Asymptotically isometric copies of ℓ10
Asymptotically isometric copies of ℓ10
Using James’ Distortion Theorems, researchers have inquired relations between spacescontaining nice copies of c0 or ℓ 1 and the failure of the fixed point property for nonexpansivemappings especially after the fact that every classical nonreflexive Banach space containsan isometric copy of either ℓ 1 or c0. For instance, finding asymptotically isometric (ai) copies of ℓ 1 or c0 inside a Banach space reveals the space’s failure of the fixed point property for nonexpansive mappings. There has been many researches done using these tools developed by James and followed by Dowling, Lennard, and Turett mainly to see if a Banach space can be renormed to have the fixed point property for nonexpansive mappings when there is failure.In this paper, we introduce the concept of Banach spaces containing ai copies of ℓ 10 andgive alternative methods of detecting them. We show the relations between spaces containing these copies and the failure of the fixed point property for nonexpansive mappings.Finally, we give some remarks and examples pointing our vital result: if a Banach spacecontains an ai copy of ℓ 10, then it contains an ai copy of ℓ 1 but the converse does not hold.
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- J. Diestel, Sequences and series in Banach spaces, Springer Science & Business Media, 2012.
- S.J. Dilworth, M. Girardi and J. Hagler, Dual Banach spaces which contain an isometric copy of L1, Bull. Polish Acad. Sci. Math. 48, 1–12, 2000.
- P.N. Dowling and C.J. Lennard, Every nonreflexive subspace of L1[0, 1] fails the fixed point property, Proc. Amer. Math. Soc. 125, 443–446, 1997.
- P.N. Dowling, C.J. Lennard and B. Turett, Asymptotically isometric copies of c0 in Banach Spaces, J. Math. Anal. Appl. 219, 377–391, 1998.
- P.N. Dowling, C.J. Lennard and B. Turett, Renormings of ℓ 1 and c0 and fixed point properties, in: Handbook of Metric Fixed Point Theory, Springer, Netherlands, 269– 297, 2001.
- P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 125, 167–174, 1997. R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (3), 542–550, 1964.
- C.J. Lennard, Personal communication, 2017.
- P.K. Lin, There is an equivalent norm on ℓ1 that has the fixed point property, Nonlinear Anal. 68, 2303–2308, 2008.
- J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: Sequence Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, 1977.
- J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II: Function Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, 1979.
- G.G. Lorentz, Some new functional spaces, Ann. Math. 51 (1), 37–55, 1950.
- V. Nezir, Fixed point properties for c0-like spaces, Ph.D., University of Pittsburgh, Pittsburgh, PA, USA, 2012.
- V. Nezir, Fixed point properties for a degenerate Lorentz-Marcinkiewicz space, Turkish J. Math. 43, 1919-1939, 2019.
- V. Nezir and N. Mustafa, On the fixed point property for a degenerate LorentzMarcinkiewicz space, in: Proceedings of the 5th International Conference on Recent Advances in Pure and Applied Mathematics (icrapam 2018), Karadeniz Technical University, Trabzon, 23–27 July 2018.
- Yayın Aralığı: Yılda 6 Sayı
- Başlangıç: 2002
- Yayıncı: Hacettepe Üniversitesi Fen Fakultesi