Some problems on narrow operators on function spaces

Mikhail Popov; Evgenii Semenov; Diana Vatsek

Open Mathematics (2014)

  • Volume: 12, Issue: 3, page 476-482
  • ISSN: 2391-5455

Abstract

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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]).

How to cite

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Mikhail Popov, Evgenii Semenov, and Diana Vatsek. "Some problems on narrow operators on function spaces." Open Mathematics 12.3 (2014): 476-482. <http://eudml.org/doc/269495>.

@article{MikhailPopov2014,
abstract = {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]).},
author = {Mikhail Popov, Evgenii Semenov, Diana Vatsek},
journal = {Open Mathematics},
keywords = {Narrow operator; Rearrangement invariant spaces; Köthe function spaces; narrow operator; rearrangement invariant space; Köthe function space},
language = {eng},
number = {3},
pages = {476-482},
title = {Some problems on narrow operators on function spaces},
url = {http://eudml.org/doc/269495},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Mikhail Popov
AU - Evgenii Semenov
AU - Diana Vatsek
TI - Some problems on narrow operators on function spaces
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 476
EP - 482
AB - 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]).
LA - eng
KW - Narrow operator; Rearrangement invariant spaces; Köthe function spaces; narrow operator; rearrangement invariant space; Köthe function space
UR - http://eudml.org/doc/269495
ER -

References

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  1. [1] Albiac F., Kalton N.J., Topics in Banach Space Theory, Grad. Texts in Math., 233, Springer, New York, 2006 Zbl1094.46002
  2. [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 Zbl0504.46022
  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 Zbl1069.46006
  4. [4] Krasikova I.V., A note on narrow operators in L 1, Mat. Stud., 2009, 31(1), 102–106 Zbl1199.47153
  5. [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. [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. [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 Zbl1183.47033
  8. [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 Zbl1316.47034
  9. [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. [10] Popov M.M., Reproducibility of sequences in Banach spaces, Naukoviĭ Vısnik Chernivets’kogo Unıversitetu, Matematika, 2003, 160, 104–108 (in Ukrainian) 
  11. [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 Zbl1258.47002
  12. [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 Zbl0315.46031

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