Cesàro function spaces fail the fixed point property. (English) Zbl 1168.46014

The Cesàro function space \(\text{Ces}_p(I)\) consists of equivalence classes of Lebesgue measurable functions \(f\) on \(I=[0,1]\) or \(I=[0,\infty)\) such that the norms
\[ \| f\|_{C(p)} = \left(\int_I \left(\frac{1}{x} \int_0^x |f(t)|\, dt\right)^p dx\right)^{1/p} \quad \text{if }1\leq p<\infty \]
\[ \| f\|_{C(\infty)}= \sup_{x\in I,\, x>0} \frac{1}{x} \int_0^x |f(t)|\, dt \quad \text{if }p=\infty \]
are finite. The authors prove that, if \(1\leq p \leq \infty\) and \(I=[0,1]\) or if \(1< p \leq \infty\) and \(I=[0,\infty)\), the Cesàro function spaces \(\text{Ces}_p(I)\) contain asymptotically isometric copies of \(\ell^1\). Consequently, for these \(p\) and \(I\), \(\text{Ces}_p(I)\) and its dual \(\text{Ces}_p(I)^*\) fail the fixed point property for nonexpansive maps. This contrasts with the known result [Y. Cui and H. Hudzik, Collect. Math. 50, No. 3, 277–288 (1999; Zbl 0955.46007)] that the sequential analogs of these spaces, the Cesàro sequence spaces, have the fixed point property for nonexpansive maps if \(1<p<\infty\).


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
46B42 Banach lattices


Zbl 0955.46007
Full Text: DOI


[1] Dale E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), no. 3, 423 – 424. · Zbl 0468.47036
[2] S. V. Astashkin and L. Maligranda, Geometry of Cesàro function spaces, in preparation. · Zbl 1271.46027
[3] Grahame Bennett, Factorizing the classical inequalities, Mem. Amer. Math. Soc. 120 (1996), no. 576, viii+130. · Zbl 0857.26009 · doi:10.1090/memo/0576
[4] S. Chen, Y. Cui, H. Hudzik, and B. Sims, Geometric properties related to fixed point theory in some Banach function lattices, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 339 – 389. · Zbl 1013.46015
[5] Yunan Cui and Henryk Hudzik, Some geometric properties related to fixed point theory in Cesàro spaces, Collect. Math. 50 (1999), no. 3, 277 – 288. · Zbl 0955.46007
[6] Yunan Cui, Henryk Hudzik, and Yanhong Li, On the García-Falset coefficient in some Banach sequence spaces, Function spaces (Poznań, 1998) Lecture Notes in Pure and Appl. Math., vol. 213, Dekker, New York, 2000, pp. 141 – 148. · Zbl 0962.46011
[7] Yunan Cui, Chenghui Meng, and Ryszard Płuciennik, Banach-Saks property and property (\?) in Cesàro sequence spaces, Southeast Asian Bull. Math. 24 (2000), no. 2, 201 – 210. · Zbl 0956.46003 · doi:10.1007/s100120070003
[8] Stephen J. Dilworth, Maria Girardi, and James Hagler, Dual Banach spaces which contain an isometric copy of \?\(_{1}\), Bull. Polish Acad. Sci. Math. 48 (2000), no. 1, 1 – 12. · Zbl 0956.46006
[9] P. N. Dowling and C. J. Lennard, Every nonreflexive subspace of \?\(_{1}\)[0,1] fails the fixed point property, Proc. Amer. Math. Soc. 125 (1997), no. 2, 443 – 446. · Zbl 0861.47032
[10] P. N. Dowling, C. J. Lennard, and B. Turett, Renormings of \?\textonesuperior and \?\(_{0}\) and fixed point properties, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 269 – 297. · Zbl 1026.47037
[11] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. · Zbl 0047.05302
[12] B. D. Hassard and D. A. Hussein, On Cesàro function spaces, Tamkang J. Math. 4 (1973), 19 – 25. · Zbl 0284.46023
[13] Alois Kufner, Lech Maligranda, and Lars-Erik Persson, The Hardy inequality, Vydavatelský Servis, Plzeň, 2007. About its history and some related results. · Zbl 1213.42001
[14] Y. Q. Liu, B. E. Wu and P. Y. Lee, Method of Sequence Spaces, Guangdong of Science and Technology Press, 1996 (in Chinese).
[15] Lech Maligranda, Narin Petrot, and Suthep Suantai, On the James constant and \?-convexity of Cesàro and Cesàro-Orlicz sequences spaces, J. Math. Anal. Appl. 326 (2007), no. 1, 312 – 331. · Zbl 1109.46026 · doi:10.1016/j.jmaa.2006.02.085
[16] Jau-shyong Shiue, A note on Cesàro function space, Tamkang J. Math. 1 (1970), no. 2, 91 – 95. · Zbl 0215.19601
[17] Polly Wee Sy, Wen Yao Zhang, and Peng Yee Lee, The dual of Cesàro function spaces, Glas. Mat. Ser. III 22(42) (1987), no. 1, 103 – 112 (English, with Serbo-Croatian summary). · Zbl 0647.46033
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