Uniqueness of unconditional basis in Lorentz sequence spaces

Albiac, F.; Leránoz, C.

doi:10.1090/S0002-9939-08-09222-8

Uniqueness of unconditional basis in Lorentz sequence spaces

Authors: F. Albiac and C. Leránoz
Journal: Proc. Amer. Math. Soc. 136 (2008), 1643-1647
MSC (2000): Primary 46A16, 46A35; Secondary 46A40, 46A45
DOI: https://doi.org/10.1090/S0002-9939-08-09222-8
Published electronically: January 3, 2008
MathSciNet review: 2373593
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Abstract: We show that the Lorentz sequence spaces $d(\omega,p)$ with and $\inf\frac{\omega_1+\cdots+\omega_n}{n^p}>0$ have unique unconditional basis. This completely settles the question of uniqueness of unconditional basis in Lorentz sequence spaces, and solves a problem raised by Popa in 1981 and Nawrocki and Ortyński in 1985.

1. F. Albiac, N. Kalton, and C. Leránoz, Uniqueness of the unconditional basis of 𝑙₁(𝑙_{𝑝}) and 𝑙_{𝑝}(𝑙₁), 0<𝑝<1, Positivity 8 (2004), no. 4, 443–454. MR 2117671, https://doi.org/10.1007/s11117-003-8542-z
2. Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
3. Zvi Altshuler, P. G. Casazza, and Bor Luh Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973), 140–155. MR 0328553, https://doi.org/10.1007/BF02764600
4. P. G. Casazza and N. J. Kalton, Uniqueness of unconditional bases in Banach spaces, Israel J. Math. 103 (1998), 141–175. MR 1613564, https://doi.org/10.1007/BF02762272
5. N. J. Kalton, Orlicz sequence spaces without local convexity, Math. Proc. Cambridge Philos. Soc. 81 (1977), no. 2, 253–277. MR 0433194, https://doi.org/10.1017/S0305004100053342
6. N. J. Kalton, C. Leránoz, and P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel J. Math. 72 (1990), no. 3, 299–311 (1991). MR 1120223, https://doi.org/10.1007/BF02773786
7. N. J. Kalton, N. T. Peck, and J. W. Rogers, An F-space sampler, London Math. Lecture Notes 89, Cambridge Univ. Press, Cambridge, 1985.
8. G. Köthe and O. Toeplitz, Lineare Raume mit unendlich vielen Koordinaten und Ringen unendlicher Matrizen, J. Reine Angew Math., 171, 1934, 193-226. (German)
9. J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in 𝐿_{𝑝}-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 0231188, https://doi.org/10.4064/sm-29-3-275-326
10. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
11. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
12. J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Functional Analysis 3 (1969), 115–125. MR 0236668
13. M. Nawrocki and A. Ortyński, The Mackey topology and complemented subspaces of Lorentz sequence spaces 𝑑(𝑤,𝑝) for 0<𝑝<1, Trans. Amer. Math. Soc. 287 (1985), no. 2, 713–722. MR 768736, https://doi.org/10.1090/S0002-9947-1985-0768736-7
14. Nicolae Popa, Basic sequences and subspaces in Lorentz sequence spaces without local convexity, Trans. Amer. Math. Soc. 263 (1981), no. 2, 431–456. MR 594418, https://doi.org/10.1090/S0002-9947-1981-0594418-7