| 1. |
- Astashkin, Sergey, et al.
(author)
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Cesaro function spaces fail the fixed point property
- 2008
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In: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 136:12, s. 4289-4294
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Journal article (peer-reviewed)abstract
- The Cesaro sequence spaces ces(p), 1 < p < infinity, are reflexive but they have the fixed point property. In this paper we prove that in contrast to these sequence spaces the corresponding Cesaro function spaces Ces(p) on both [0, 1] and [0, infinity) for 1 < p < infinity are not reflexive and they fail to have the fixed point property.
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| 2. |
- Astashkin, Sergey, et al.
(author)
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Interpolation between L1 and Lp, 1
- 2004
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In: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 132:10, s. 2929-2938
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Journal article (peer-reviewed)abstract
- The main result of this paper is that if $X$ is an interpolation rearrangement invariant space on $[0,1]$ between $L_1$ and $L_\infty$, for which the Boyd index $\alpha(X)>1/p$, $1
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| 3. |
- Astashkin, Sergey, et al.
(author)
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Isomorphic structure of Cesàro and Tandori spaces
- 2019
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - : Cambridge University Press. - 0008-414X .- 1496-4279. ; 71:3, s. 501-532
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Journal article (peer-reviewed)abstract
- We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space Ces∞ and its sequence counterpart ces∞ are isomorphic. This is rather surprising since Ces∞ (like Talagrand’s example) has no natural lattice predual. We prove that ces∞ is not isomorphic to ℓ∞ nor is Ces∞ isomorphic to the Tandori space L1 with the norm ∥f∥L1 = ∥f∥L1, where f(t) = esssups≥tf(s). Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.
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| 4. |
- Astashkin, Sergey, et al.
(author)
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Rademacher functions in Cesaro type spaces
- 2010
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In: Studia Mathematica. - : Institute of Mathematics, Polish Academy of Sciences. - 0039-3223 .- 1730-6337. ; 198:3, s. 235-247
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Journal article (peer-reviewed)abstract
- The Rademacher sums are investigated in the Cesaro spaces Ces(p) (1 <= p <= infinity) and in the weighted Korenblyum-Krein-Levin spaces K-p,K-w on [0,1]. They span l(2) space in Ces(p) for any 1 <= p < infinity and in K-p,K-w if and only if the weight w is larger than t log(2)(p/2)(2/t) on (0,1). Moreover, the span of the Rademachers is not complemented in Ces(p) for any 1 <= p < infinity or in K-1,K-w for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l(2), this span is a complemented subspace in K-p,K-w.
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| 5. |
- Astashkin, Sergey, et al.
(author)
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Structure of Cesaro function spaces
- 2009
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In: Indagationes mathematicae. - 0019-3577 .- 1872-6100. ; 20:3, s. 329-379
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Journal article (peer-reviewed)abstract
- The structure of the Cesàro function spaces Cesp on both [0,1] and [0, ∞) for 1 < p ≤ ∞ is investigated. We find their dual spaces, which equivalent norms have different description on [0, 1] and [0, ∞).The spaces Cesp for 1 < p < ∞ are not reflexive but strictly convex. They are not isomorphic to any Lq space with 1 ≤ q ≤ ∞. They have "near zero" complemented subspaces isomorphic to lp and "in the middle" contain an asymptotically isometric copy of l1 and also a copy of L1[0, 1]. They do not have Dunford-Pettis property but they do have the weak Banach-Saks property. Cesàro function spaces on [0, 1] and [0, ∞) are isomorphic for 1 < p ≤ ∞. Moreover, we give characterizations in terms of p and q when Cesp[0, 1] contains an isomorphic copy of lq.
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| 6. |
- Astashkin, Sergey, et al.
(author)
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Ultrasymmetric Orlicz spaces
- 2008
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 347:1, s. 273-285
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Journal article (peer-reviewed)abstract
- It is proved that ultrasymmetric reflexive Orlicz spaces can be described exactly as all those Orlicz spaces which can be written as some Lorentz spaces. This description is an answer to the problem posed by Pustylnik in [E. Pustylnik, Ultrasymmetric spaces, J. London Math. Soc. (2) 68 (1) (2003) 165-182]. On the other hand, the Lorentz-Orlicz spaces with non-trivial indices of their fundamental functions are ultrasymmetric.
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| 7. |
- Astashkin, Sergey V., et al.
(author)
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A short proof of some recent results related to Cesàro function spaces
- 2013
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In: Indagationes mathematicae. - : Elsevier BV. - 0019-3577 .- 1872-6100. ; 24:3, s. 589-592
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Journal article (peer-reviewed)abstract
- We give a short proof of the recent results that, for every 1≤p<∞, the Cesàro function space Cesp(I) is not a dual space, has the weak Banach–Saks property and does not have the Radon–Nikodym property.
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| 8. |
- Astashkin, Sergey V., et al.
(author)
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Geometry of Cesaro function spaces
- 2011
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In: Functional analysis and its applications. - : Springer Science and Business Media LLC. - 0016-2663 .- 1573-8485. ; 45:1, s. 64-68
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Journal article (peer-reviewed)abstract
- Geometric properties of Cesàro function spaces Ces p (I), where I = [0,∞) or I = [0, 1], are investigated. In both cases, a description of their dual spaces for 1 < p < ∞ is given. We find the type and the cotype of Cesàro spaces and present a complete characterization of the spaces l q that have isomorphic copies in Ces p [0, 1] (1 ⩽ p < ∞).
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| 9. |
- Astashkin, Sergey V., et al.
(author)
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Interpolation of Cesaro and Copson spaces
- 2014
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In: Proceedings of the Fourth International Symposium on Banach and Function Spaces IV (ISBFS 2012). - Yokohama : Yokohama Publishers. - 9784946552489 ; , s. 123-133
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Conference paper (peer-reviewed)abstract
- Summary. Interpolation properties of Cesàro and Copson spaces are investigated. It is shown that the Cesàro function space Ces_p(I), where I = [0, 1] or [0, \infty), is an interpolation space between Ces_{p_0}(I) and Ces_{p_1}(I) for 1 < p_0 < p_1 \leq \infty and 1/p = (1 - \theta)/p_0 + \theta /p_1 with 0 < \theta < 1. The same result is true for Cesàro sequence spaces. For Copson function and sequence spaces a similar result holds even in the case when 1 \leq p_0 < p_1 \leq \infty. At the same time, $Ces_p[0, 1]$ is not an interpolation space between Ces_1[0, 1] and Ces_{\infty}[0, 1] for any 1
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| 10. |
- Astashkin, Sergey V., et al.
(author)
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Interpolation of cesàro sequence and function spaces
- 2013
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In: Studia Mathematica. - : Institute of Mathematics, Polish Academy of Sciences. - 0039-3223 .- 1730-6337. ; 215:1, s. 39-69
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Journal article (peer-reviewed)abstract
- The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that Cesp(I) is an interpolation space between Cesp0 (I) and Cesp1 (I) for 1 < p0 < p1 ≤ ∞ and 1/p = (1 - θ)/p0 + θ/p1 with 0 < θ < 1, where I = [0,∞) or [0, 1]. The same result is true for Cesàro sequence spaces. On the other hand, Cesp[0; 1] is not an interpolation space between Ces 1[0; 1] and Ces∞[0; 1].
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