Lipschitz slices and the Daugavet equation for Lipschitz operators
HTML articles powered by AMS MathViewer
- by Vladimir Kadets, Miguel Martín, Javier Merí and Dirk Werner PDF
- Proc. Amer. Math. Soc. 143 (2015), 5281-5292 Request permission
Abstract:
We introduce a substitute for the concept of slice for the case of non-linear Lipschitz functionals and transfer to the non-linear case some results about the Daugavet and the alternative Daugavet equations previously known only for linear operators.References
- Antonio Avilés, Vladimir Kadets, Miguel Martín, Javier Merí, and Varvara Shepelska, Slicely countably determined Banach spaces, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4871–4900. MR 2645054, DOI 10.1090/S0002-9947-10-05038-5
- Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, and Javier Merí, Properties of lush spaces and applications to Banach spaces with numerical index 1, Studia Math. 190 (2009), no. 2, 117–133. MR 2461290, DOI 10.4064/sm190-2-2
- Kostyantyn Boyko, Vladimir Kadets, Miguel Martín, and Dirk Werner, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 93–102. MR 2296393, DOI 10.1017/S0305004106009650
- Gustave Choquet, Lectures on analysis. Vol. II: Representation theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Edited by J. Marsden, T. Lance and S. Gelbart. MR 0250012
- J. Duncan, C. M. McGregor, J. D. Pryce, and A. J. White, The numerical index of a normed space, J. London Math. Soc. (2) 2 (1970), 481–488. MR 264371, DOI 10.1112/jlms/2.Part_{3}.481
- Vladimir Kadets, Miguel Martín, Javier Merí, and Varvara Shepelska, Lushness, numerical index one and duality, J. Math. Anal. Appl. 357 (2009), no. 1, 15–24. MR 2526802, DOI 10.1016/j.jmaa.2009.03.055
- Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855–873. MR 1621757, DOI 10.1090/S0002-9947-99-02377-6
- Vladimir Kadets, Miguel Martín, and Rafael Payá, Recent progress and open questions on the numerical index of Banach spaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 155–182 (English, with English and Spanish summaries). MR 2267407
- Vladimir Kadets, Miguel Martín, Javier Merí, and Rafael Payá, Convexity and smoothness of Banach spaces with numerical index one, Illinois J. Math. 53 (2009), no. 1, 163–182. MR 2584940
- Vladimir Kadets, Miguel Martín, Javier Merí, and Dirk Werner, Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces, Canad. J. Math. 65 (2013), no. 2, 331–348. MR 3028566, DOI 10.4153/CJM-2011-096-2
- Vladimir M. Kadets, Roman V. Shvidkoy, and Dirk Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math. 147 (2001), no. 3, 269–298. MR 1853772, DOI 10.4064/sm147-3-5
- Miguel Martín and Timur Oikhberg, An alternative Daugavet property, J. Math. Anal. Appl. 294 (2004), no. 1, 158–180. MR 2059797, DOI 10.1016/j.jmaa.2004.02.006
- Enrique A. Sánchez Pérez and Dirk Werner, Slice continuity for operators and the Daugavet property for bilinear maps, Funct. Approx. Comment. Math. 50 (2014), no. 2, 251–269. MR 3229060, DOI 10.7169/facm/2014.50.2.4
- R. V. Shvydkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), no. 2, 198–212. MR 1784413, DOI 10.1006/jfan.2000.3626
- Ruidong Wang, The numerical radius of Lipschitz operators on Banach spaces, Studia Math. 209 (2012), no. 1, 43–52. MR 2914928, DOI 10.4064/sm209-1-4
- Ruidong Wang, Xujian Huang, and Dongni Tan, On the numerical radius of Lipschitz operators in Banach spaces, J. Math. Anal. Appl. 411 (2014), no. 1, 1–18. MR 3118463, DOI 10.1016/j.jmaa.2013.08.054
Additional Information
- Vladimir Kadets
- Affiliation: Department of Mechanics and Mathematics, Kharkiv National University, pl. Svobody 4, 61077 Kharkiv, Ukraine
- MR Author ID: 202226
- ORCID: 0000-0002-5606-2679
- Email: vova1kadets@yahoo.com
- Miguel Martín
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- MR Author ID: 643000
- ORCID: 0000-0003-4502-798X
- Email: mmartins@ugr.es
- Javier Merí
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
- MR Author ID: 739081
- Email: jmeri@ugr.es
- Dirk Werner
- Affiliation: Department of Mathematics, Freie Universität Berlin, Arnimallee 6, D-14 195 Berlin, Germany
- Email: werner@math.fu-berlin.de
- Received by editor(s): September 25, 2014
- Published electronically: July 30, 2015
- Additional Notes: The work of the first named author was partially done during his visit to the University of Granada in June and July 2013 under the support of Spanish MINECO and FEDER project no. MTM2012-31755. The second and third authors were partially supported by Spanish MICINN and FEDER project no. MTM2012-31755 and by Junta de Andalucía and FEDER grants FQM-185 and P09-FQM-4911.
- Communicated by: Thomas Schlumprecht
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5281-5292
- MSC (2010): Primary 46B04; Secondary 46B80, 46B22, 47A12
- DOI: https://doi.org/10.1090/proc/12818
- MathSciNet review: 3411146