Quotients of Banach spaces with the Daugavet property

• Autorzy: Kadets V. , Shepelska V. , Werner D.
• Języki publikacji: EN

Abstrakty

EN

We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak* analogue. We introduce and study analogues of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L1[0,1] by an l1-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.

Słowa kluczowe

EN

Daugavet property; Daugavet equation; rich subspace; poor subspace; narrow operator; Radon-Nikodym property

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: Vol. 56, no 2
• Strony: 131--147
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Kadets V.

autor

Shepelska V.

autor

Werner D.

• Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine,

Bibliografia

• [1] D. Bilik, V. M. Kadets, R. V. Shvidkoy and D. Werner, Narrow operators and the Daugavet property for ultraproducts, Positivity 9 (2005) , 45-62.
• [2] J. Bourgain, La propriete de Radon-Nikodym, Publ. Math. Univ. Pierre et Marie Curie 36 (1979).
• [3] J. Bourgain and H. P. Rosenthal, Martingales valued in certain subspaces of L1 , Israel J. Math. 37 (1980), 54-75.
• [4] R. D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodym Property, Lecture Notes in Math. 993, Springer, Berlin, 1983.
• [5] G. Choquet, Lectures on Analysis, Vol. II, W. A. Benjamin, New York, 1969.
• [6] N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience Publ., New York, 1958.
• [7] V. M. Kadets, N. Kalton, and D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003), 195-206.
• [8] V. M. Kadets and M. M. Popov, The Daugavet property for narrow operators in rich subspaces of C[0, 1] and L1[0, 1], St. Petersburg Math. J. 8 (1997), 571-584.
• [9] V. M. Kadets, R. V. Shvidkoy, G. G. Sirotkin, and D. Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), 855-873.
• [10] V. M. Kadets, R. V. Shvidkoy, and D. Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), 269-298.
• [11] V. M. Kadets and D. Werner, A Banach space with the Schur and the Daugavet property, Proc. Amer. Math. Soc. 132 (2004), 1765-1773.
• [12] A. M. Plichko and M. M. Popov, Symmetric function spaces on atomless probability spaces, Dissertationes Math. 306 (1990).
• [13] R. V. Shvydkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), 198-212.
• [14] M. Talagrand, The three-space problem for L1, J. Amer. Math. Soc. 3 (1990), 9-29.