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  • 1.
    Astashkin, Sergey
    et al.
    Samara State Univ, Dept Math & Mech, , Samara, Russia.Samara State Aerosp Univ, Samara , Russia .
    Lesnik, Karol
    Poznan Univ Tech, Inst Math, Poznan, Poland.
    Maligranda, Lech
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Isomorphic structure of Cesàro and Tandori spaces2019In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 71, no 3, p. 501-532Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    Berezhoi, Evgenii I.
    et al.
    Department of Mathematics, Yaroslavl State University.
    Maligranda, Lech
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Representation of Banach ideal spaces and factorization of operators2005In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 57, no 5, p. 897-940Article in journal (Refereed)
    Abstract [en]

    Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón-Lozanovskiǐ construction. Factorization theorems for operators in spaces more general than the Lebesgue Lp spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt Ap weights to reflexive Orlicz spaces. However, it turns out that for the scales far from Lp-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón-Lozanovskiǐ construction are involved in the proofs

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