Some problems on narrow operators on function spaces

• Autorzy: Mikhail Popov , Evgenii Semenov , Diana Vatsek
• Języki publikacji: 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]).

Słowa kluczowe

EN

Narrow operator; Rearrangement invariant spaces; Köthe function spaces

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 46B42: Banach lattices

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 3
• Strony: 476-482

Daty

wydano

2014-03-01

online

2013-12-21

Twórcy

autor

Mikhail Popov

• Chernivtsi National University

autor

Evgenii Semenov

• Voronezh State University

autor

Diana Vatsek

• Chernivtsi National University

Bibliografia

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• [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

Identyfikatory

DOI

10.2478/s11533-013-0358-x