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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
Published electronically:
January 3, 2008
MathSciNet review:
2373593
Full-text PDF Free Access
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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
𝑙₁(𝑙_{𝑝}) and
𝑙_{𝑝}(𝑙₁), 0<𝑝<1,
Positivity 8 (2004), no. 4, 443–454. MR 2117671
(2005k:46008), http://dx.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
(2006h:46005)
- 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
(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), http://dx.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
(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), no. 3, 299–311 (1991). MR 1120223
(92f:46004), http://dx.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
(37 #6743)
- 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
(58 #17766)
- 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 (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.
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
(86c:46007), http://dx.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
(82f:46012), http://dx.doi.org/10.1090/S0002-9947-1981-0594418-7
- 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)
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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:
http://dx.doi.org/10.1090/S0002-9939-08-09222-8
PII:
S 0002-9939(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.
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