Asymptotically isometric and isometric copies of ℓ₁ in some Banach function lattices

• Autorzy: Kamińska A. , Mastyło M.

Identyfikatory

DOI

10.14708/cm.v53i2.798

• Języki publikacji: EN

Abstrakty

EN

We identify the class of Calderón-Lozanovskii spaces that do not contain an asymptotically isometric copy of ℓ₁, and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of ℓ₁⁽ⁿ⁾ for each integer n≥2. As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of ℓ₁. In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of ℓ₁⁽ⁿ⁾ and ℓ₁. The results are applied to obtain the description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.

Słowa kluczowe

EN

Calderón-Lozanovskii spaces; Orlicz-Lorentz spaces; Orlicz spaces; Lorentz spaces; Marcinkiewicz spaces; isometric copies of ℓ(n)1 and ℓ1; asymptotically isometric copies of ℓ1; fixed-point property

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2013
• Tom: Vol. 53, No. 2
• Strony: 283--300
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Kamińska A.

• Department of Mathematical Sciences, The University of Memphis, Memphis, TN 3815, USA

autor

Mastyło M.

• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Polish Academy of Sciences (Poznan branch), Institute of Mathematics, Umultowska 87, 61-614, Poznan, Poland

Bibliografia

• [1] Y. Abramovich, Operators preserving disjointness on rearrangement invariant spaces, Pacific J. Math. 148 (1991), no. 2, 201–206.
• [2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
• [3] I. Briskin and E. M. Semenov, Some geometrical properties of r.i. spaces, Progress in Nonlinear Differential Equations and their Applications 40 (2000), 47–54.
• [4] C. Choi, A. Kaminska and H. J. Lee, Complex convexity of Orlicz-Lorentz spaces and its applications, Bull. Pol. Acad. Sci. Math. 52 (2004), no. 1, 19–38.
• [5] S. J. Dilworth, Maria Girardi and J. Hagler, Dual Banach spaces which contain an isometric copy of L1, Bull. Polish. Acad. Sci. 48 (2000), 1–12.
• [6] P. N. Dowling, C. J. Lennard and B. Turrett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996), 652–662.
• [7] P. N. Dowling and C. J. Lennard, Every nonreflexive subspace of L1 fails the fixed point property, Proc. Amer. Math. Soc. 125 (1997), 443–174.
• [8] T. S. Ferguson, A representation of the symmetric bivariate Cauchy distribution, Ann. Math. Statist. 33 (1962), 1256–1266.
• [9] S. Guerre-Delabrière, Classical Sequences in Banach Spaces, Monographs and Textbooks in Pure and Applied Mathematics 166, Marcel Dekker, New York, 1992.
• [10] C. S. Herz, A class of negative-definite functions, Proc. Amer. Math. Soc. 14 (1963), 670–676.
• [11] A. Kaminska, Some remarks on Orlicz-Lorentz spaces, Math. Nachrichten 147 (1990), 29–38.
• [12] S. G. Krein, Yu. I. Petunin and E. M. Semenov, Interpolation of linear operators. Translations of Mathematical Monographs, Vol. 54, American Mathematical Society, Providence R.I., 1982.
• [13] J. Kolmos, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229.
• [14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer-Verlag, 1979.
• [15] G. J. Lozanovski˘ı, Mappings of Banach lattices of measurable functions, Soviet Math. (Iz. VUZ) 22 (1978), 61–63.
• [16] M. Mastyło, Interpolation of linear operators in Calderón-Lozanovskii spaces, Comment. Math. Prace Mat. 26 (1986), 247–256.
• [17] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983.
• [18] S. Reisner, On two theorems of Lozanovskii concerning intermediate Banach lattices, Lecture Notes in Mathematics 1317, Springer Verlag, Israel Seminar 1986 - 87 (Gafa), 57–83.
• [19] M. Wójtowicz, Contractive projections onto isometric copies of L1(v), Bull. Polish Math. Acad. Sci. Math. 51 (2003), 1–12.