Integral polynomials on Banach spaces not containing ℓ 1

Raffaella Cilia; Joaquín M. Gutiérrez

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 1, page 221-231
  • ISSN: 0011-4642

Abstract

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We give new characterizations of Banach spaces not containing ℓ 1 in terms of integral and p -dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.

How to cite

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Cilia, Raffaella, and Gutiérrez, Joaquín M.. "Integral polynomials on Banach spaces not containing $\ell _1$." Czechoslovak Mathematical Journal 60.1 (2010): 221-231. <http://eudml.org/doc/38003>.

@article{Cilia2010,
abstract = {We give new characterizations of Banach spaces not containing $\ell _1$ in terms of integral and $p$-dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.},
author = {Cilia, Raffaella, Gutiérrez, Joaquín M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {(Pietsch) integral polynomial; Banach space not containing $\ell _1$; $p$-dominated polynomial; (Pietsch) integral polynomial; Banach space not containing , -dominated polynomial},
language = {eng},
number = {1},
pages = {221-231},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Integral polynomials on Banach spaces not containing $\ell _1$},
url = {http://eudml.org/doc/38003},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Cilia, Raffaella
AU - Gutiérrez, Joaquín M.
TI - Integral polynomials on Banach spaces not containing $\ell _1$
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 221
EP - 231
AB - We give new characterizations of Banach spaces not containing $\ell _1$ in terms of integral and $p$-dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.
LA - eng
KW - (Pietsch) integral polynomial; Banach space not containing $\ell _1$; $p$-dominated polynomial; (Pietsch) integral polynomial; Banach space not containing , -dominated polynomial
UR - http://eudml.org/doc/38003
ER -

References

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  1. Alencar, R., On reflexivity and basis for 𝒫 ( m E ) , Proc. Roy. Irish Acad. Sect. A 85 (1985), 131-138. (1985) MR0845536
  2. Ansemil, J. M., Floret, K., The symmetric tensor product of a direct sum of locally convex spaces, Studia Math. 129 (1998), 285-295. (1998) Zbl0931.46005MR1609655
  3. Aron, R. M., Berner, P. D., 10.24033/bsmf.1862, Bull. Soc. Math. France 106 (1978), 3-24. (1978) Zbl0378.46043MR0508947DOI10.24033/bsmf.1862
  4. Aron, R. M., Hervés, C., Valdivia, M., 10.1016/0022-1236(83)90081-2, J. Funct. Anal. 52 (1983), 189-204. (1983) MR0707203DOI10.1016/0022-1236(83)90081-2
  5. Aron, R. M., Prolla, J. B., Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195-216. (1980) Zbl0413.41022MR0552473
  6. Blasco, F., 10.4064/sm-123-2-165-173, Studia Math. 123 (1997), 165-173. (1997) Zbl0870.46028MR1439028DOI10.4064/sm-123-2-165-173
  7. Carando, D., Lassalle, S., 10.1016/j.jmaa.2004.10.020, J. Math. Anal. Appl. 307 (2005), 77-85. (2005) Zbl1082.46034MR2138976DOI10.1016/j.jmaa.2004.10.020
  8. Cardassi, C. S., Strictly p -integral and p -nuclear operators, in: Analyse harmonique: Groupe de travail sur les espaces de Banach invariants par translation, Exp. II, Publ. Math. Orsay (1989). (1989) MR1026052
  9. Cilia, R., D'Anna, M., Gutiérrez, J. M., 10.1016/S0022-247X(02)00461-4, J. Math. Anal. Appl. 275 (2002), 900-912. (2002) MR1943787DOI10.1016/S0022-247X(02)00461-4
  10. Cilia, R., Gutiérrez, J. M., 10.36045/bbms/1126195343, Bull. Belgian Math. Soc. Simon Stevin {12} (2005), 393-400. (2005) MR2173701DOI10.36045/bbms/1126195343
  11. Cilia, R., Gutiérrez, J. M., Integral and S-factorizable multilinear mappings, Proc. Roy. Soc. Edinburgh 136A (2006), 115-137. (2006) MR2217510
  12. Cilia, R., Gutiérrez, J. M., Ideals of integral and r -factorable polynomials, Bol. Soc. Mat. Mexicana 14, 3a Serie (2008). (2008) MR2667159
  13. Defant, A., Floret, K., Tensor Norms and Operator Ideals, Math. Studies, vol. 176, North-Holland, Amsterdam (1993). (1993) Zbl0774.46018MR1209438
  14. Diestel, J., Jarchow, H., Tonge, A., Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43 Cambridge University Press, Cambridge, 1995. Zbl1139.47021MR1342297
  15. J. Diestel, J. J. Uhl, Jr., Vector Measures, Math. Surveys Monographs 15, American Mathematical Society, Providence RI (1977). (1977) Zbl0369.46039MR0453964
  16. Dineen, S., Complex Analysis on Infinite Dimensional Spaces, Springer Monogr. Math., Springer, Berlin (1999). (1999) Zbl1034.46504MR1705327
  17. Emmanuele, G., Dominated operators on C [ 0 , 1 ] and the (CRP), {Collect. Math.} 41 (1990), 21-25. (1990) Zbl0752.47006MR1134442
  18. Emmanuele, G., 10.5186/aasfm.1991.1608, Ann. Acad. Sci. Fenn. Math. 16 (1991), 243-248. (1991) Zbl0707.47021MR1139796DOI10.5186/aasfm.1991.1608
  19. Floret, K., Natural norms on symmetric tensor products of normed spaces, Note Mat. 17 (1997), 153-188. (1997) Zbl0961.46013MR1749787
  20. Floret, K., On ideals of n -homogeneous polynomials on Banach spaces, Topological algebras with applications to differential geometry and mathematical physics (Athens, 1999), 19-38, Univ. Athens, Athens (2002). (2002) Zbl1095.46026MR2000732
  21. Floret, K., Hunfeld, S., 10.1090/S0002-9939-01-06228-1, Proc. Amer. Math. Soc. 130 (2002), 1425-1435. (2002) Zbl1027.46054MR1879966DOI10.1090/S0002-9939-01-06228-1
  22. Gutiérrez, J. M., 10.2307/2159836, Proc. Amer. Math. Soc. 119 (1993), 147-152. (1993) MR1158000DOI10.2307/2159836
  23. Gutiérrez, J. M., Villanueva, I., Extensions of multilinear operators and Banach space properties, Proc. Roy. Soc. Edinburgh 133A (2003), 549-566. (2003) MR1983686
  24. Matos, M. C., Absolutely summing holomorphic mappings, An. Acad. Bras. Ci. 68 (1996), 1-13. (1996) Zbl0854.46042MR1752625
  25. Meléndez, Y., Tonge, A., Polynomials and the Pietsch domination theorem, Proc. Roy. Irish Acad. Sect. A 99 (1999), 195-212. (1999) MR1881812
  26. Mujica, J., Complex Analysis in Banach Spaces, Math. Studies, vol. 120, North-Holland, Amsterdam (1986). (1986) Zbl0586.46040MR0842435
  27. Musial, K., 10.1007/BFb0088234, D. Kölzow Measure Theory. Oberwolfach 1979, Lecture Notes in Math. 794, Springer, Berlin (1980), 324-339. (1980) Zbl0433.28010MR0577981DOI10.1007/BFb0088234
  28. Pisier, G., Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Reg. Conf. Ser. Math. 60, American Mathematical Society, Providence RI (1987). (1987) MR0829919
  29. Rosenthal, H. P., Some new characterizations of Banach spaces containing ℓ 1 , preprint, . 
  30. Ryan, R. A., Applications of topological tensor products to infinite dimensional holomorphy, Ph. D. Thesis, Trinity College, Dublin (1980). (1980) 
  31. Talagrand, M., Pettis Integral and Measure Theory, Mem. Amer. Math. Soc. 307, American Mathematical Society, Providence RI (1984). (1984) Zbl0582.46049MR0756174
  32. Villanueva, I., 10.1016/S0022-247X(02)00362-1, J. Math. Anal. Appl. 279 (2003), 56-70. (2003) Zbl1030.47503MR1970490DOI10.1016/S0022-247X(02)00362-1

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