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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We show that the Lorentz sequence spaces with
and
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
and
,
, Positivity, 8, 2004, 4, 443-454. MR 2117671 (2005k:46008)
- 2. F. Albiac and N. J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298 (2006h:46005)
- 3. Z. 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 (48:6895)
- 4. P. G. Casazza and N. J. Kalton, Uniqueness of unconditional bases in Banach spaces, Israel J. Math., 103, 1998, 141-175. MR 1613564 (99d:46007)
- 5. N. J. Kalton, Orlicz sequence spaces without local convexity, Math. Proc. Cambridge Philos. Soc., 81, 1977, 2, 253-277. MR 0433194 (55:6173)
- 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, 3, 299-311. MR 1120223 (92f:46004)
- 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
-spaces and their applications, Studia Math., 29, 1968, 275-326. MR 0231188 (37:6743)
- 10. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I, Sequence spaces, Springer-Verlag, Berlin, 1977. MR 0500056 (58:17766)
- 11. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, vol. 97, Function spaces, Springer-Verlag, Berlin, 1979. MR 540367 (81c:46001)
- 12. J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Functional Analysis, 3, 1969, 115-125. MR 0236668 (38:4963)
- 13.
N. Nawrocki and A. Ortyński, The Mackey topology and complemented subspaces of Lorentz sequence spaces
for
, Trans. Amer. Math. Soc., 287, 1985, 713-722. MR 768736 (86c:46007)
- 14. N. Popa, Basic sequences and subspaces in Lorentz sequence spaces without local convexity, Trans. Amer. Math. Soc., 263, 1981, 2, 431-456. MR 594418 (82f:46012)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46A16, 46A35, 46A40, 46A45
Retrieve articles in all journals with MSC (2000): 46A16, 46A35, 46A40, 46A45
Additional Information
F. Albiac
Affiliation:
Departamento de Matemática e Informática, Universidad Pública de Navarra, Pamplona 31006, Spain
Email:
fernando.albiac@unavarra.es
C. Leránoz
Affiliation:
Departamento de Matemática e Informática, Universidad Pública de Navarra, Pamplona 31006, Spain
Email:
camino@unavarra.es
DOI:
https://doi.org/10.1090/S0002-9939-08-09222-8
Received by editor(s):
October 23, 2006
Published electronically:
January 3, 2008
Communicated by:
N. Tomczak-Jaegermann
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.