Contractive projections onto isometric copies of L1(ν) in strictly monotone Banach lattices

• Autorzy: Wójtowicz, M.
• Języki publikacji: EN

Abstrakty

EN

Let Y be a Banach lattice and X a strictly monotone sublattice of Y. Every isometric copy of an L1-space in X is 1-complemented in Y (Theorem l). This is an extension of the classical result of Pełczyński for Y = X = L1(my), and of Dor for Y = X being q-concave. We also study the consequences of the existence of isometric copies of l1 in strictly monotone E[phi](my)-spaces, where E[phi](my) denotes the ideal of an Orlicz space L[phi](my) of the elements with absolutely continuous norm.

Słowa kluczowe

PL

krata Banacha ściśle monotoniczna; izometria; rzutowanie zawężające; przestrzeń Orlicza; przestrzeń L1

EN

strictly monotone Banach; isometry; contractive projection; Orlicz space; L1-space

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 1
• Strony: 1-12
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Wójtowicz, M.

• Institute of Mathematics, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra, Poland,

Bibliografia

• [1] Y. A. Abramovich, Multiplicative representation of disjointness preserving operators, Indag. Math., 45 (1983) 265-279.
• [2] C. D. Aliprantis, O. Burkishaw, Positive operators, Academic Press, New York 1985.
• [3] S. Banach, S. Mazur, Zur Theorie der linearen Dimensionen, Studia Math., 4 (1933) 100-112.
• [4] J. Bourgain, A counterexample to a complementation problem, Compositio Math., 43 (1981) 133-144.
• [5] I. Briskin, E. M. Semenov, Some geometrical properties of rearrangement invariant spaces, Progr. Nonlinear Differential Equations Appl., 40 (2000) 47-54.
• [6] S. Chen, Geometry of Orlicz spaces, Dissertationes Math., 356 (1995).
• [7] L. E. Dor, On projections in L1, Ann. of Math., 102 (1975) 463-474.
• [8] L. E. Dor, On embedding of Lp-spaces in Lp-spaces, Dissertation, Ohio State University, 1975.
• [9] M. Duhoux, M. Meyer, A new proof of the lattice structure of orthomorphisms, J. London Math. Soc., 25 (1982) 375-378.
• [10] R. J. Fleming, J. E. Jamison, Isometries on Banach spaces: a survey, in: Analysis, geometry and groups: A Riemann legacy volume, 52-123; Hadronic Press, Palm Harbor 1993.
• [11] P. Foralewski, On some geometric properties of generalized Calderón-Lozanovskiĭ spaces, Acta Math. Hungar., 80 (1-2) (1998) 35-49.
• [12] H. Hudzik, Orlicz spaces containing a copy of l1, Math. Japonica, 34 (1989) 747-759.
• [13] H. Hudzik, A. Kamińska, Monotonicity properties of Lorentz spaces, Proc. Amer. Math. Soc., 123 (1995) 2715-2721.
• [14] H. Hudzik, A. Kamińska, M. Mastyło, Monotonicity and rotundity properties in Banach lattices, Rocky Mountain J. Math., 30 (2000) 933-949.
• [15] H. Hudzik, W. Kurc, Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices, J. Approx. Theory, 95 (1998) 353-386.
• [16] M. A. Krasnoselskiĭ, Ya. B. Rutickiĭ, Convex functions and Orlicz spaces, Groningen 1961.
• [17] W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J. Approx. Theory, 69 (1992) 173-187.
• [18] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I, Springer Verlag, Berlin Heidelberg New York 1977.
• [19] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Springer Verlag, Berlin Heidelberg New York 1979.
• [20] H. P. Lotz, Extensions and liftings of positive linear mappings on Banach lattices, Trans. Amer. Math. Soc., 211 (1975) 85-100.
• [21] W. A. J. Luxemburg, A. C. Zaanen, Riesz spaces I, North Holland, Amsterdam London 1971.
• [22] P. Meyer-Nieberg, Banach Lattices, Springer Verlag, Berlin Heidelberg New York 1991.
• [23] J. R. Partington, Subspaces of certain Banach sequence spaces, Bull. London Math. Soc., 13 (1981) 162-166.
• [24] A. Pełczyński, Projections in certain Banach spaces, Studia Math., 19 (1960) 209-228.
• [25] A. M. Plichko, D. Yost, Complemented and uncomplemented subspaces of Banach spaces, Extracta Math., 15 (2000) 335-371.
• [26] B. Randrianantoanina, Norm-one projections in Banach spaces, Taiwanese J. Math., 5 (2001) 25-95.
• [27] H. H. Schaefer, Banach lattices and positive operators, Springer Verlag, Berlin Heidelberg New York 1974.
• [28] W. Wnuk, On the order-topological properties of the quotient space L/LA, Studia Math., 79 (1984) 139-149.
• [29] W. Wnuk, Representations of Orlicz lattices, Dissertationes Math., 235 (1984) 1-66.
• [30] A. C. Zaanen, Riesz spaces II, North Holland, Amsterdam London 1983.