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Structure of Cesàro function spaces. (English) Zbl 1200.46027
Let $$1\leq p \leq \infty$$. The Cesàro sequence space $$\text{ces}_p$$ is the set of real sequences $$x = \{x_k\}$$ such that
$\|x\|_{c(p)} = \left[\sum_{n=1}^\infty\left(\frac{1}{n} \sum_{k=1}^n|x_k|\right)^p\right]^{1/p} < \infty,\quad 1\leq p < \infty,$
and
$\|x\|_{c(\infty)} = \sup_{n\in\mathbb{N}}\frac{1}{n} \sum_{n=1}^n |x_k| < \infty, \quad p=\infty.$
The Cesàro function spaces Ces$$_p = \text{Ces}_p(I)$$ are classes of Lebesgue measurable real functions $$f$$ on $$I =[0,1]$$ or $$I=[0,\infty)$$ such that
$\|f\|_{c(p)} = \left[\int_I\left(\frac{1}{x} \int_{0}^x|f(t)|\,dt\right)^p \,dx\right]^{1/p} < \infty,\quad 1\leq p < \infty,$
and
$\|f\|_{c(\infty)} = \sup_{x\in I, x>0}\frac{1}{x} \int_{0}^x |f(t)|\,dt < \infty, \quad p=\infty.$
The authors investigate several isomorphic structure properties of Cesàro sequence and function spaces. They find their dual spaces up to equivalence of norm, with different description for $$[0,1]$$ and $$[0,\infty)$$. They prove that Ces$$_p$$, $$1<p<\infty$$, is strictly convex but not reflexive. It is also shown that these spaces are not isomorphic to any space $$L^q$$ with $$1\leq q\leq \infty$$, and do not enjoy the Dunford-Pettis property while they have the weak Banach-Saks property. It is characterized when the Cesàro spaces contain isomorphic and complemented copies of $$\ell^p$$, and a description when Ces$$_p[0,1]$$ contains an isomorphic subspace of $$\ell^q$$ is given. Finally, the authors prove that Ces$$_p[0,1]$$ and Ces$$_p[0,\infty)$$ are isomorphic for $$1<p\leq\infty$$.

MSC:
 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
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References:
 [1] Albiac, F.; Kalton, N.J., () [2] Aliprantis, C.D.; Burkinshaw, O., () [3] Alspach, D.E., A fixed point free nonexpansive map, Proc. amer. math. soc., 82, 423-424, (1981) · Zbl 0468.47036 [4] Astashkin, S.V.; Maligranda, L., Cesàro function spaces fail the fixed point property, Proc. amer. math. soc., 136, 12, 4289-4294, (2008) · Zbl 1168.46014 [5] Astashkin S.V., Maligranda L. — Rademacher functions in Cesàro type spaces, Studia Math., to appear. · Zbl 1202.46031 [6] Astashkin, S.V.; Sukochev, F.A., Banach-Saks property in Marcinkiewicz spaces, J. math. anal. appl., 336, 2, 1231-1258, (2007) · Zbl 1126.46005 [7] Banach, S.; Banach, S., Theory of linear operations, (), 13-219, English transl. · Zbl 0613.46001 [8] Bennett, G., Factorizing the classical inequalities, () · Zbl 0857.26009 [9] Bennett, C.; Sharpley, R., () [10] Chen, S.; Cui, Y.; Hudzik, H.; Sims, B., Geometric properties related to fixed point theory in some Banach function lattices, (), 339-389 · Zbl 1013.46015 [11] Cui, Y.; Hudzik, H., Some geometric properties related to fixed point theory in Cesàro sequence spaces, Collect. math., 50, 277-288, (1999) · Zbl 0955.46007 [12] Cui, Y.; Hudzik, H., On the Banach-Saks and weak Banach-Saks properties of some Banach sequence spaces, Acta sci. math. (Szeged), 65, 179-187, (1999) · Zbl 0938.46008 [13] Cui, Y.; Hudzik, H., Packing constant for Cesàro sequence spaces, Nonlinear anal., 47, 2695-2702, (2001) · Zbl 1042.46505 [14] Cui, Y.; Hudzik, H.; Li, Y., On the garcía-falset coefficient in some Banach sequence spaces, (), 141-148 · Zbl 0962.46011 [15] Cui, Y.; Jie, L.; Pluciennik, R., Local uniform nonsquareness in Cesàro sequence spaces, Commentat. math. prace mat., 37, 47-58, (1997) · Zbl 0898.46006 [16] Cui, Y.; Meng, C.; Pluciennik, R., Banach-Saks property and property (β) in Cesàro sequence spaces, Southeast Asian bull. math., 24, 201-210, (2000) · Zbl 0956.46003 [17] Diestel, J.; Jarchow, H.; Tonge, A., () [18] Dilworth, S.; Girardi, M.; Hagler, J., Dual Banach spaces which contain an isometric copy of L1, Bull. Polish acad. sci. math., 48, 1, 1-12, (2000) · Zbl 0956.46006 [19] Dodds, P.G.; Semenov, E.M.; Sukochev, F.A., The Banach-Saks property in rearrangement invariant spaces, Studia math., 162, 3, 263-294, (2004) · Zbl 1057.46016 [20] Dowling, P.N.; Lennard, C.J., Every nonreflexive subspace of L1 [0, 1] fails the fixed point property, Proc. amer. math. soc., 125, 2, 443-446, (1997) · Zbl 0861.47032 [21] Dowling, P.N.; Lennard, C.J.; Turett, B., Renormings of l1 and c0 and fixed point properties, (), 269-297 · Zbl 1026.47037 [22] Edwards, R.E., Functional analysis, () · Zbl 0182.16101 [23] Fabian, M.; Habala, P.; Hájek, P.; Montesinos Santalucía, V.; Pelant, J.; Zizler, V., Functional analysis and infinite-dimensional geometry, () · Zbl 0981.46001 [24] Hardy, G.H.; Littlewood, J.E.; Pólya, G., () [25] Hassard, B.D.; Hussein, D.A., On Cesàro function spaces, Tamkang J. math., 4, 19-25, (1973) · Zbl 0284.46023 [26] Jagers, A.A., A note on Cesàro sequence spaces, Nieuw arch. wiskund. (3), 22, 113-124, (1974) · Zbl 0286.46017 [27] Kadec, M.I.; Pełczyński, A., Bases, lacunary sequences and complemented subspaces in the spaces L_{p}, Studia math., 21, 161-176, (1962) · Zbl 0102.32202 [28] Kantorovich, L.V.; Akilov, G.P., () [29] Kashin, B.S.; Saakyan, A.A., () [30] Kerman, R.; Milman, M.; Sinnamon, G., On the brudnyĭ-krugljak duality theory of spaces formed by the K-method of interpolation, Rev. mat. complut., 20, 2, 367-389, (2007) · Zbl 1144.46058 [31] Korenblyum, B.I.; Kreĭn, S.G.; Levin, B.Ya., On certain nonlinear questions of the theory of singular integrals, Doklady akad. nauk SSSR (N.S.), 62, 17-20, (1948), (in Russian) · Zbl 0031.16801 [32] Krein, S.G.; Petunin, Yu.I.; Semenov, E.M., () [33] Kufner, A.; Maligranda, L.; Persson, L.-E., () [34] Lee, P.Y., Cesàro sequence spaces, Math. chronicle, New Zealand, 13, 29-45, (1984) · Zbl 0568.46006 [35] Lee, P.Y., Cesàro sequence spaces, Manuscript, Singapore, 1-17, (1999) [36] Leibowitz, G.M., A note on the Cesàro sequence spaces, Tamkang J. math., 2, 151-157, (1971) · Zbl 0236.46012 [37] Lindenstrauss, J.; Tzafriri, L., () [38] Lindenstrauss, J.; Tzafriri, L., () [39] Liu, Y.Q.; Wu, B.E.; Lee, P.Y., (), (in Chinese) [40] Lozanovskiĭ, G.Ja.; Lozanovskiĭ, G.Ja., Isomorphic Banach lattices, Sibirsk. mat. zh., Siberian math. J., 10, 1, 64-68, (1969), English transl.: · Zbl 0269.46009 [41] Lozanovskiĭ, G.Ja.; Lozanovskiĭ, G.Ja., Certain Banach lattices, Sibirsk. mat. zh., Siberian math. J., 10, 3, 419-431, (1969), English transl.: · Zbl 0269.46009 [42] Luxemburg, W.A.J.; Zaanen, A.C., Some examples of normed Köthe spaces, Math. ann., 162, 337-350, (1965/1966) · Zbl 0132.35002 [43] Maligranda, L., () [44] Maligranda, L., Type, cotype and convexity properties of quasi-Banach spaces, (), 83-120 · Zbl 1087.46024 [45] Maligranda, L.; Petrot, N.; Suantai, S., On the James constant and B-convexity of Cesàro and Cesàro-Orlicz sequence spaces, J. math. anal. appl., 326, 1, 312-331, (2007) · Zbl 1109.46026 [46] Novikov, S.Ya.; Semenov, E.M.; Tokarev, E.V.; Novikov, S.Ya.; Semenov, E.M.; Tokarev, E.V., The structure of subspaces of the space A_{p}(μ), Dokl. akad. nauk SSSR, Sov. math. dokl., 20, 760-761, (1979), English transl.: · Zbl 0436.46019 [47] Novikov, S.Ya.; Semenov, E.M.; Tokarev, E.V.; Novikov, S.Ya.; Semenov, E.M.; Tokarev, E.V., On the structure of subspaces of the spaces A_{p}(μ), Teor. funkts., funkts. anal. prilozh., Ser. 2, amer. math. soc., 136, 121-127, (1984), English transl.: · Zbl 0621.46033 [48] Rakov, S.A.; Rakov, S.A., The Banach-Saks exponent of some Banach spaces of sequences, Mat. zametki, Math. notes, 32, 5,6, 791-797, (1982), English transl.: · Zbl 0535.46005 [49] Sawyer, E., Boundedness of classical operators on classical Lorentz spaces, Studia math., 96, 2, 145-158, (1990) · Zbl 0705.42014 [50] Shiue, J.S., Cesàro sequence spaces, Tamkang J. math., 1, 19-25, (1970) · Zbl 0215.19504 [51] Shiue, J.S., A note on Cesàro function space, Tamkang J. math., 1, 91-95, (1970) · Zbl 0215.19601 [52] Sinnamon, G., The level function in rearrangement invariant spaces, Publ. mat., 45, 1, 175-198, (2001) · Zbl 0987.46033 [53] Sinnamon, G., Transferring monotonicity in weighted norm inequalities, Collect. math., 54, 2, 181-216, (2003) · Zbl 1093.26025 [54] Sy, P.W.; Zhang, W.Y.; Lee, P.Y., The dual of Cesàro function spaces, Glas. mat. ser. III, 22, 1, 103-112, (1987) · Zbl 0647.46033 [55] Szlenk, W., Sur LES suites faiblement convergentes dans l’espace L, Studia math., 25, 337-341, (1965) · Zbl 0131.11505 [56] Tandori, K., Über einen speziellen banachschen raum, Publ. math. debrecen, 3, 263-268, (1954), (1955) · Zbl 0057.33804 [57] Tokarev, E.V., The banach—saks property in Banach lattices, Sibirsk. mat. zh., 24, 1, 187-189, (1983), (in Russian) · Zbl 0541.46021 [58] Wnuk, W., l^{(pn)} spaces with the dunford—pettis property, Comment. math. prace mat., 30, 2, 483-489, (1991) · Zbl 0764.46019 [59] Wnuk, W., Banach lattices with order continuous norms, Polish scientific publishers PWN, warszawa, (1999)
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