On strong asymptotic uniform smoothness and convexity
Tóm tắt
We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth et al. (J Math Anal Appl 402(1):297–307, 2013). Our techniques also provide a characterisation of Orlicz functions M, N such that the space of compact operators
$$\mathscr { K}(h_M,h_N)$$
is asymptotically uniformly smooth. Finally we show that
$$\mathscr { K}(X, Y)$$
is not strictly convex whenever X and Y are at least two-dimensional, which extends a result by Dilworth and Kutzarova (Function Spaces (Edwardsville, IL, 1994), Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1995).
Tài liệu tham khảo
Altshuler, Z.: Uniform convexity in Lorentz sequence spaces. Isr. J. Math. 20(3–4), 260–274 (1975)
Ausekle, E.A., Oya, È.F.: Pitt’s theorem for Lorentz and Orlicz sequence spaces. Mat. Zametki 61(1), 18–25 (1997)
Besbes, M.: Points fixes dans les espaces des opérateurs nucléaires. Bull. Aust. Math. Soc. 46(2), 287–294 (1992)
Borel-Mathurin, L.: Isomorphismes non linéaires entre espaces de Banach. Ph.D. thesis, Université Paris 6 (2010)
Causey, R.M.: The Szlenk index of injective tensor products and convex hulls. J. Funct. Anal. 272(8), 3375–3409 (2017)
Delpech, S.: Asymptotic uniform moduli and Kottman constant of Orlicz sequence spaces. Rev. Mat. Complut. 22(2), 455–467 (2009)
Dilworth, S.J., Kutzarova, D.: Kadec-Klee properties for \(L(l_p,l_q)\). Function Spaces (Edwardsville, IL, 1994). Lecture Notes in Pure and Applied Mathematics, vol. 172, pp. 71–83. Dekker, New York (1995)
Dilworth, S.J., Kutzarova, D., Lancien, G., Randrianarivony, N.L.: Equivalent norms with the property (\(\beta \)) of Rolewicz. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 111, 101–113 (2016)
Dilworth, S.J., Kutzarova, D., Lovasoa Randrianarivony, N., Revalski, J.P., Zhivkov, N.V.: Compactly uniformly convex spaces and property \((\beta )\) of Rolewicz. J. Math. Anal. Appl. 402(1), 297–307 (2013)
Draga, S., Kochanek, T.: The Szlenk power type and tensor products of Banach spaces. Proc. Am. Math. Soc. 145, 1685–1698 (2017)
van Dulst, D., Sims, B.: Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK). Banach Space Theory and Its Applications (Bucharest, 1981). Lecture Notes in Mathematics, pp. 35–43. Springer, Berlin-NewYork (1983)
Dutriex, Y.: Géométrie non linéraire des espaces de Banach. Ph.D. thesis, Université Paris 6 (2002)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011)
Girardi, M.: The dual of the James tree space is asymptotically uniformly convex. Studia Math. 147(2), 119–130 (2001)
Gonzalo, R., Jaramillo, J.A., Troyanski, S.L.: High order smoothness and asymptotic structure in Banach spaces. J. Convex Anal. 14(2), 249–269 (2007)
Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mt. J. Math. 10(4), 743–749 (1980)
Johnson, W.B., Lindenstrauss, J., Preiss, D., Schechtman, G.: Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces. Proc. Lond. Math. Soc.(3) 84(3), 711–746 (2002)
Johnson, W.B., Oikhberg, T.: Separable lifting property and extensions of local reflexivity. Ill. J. Math. 45(1), 123–137 (2001)
Knaust, H., Odell, E., Schlumprecht, T.: On asymptotic structure, the Szlenk index and UKK properties in Banach spaces. Positivity 3(2), 173–199 (1999)
Lancien, G.: Théorie de l’indice et problèmes de renormage en géometie des espaces de banach. Ph.D. thesis, Université Paris 6 (1992)
Lennard, C.: \({C}_1\) is uniformly Kadec-Klee. Proc. Am. Math. Soc. 109(1), 71–77 (1990)
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, Berlin-New York (1977)
Prus, S.: Nearly uniformly smooth Banach spaces. Boll. Un. Mat. Ital. B(7) 3(3), 507–521 (1989)
Prus, S.: On infinite-dimensional uniform smoothness of Banach spaces. Comment. Math. Univ. Carolin. 40(1), 97–105 (1999)
Raja, M.: On asymptotically uniformly smooth Banach spaces. J. Funct. Anal. 264(2), 479–492 (2013)
Ruess, W.M., Stegall, C.P.: Exposed and denting points in duals of operator spaces. Isr. J. Math. 53(2), 163–190 (1986)
Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer, London (2002)
Schlumprecht, T.: On Zippin’s embedding theorem of Banach spaces into Banach spaces with bases. Adv. Math. 274, 833–880 (2015)