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2014 | 12 | 3 | 476-482
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Some problems on narrow operators on function spaces

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EN
Abstrakty
EN
It is known that if a rearrangement invariant (r.i.) space E on [0, 1] has an unconditional basis then every linear bounded operator on E is a sum of two narrow operators. On the other hand, for the classical space E = L 1[0, 1] having no unconditional basis the sum of two narrow operators is a narrow operator. We show that a Köthe space on [0, 1] having “lots” of nonnarrow operators that are sum of two narrow operators need not have an unconditional basis. However, we do not know if such an r.i. space exists. Another result establishes sufficient conditions on an r.i. space E under which the orthogonal projection onto the closed linear span of the Rademacher system is a hereditarily narrow operator. This, in particular, answers a question of the first named author and Randrianantoanina (Problem 11.9 in [Popov M., Randrianantoanina B., Narrow Operators on Function Spaces and Vector Lattices, de Gruyter Stud. Math., 45, Walter de Gruyter, Berlin, 2013]).
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
3
Strony
476-482
Opis fizyczny
Daty
wydano
2014-03-01
online
2013-12-21
Twórcy
autor
Bibliografia
  • [1] Albiac F., Kalton N.J., Topics in Banach Space Theory, Grad. Texts in Math., 233, Springer, New York, 2006
  • [2] Burkholder D.L., A nonlinear partial differential equation and the unconditional constant of the Haar system in L p, Bull. Amer. Math. Soc., 1982, 7(3), 591–595 http://dx.doi.org/10.1090/S0273-0979-1982-15061-3
  • [3] Kadets V.M., Popov M.M., Some stability theorems on narrow operators acting in L 1 and C(K), Mat. Fiz. Anal. Geom., 2003, 10(1), 49–60
  • [4] Krasikova I.V., A note on narrow operators in L 1, Mat. Stud., 2009, 31(1), 102–106
  • [5] Lindenstrauss J., Tzafriri L., Classical Banach Spaces, I, Ergeb. Math. Grenzgeb., 92, Springer, Berlin, 1977 http://dx.doi.org/10.1007/978-3-642-66557-8
  • [6] Lindenstrauss J., Tzafriri L., Classical Banach Spaces, II, Ergeb. Math. Grenzgeb., 97, Springer, Berlin, 1979 http://dx.doi.org/10.1007/978-3-662-35347-9
  • [7] Maslyuchenko O.V., Mykhaylyuk V.V., Popov M.M., A lattice approach to narrow operators, Positivity, 2009, 13(3), 459–495 http://dx.doi.org/10.1007/s11117-008-2193-z
  • [8] Mykhaylyuk V.V., Popov M.M., On sums of narrow operators on Köthe function spaces, J. Math. Anal. Appl., 2013, 404(2), 554–561 http://dx.doi.org/10.1016/j.jmaa.2013.03.008
  • [9] Plichko A.M., Popov M.M., Symmetric Function Spaces on Atomless Probability Spaces, Dissertationes Math. (Rozprawy Mat.), 306, Polish Academy of Sciences, Warsaw, 1990
  • [10] Popov M.M., Reproducibility of sequences in Banach spaces, Naukoviĭ Vısnik Chernivets’kogo Unıversitetu, Matematika, 2003, 160, 104–108 (in Ukrainian)
  • [11] Popov M., Randrianantoanina B., Narrow Operators on Function Spaces and Vector Lattices, de Gruyter Stud. Math., 45, Walter de Gruyter, Berlin, 2013 http://dx.doi.org/10.1515/9783110263343
  • [12] Rodin V.A., Semyonov E.M., Rademacher series in symmetric spaces, Anal. Math., 1975, 1(3), 207–222 http://dx.doi.org/10.1007/BF01930966
Typ dokumentu
Bibliografia
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