Banach Journal of Mathematical Analysis

The geometry of L^p-spaces over atomless measure spaces and the Daugavet property

Enrique A. Sanchez Perez and Dirk Werner

Full-text: Open access

Abstract

We show that $L^p$-spaces over atomless measure spaces can be characterized in terms of a $p$-concavity type geometric property that is related with the Daugavet property.

Article information

Source
Banach J. Math. Anal., Volume 5, Number 1 (2011), 167-180.

Dates
First available in Project Euclid: 14 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1313362988

Digital Object Identifier
doi:10.15352/bjma/1313362988

Mathematical Reviews number (MathSciNet)
MR2738528

Zentralblatt MATH identifier
1215.46011

Subjects
Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B25: Classical Banach spaces in the general theory

Keywords
Daugavet property L_p-space

Citation

Sanchez Perez, Enrique A.; Werner, Dirk. The geometry of L^p-spaces over atomless measure spaces and the Daugavet property. Banach J. Math. Anal. 5 (2011), no. 1, 167--180. doi:10.15352/bjma/1313362988. https://projecteuclid.org/euclid.bjma/1313362988


Export citation

References

  • Yu.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, The Daugavet equation in uniformly convex Banach spaces, J. Funct. Anal. 97 (1991), 215–230.
  • Yu.A. Abramovich and C.D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics, Vol. 50. Amer. Math. Soc., Providence RI, 2002.
  • M.D. Acosta, A. Kamińska and M. Mastyło, The Daugavet property and weak neighborhoods in Banach lattices, Preprint 2009.
  • Y. Benyamini and P.K. Lin, An operator on $L^p$ without best compact approximation, Israel J. Math. 51 (1985), 298–304.
  • D. Bilik, V. Kadets, R. Shvidkoy and D. Werner, Narrow operators and the Daugavet property for ultraproducts, Positivity 9 (2005), 45–62.
  • K. Boyko and V. Kadets, Daugavet equation in $L_1$ as a limiting case of the Benyamini-Lin $L_p$ theorem, Kharkov National University Vestnik 645 (2004), 22–29.
  • V. Kadets, M. Martí n and J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007), 2385–2411.
  • V. Kadets, V. Shepelska and D. Werner, Quotients of Banach spaces with the Daugavet property, Bull. Pol. Acad. Sci. 56 (2008), 131–147.
  • V. Kadets, R. Shvidkoy, G. Sirotkin and D. Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), 855–873.
  • V. Kadets, R. Shvidkoy and D. Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), 269–298.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin, 1979.
  • M. Martí n, The Daugavetian index of a Banach space, Taiwan. J. Math. 7 (2003), 631–640.
  • M. Martí n and T. Oikhberg, An alternative Daugavet property, J. Math Anal. Appl. 294 (2004), 158–180.
  • P. Meyer-Nieberg, Banach Lattices, Springer, Berlin, 1991.
  • T. Oikhberg, Spaces of operators, the $\psi$-Daugavet property, and numerical indices, Positivity 9 (2005), 607–623.
  • S. Okada, W.J. Ricker and E.A. Sánchez Pérez, Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel, 2008.
  • M.M. Popov and B. Randrianantoanina, A pseudo-Daugavet property for narrow projections in Lorentz spaces, Illinois J. Math. 46 (2002), 1313–1338.
  • E.A. Sánchez Pérez and D. Werner, The $p$-Daugavet property for function spaces, Preprint 2010.
  • A. Schep, Daugavet type inequalities for operators on $L^p$-spaces, Positivity 7 (2003), 103–111.
  • D. Werner, Recent progress on the Daugavet property, Irish Math. Soc. Bulletin 46 (2001), 77–97.