Proc. Am. Math. Soc. 123 (1995), 1379–1384. | MR

[26] A.  Pietsch: Operator Ideals. North Holland, Amsterdam, 1980. | MR | Zbl

[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. | MR | Zbl

[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. | DOI | MR | Zbl