| Title:
|
Integral polynomials on Banach spaces not containing $\ell _1$ (English) |
| Author:
|
Cilia, Raffaella |
| Author:
|
Gutiérrez, Joaquín M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
221-231 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
(Pietsch) integral polynomial |
| Keyword:
|
Banach space not containing $\ell _1$ |
| Keyword:
|
$p$-dominated polynomial |
| MSC:
|
46B20 |
| MSC:
|
46G25 |
| MSC:
|
47H60 |
| idZBL:
|
Zbl 1224.46088 |
| idMR:
|
MR2595085 |
| . |
| Date available:
|
2010-07-20T16:30:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140564 |
| . |
| Reference:
|
[1] Alencar, R.: On reflexivity and basis for ${\mathcal P}(^m E)$.Proc. Roy. Irish Acad. Sect. A 85 (1985), 131-138. MR 0845536 |
| Reference:
|
[2] Ansemil, J. M., Floret, K.: The symmetric tensor product of a direct sum of locally convex spaces.Studia Math. 129 (1998), 285-295. Zbl 0931.46005, MR 1609655 |
| Reference:
|
[3] Aron, R. M., Berner, P. D.: A Hahn-Banach extension theorem for analytic mappings.Bull. Soc. Math. France 106 (1978), 3-24. Zbl 0378.46043, MR 0508947, 10.24033/bsmf.1862 |
| Reference:
|
[4] Aron, R. M., Hervés, C., Valdivia, M.: Weakly continuous mappings on Banach spaces.J. Funct. Anal. 52 (1983), 189-204. MR 0707203, 10.1016/0022-1236(83)90081-2 |
| Reference:
|
[5] Aron, R. M., Prolla, J. B.: Polynomial approximation of differentiable functions on Banach spaces.J. Reine Angew. Math. 313 (1980), 195-216. Zbl 0413.41022, MR 0552473 |
| Reference:
|
[6] Blasco, F.: Complementation in spaces of symmetric tensor products and polynomials.Studia Math. 123 (1997), 165-173. Zbl 0870.46028, MR 1439028, 10.4064/sm-123-2-165-173 |
| Reference:
|
[7] Carando, D., Lassalle, S.: Extension of vector-valued integral polynomials.J. Math. Anal. Appl. 307 (2005), 77-85. Zbl 1082.46034, MR 2138976, 10.1016/j.jmaa.2004.10.020 |
| Reference:
|
[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). MR 1026052 |
| Reference:
|
[9] Cilia, R., D'Anna, M., Gutiérrez, J. M.: Polynomial characterization of ${\mathscr L}_\infty$-spaces.J. Math. Anal. Appl. 275 (2002), 900-912. MR 1943787, 10.1016/S0022-247X(02)00461-4 |
| Reference:
|
[10] Cilia, R., Gutiérrez, J. M.: Polynomial characterization of Asplund spaces.Bull. Belgian Math. Soc. Simon Stevin {12} (2005), 393-400. MR 2173701, 10.36045/bbms/1126195343 |
| Reference:
|
[11] Cilia, R., Gutiérrez, J. M.: Integral and S-factorizable multilinear mappings.Proc. Roy. Soc. Edinburgh 136A (2006), 115-137. MR 2217510 |
| Reference:
|
[12] Cilia, R., Gutiérrez, J. M.: Ideals of integral and $r$-factorable polynomials.Bol. Soc. Mat. Mexicana 14, 3a Serie (2008). MR 2667159 |
| Reference:
|
[13] Defant, A., Floret, K.: Tensor Norms and Operator Ideals.Math. Studies, vol. 176, North-Holland, Amsterdam (1993). Zbl 0774.46018, MR 1209438 |
| Reference:
|
[14] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators.Cambridge Stud. Adv. Math. 43 Cambridge University Press, Cambridge, 1995. Zbl 1139.47021, MR 1342297 |
| Reference:
|
[15] J. Diestel, J. J. Uhl, Jr.: Vector Measures.Math. Surveys Monographs 15, American Mathematical Society, Providence RI (1977). Zbl 0369.46039, MR 0453964 |
| Reference:
|
[16] Dineen, S.: Complex Analysis on Infinite Dimensional Spaces.Springer Monogr. Math., Springer, Berlin (1999). Zbl 1034.46504, MR 1705327 |
| Reference:
|
[17] Emmanuele, G.: Dominated operators on $C[0,1]$ and the (CRP).{Collect. Math.} 41 (1990), 21-25. Zbl 0752.47006, MR 1134442 |
| Reference:
|
[18] Emmanuele, G.: Banach spaces with the (CRP) and dominated operators on $C(K)$.Ann. Acad. Sci. Fenn. Math. 16 (1991), 243-248. Zbl 0707.47021, MR 1139796, 10.5186/aasfm.1991.1608 |
| Reference:
|
[19] Floret, K.: Natural norms on symmetric tensor products of normed spaces.Note Mat. 17 (1997), 153-188. Zbl 0961.46013, MR 1749787 |
| Reference:
|
[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). Zbl 1095.46026, MR 2000732 |
| Reference:
|
[21] Floret, K., Hunfeld, S.: Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces.Proc. Amer. Math. Soc. 130 (2002), 1425-1435. Zbl 1027.46054, MR 1879966, 10.1090/S0002-9939-01-06228-1 |
| Reference:
|
[22] Gutiérrez, J. M.: Weakly continuous functions on Banach spaces not containing $\ell_1$.Proc. Amer. Math. Soc. 119 (1993), 147-152. MR 1158000, 10.2307/2159836 |
| Reference:
|
[23] Gutiérrez, J. M., Villanueva, I.: Extensions of multilinear operators and Banach space properties.Proc. Roy. Soc. Edinburgh 133A (2003), 549-566. MR 1983686 |
| Reference:
|
[24] Matos, M. C.: Absolutely summing holomorphic mappings.An. Acad. Bras. Ci. 68 (1996), 1-13. Zbl 0854.46042, MR 1752625 |
| Reference:
|
[25] Meléndez, Y., Tonge, A.: Polynomials and the Pietsch domination theorem.Proc. Roy. Irish Acad. Sect. A 99 (1999), 195-212. MR 1881812 |
| Reference:
|
[26] Mujica, J.: Complex Analysis in Banach Spaces.Math. Studies, vol. 120, North-Holland, Amsterdam (1986). Zbl 0586.46040, MR 0842435 |
| Reference:
|
[27] Musial, K.: Martingales of Pettis integrable functions.D. Kölzow Measure Theory. Oberwolfach 1979, Lecture Notes in Math. 794, Springer, Berlin (1980), 324-339. Zbl 0433.28010, MR 0577981, 10.1007/BFb0088234 |
| Reference:
|
[28] Pisier, G.: Factorization of Linear Operators and Geometry of Banach Spaces.CBMS Reg. Conf. Ser. Math. 60, American Mathematical Society, Providence RI (1987). MR 0829919 |
| Reference:
|
[29] Rosenthal, H. P.: Some new characterizations of Banach spaces containing $\ell_1$, preprint.. |
| Reference:
|
[30] Ryan, R. A.: Applications of topological tensor products to infinite dimensional holomorphy.Ph. D. Thesis, Trinity College, Dublin (1980). |
| Reference:
|
[31] Talagrand, M.: Pettis Integral and Measure Theory.Mem. Amer. Math. Soc. 307, American Mathematical Society, Providence RI (1984). Zbl 0582.46049, MR 0756174 |
| Reference:
|
[32] Villanueva, I.: Integral mappings between Banach spaces.J. Math. Anal. Appl. 279 (2003), 56-70. Zbl 1030.47503, MR 1970490, 10.1016/S0022-247X(02)00362-1 |
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