On a Property of Rearrangement Invariant Spaces whose Second Köthe Dual is Nonseparable
We study the family of rearrangement invariant spaces E containing subspaces on which the E -norm is equivalent to the L 1 -norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonsepar...
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| Published in | Mathematical Notes Vol. 107; no. 1-2; pp. 10 - 19 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Moscow
Pleiades Publishing
2020
Springer Nature B.V |
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| Abstract | We study the family of rearrangement invariant spaces
E
containing subspaces on which the
E
-norm is equivalent to the
L
1
-norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Köthe dual belongs to this family. In the course of the proof, we show that every nonseparable rearrangement invariant space
E
can be equipped with an equivalent norm with respect to which
E
contains a nonzero function orthogonal to the separable part of
E
. |
|---|---|
| AbstractList | We study the family of rearrangement invariant spaces
E
containing subspaces on which the
E
-norm is equivalent to the
L
1
-norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Köthe dual belongs to this family. In the course of the proof, we show that every nonseparable rearrangement invariant space
E
can be equipped with an equivalent norm with respect to which
E
contains a nonzero function orthogonal to the separable part of
E
. We study the family of rearrangement invariant spaces E containing subspaces on which the E-norm is equivalent to the L1-norm and a certain geometric characteristic related to the Kadec–Pełcziński alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Köthe dual belongs to this family. In the course of the proof, we show that every nonseparable rearrangement invariant space E can be equipped with an equivalent norm with respect to which E contains a nonzero function orthogonal to the separable part of E. |
| Author | Semenov, E. M. Astashkin, S. V. |
| Author_xml | – sequence: 1 givenname: S. V. surname: Astashkin fullname: Astashkin, S. V. email: astash56@mail.ru organization: Korolev Samara National Research University – sequence: 2 givenname: E. M. surname: Semenov fullname: Semenov, E. M. email: nadezhka_ssm@geophys.vsu.ru organization: Voronezh State University |
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| Cites_doi | 10.1007/978-3-662-35347-9 10.1016/j.jfa.2014.02.011 10.1007/BF01429812 10.1016/0021-9045(75)90023-4 10.1215/ijm/1255986545 10.4064/sm-71-3-297-304 10.4064/sm-21-2-161-176 |
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| Keywords | subspace Marcinkiewicz space Köthe dual space rearrangement invariant space |
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| References | AstashkinS VRademacher System in Function Spaces2017MoscowFizmatlit1387.46028[in Russian] AstashkinS VSukochevF ASequences of independent identically distributed functions in rearrangement invariant spacesBanach Center Publ., Vol. 79: Function Spaces VIII2008WarszawaPolish Acad. Sci. Inst. Math.2737 VakhaniyaN NTarieladzeV IChobanyanS AProbability Distributions in Banach Spaces1985MoscowNauka0572.60003[in Russian] KreinS GPetuninJu ISemenovE MInterpolation of Linear Operators1978MoscowNaukaAm. Math. Soc., Providence, RI, 2002 Ya. LozanovskiiGMath. Notes19684151451610.1007/BF01429812 AlbiacFKaltonN JTopics in Banach Space Theory, in Grad. Texts in Math.2006New YorkSpringer1094.46002Vol. 233 LindenstraussJTzafririLClassical Banach Spaces. II. Function Spaces1979BerlinSpringer-Verlag10.1007/978-3-662-35347-9 FigielTJohnsonW BTzafririLOn Banach lattices and spaces having local unconditional structure with applications to Lorentz function spacesJ. Approximation Theory19751339541236762410.1016/0021-9045(75)90023-4 BennettCSharpleyRInterpolation of OperatorsPure Appl. Math.1988BostonAcademic PressVol. 129 TokarevE VSubspaces of certain rearrangement invariant spacesTeor. Funktsii Funktsional. Anal. iPrilozhen197524156161Kharkov KadecM IPełczyńskiABases, lacunary sequences and complemented subspaces in the spaces LpStudia Math.19622116117610.4064/sm-21-2-161-176 AstashkinS VΛ(p)-spacesJ. Funct. Anal.2014266851745198317733410.1016/j.jfa.2014.02.011 NovikovS YaSemenovE MTokarevE VSoviet Math. Dokl.197920760761 Ya. LozanovskiiGProjectors in certain Banach latticesMat. Zametki1968414144235402 AldousD JFremlinD HColacunary sequences in L-spacesStud. Math.19827129730466731810.4064/sm-71-3-297-304 S. Ya. Novikov, E. M. Semenov, and E. V. Tokarev, “The structure of subspaces of the spaces Λp(μ),” in Teor. Funktsii Funktsional. Anal. i Prilozhen, Vol. 42 (Kharkov, 1984), pp. 91–97. NovikovS YaSemenovE MTokarevE VThe structure of subspaces of the space Λp(φ)Dokl. Akad. Nauk SSSR19792473552554545690 P. P. Zabreiko, “Ideal function spaces. I,” Vestn. Yaroslavl. Gos. Univ., No. 8, 12–52 (1974). S. Ya. Novikov, Geometric Properties of Symmetric Spaces, Cand. Sci. (Phys.–Math.) Dissertation (Voronezh, 1980) [in Russian]. RaynaudYComplemented Hilbertian subspaces in rearrangement invariant function spacesIllinois J. Math.1995392212250131653410.1215/ijm/1255986545 G Ya. Lozanovskii (1246_CR15) 1968; 4 M I Kadec (1246_CR3) 1962; 21 S V Astashkin (1246_CR13) 2017 S G Krein (1246_CR1) 1978 S V Astashkin (1246_CR10) 2014; 266 1246_CR16 J Lindenstrauss (1246_CR8) 1979 Y Raynaud (1246_CR20) 1995; 39 S V Astashkin (1246_CR17) 2008 C Bennett (1246_CR12) 1988 D J Aldous (1246_CR19) 1982; 71 1246_CR11 T Figiel (1246_CR7) 1975; 13 F Albiac (1246_CR9) 2006 G Ya. Lozanovskii (1246_CR14) 1968; 4 E V Tokarev (1246_CR2) 1975; 24 S Ya Novikov (1246_CR4) 1979; 247 1246_CR6 S Ya Novikov (1246_CR5) 1979; 20 N N Vakhaniya (1246_CR18) 1985 |
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| Snippet | We study the family of rearrangement invariant spaces
E
containing subspaces on which the
E
-norm is equivalent to the
L
1
-norm and a certain geometric... We study the family of rearrangement invariant spaces E containing subspaces on which the E-norm is equivalent to the L1-norm and a certain geometric... |
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| SubjectTerms | Equivalence Invariants Mathematics Mathematics and Statistics Subspaces |
| Title | On a Property of Rearrangement Invariant Spaces whose Second Köthe Dual is Nonseparable |
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