Previous |  Up |  Next

Article

Keywords:
compact approximation property; weakly compact approximation property; ideals of homogeneous polynomials
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 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.
References:
[1] R. M.  Aron, G. Galindo: Weakly compact multilinear mappings. Proc. Edinb. Math. Soc. 40 (1997), 181–192. DOI 10.1017/S0013091500023543 | MR 1437822
[2] R. M.  Aron, C.  Hervés, and M.  Valdivia: Weakly continuous mappings on Banach spaces. J.  Funct. Anal. 52 (1983), 189–204. DOI 10.1016/0022-1236(83)90081-2 | MR 0707203
[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 10.1017/S0305004100068638 | MR 1027789
[4] G.  Botelho: Weakly compact and absolutely summing polynomials. J. Math. Anal. Appl. 265 (2002), 458–462. DOI 10.1006/jmaa.2001.7674 | MR 1876152 | Zbl 1036.46034
[5] G.  Botelho: Ideals of polynomials generated by weakly compact operators. Note Mat. 25 (2005/2006), 69–102. MR 2220454
[6] G.  Botelho, D. M.  Pellegrino: Two new properties of ideals of polynomials and applications. Indag. Math. 16 (2005), 157–169. DOI 10.1016/S0019-3577(05)80019-9 | MR 2319290
[7] C.  Boyd: Montel and reflexive preduals of the space of holomorphic functions. Stud. Math. 107 (1993), 305–315. MR 1247205
[8] H. A.  Braunss, H.  Junek: Factorization of injective ideals by interpolation. J.  Math. Anal. Appl. 297 (2004), 740–750. DOI 10.1016/j.jmaa.2004.04.045 | MR 2088691
[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 1863695 | Zbl 1067.46025
[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 2040241
[12] E. Çalışkan: Approximation of holomorphic mappings on infinite dimensional spaces. Rev. Mat. Complut. 17 (2004), 411–434. MR 2083963
[13] A. M.  Davie: The approximation problem for Banach spaces. Bull. London Math. Soc. 5 (1973), 261–266. DOI 10.1112/blms/5.3.261 | MR 0338735 | Zbl 0267.46013
[14] W.  Davis, T.  Figiel, W.  Johnson, and A. Pełczyński: Factoring weakly compact operators. J.  Funct. Anal. 17 (1974), 311–327. DOI 10.1016/0022-1236(74)90044-5 | MR 0355536
[15] S.  Dineen: Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Math. Springer-Verlag, Berlin, 1999. MR 1705327
[16] T.  Figiel: Factorization of compact opertors and applications to the approximation problem. Stud. Math. 45 (1973), 191–210. MR 0336294
[17] N.  Grønbæk, G. A.  Willis: Approximate identities in Banach algebras of compact operators. Can. Math. Bull. 36 (1993), 45–53. DOI 10.4153/CMB-1993-008-8 | MR 1205894
[18] S.  Heinrich: Closed operator ideals and interpolation. J.  Funct. Anal. 35 (1980), 397–411. DOI 10.1016/0022-1236(80)90089-0 | MR 0563562 | 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. DOI 10.1007/BF02810673 | MR 1802659
[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 0417761 | Zbl 0232.46019
[21] J.  Lindenstrauss, L.  Tzafriri: Classical Banach Spaces  I. Sequence Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0500056
[22] J.  Mujica: Complex Analysis in Banach Spaces. North-Holland Math. Stud. North-Holland, Amsterdam, 1986. MR 0842435
[23] J.  Mujica: Linearization of bounded holomorphic mappings on Banach spaces. Trans. Am. Math. Soc. 324 (1991), 867–887. DOI 10.1090/S0002-9947-1991-1000146-2 | MR 1000146 | Zbl 0747.46038
[24] J.  Mujica: Reflexive spaces of homogeneous polynomials. Bull. Pol. Acad. Sci. Math. 49 (2001), 211–223. MR 1863260 | Zbl 1068.46027
[25] J.  Mujica, M.  Valdivia: Holomorphic germs on Tsirelson’s space. Proc. Am. Math. Soc. 123 (1995), 1379–1384. MR 1219730
[26] A.  Pietsch: Operator Ideals. North Holland, Amsterdam, 1980. MR 0582655 | Zbl 0455.47032
[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 0763541 | 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. MR 1184105 | Zbl 0814.46017
Partner of
EuDML logo