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