On total incomparability of mixed Tsirelson spaces

Julio Bernués; Javier Pascual

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 4, page 841-859
  • ISSN: 0011-4642

Abstract

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We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form T [ ( ℳ k , θ k ) k = 1 l ] with index i ( ℳ k ) finite are either c 0 or ℓ p saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index i ( ℳ k ) and the parameter θ k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form T [ ( 𝒜 k , θ k ) k = 1 ∞ ] in terms of the asymptotic behaviour of the sequence ∑ i = 1 n e i where ( e i ) is the canonical basis.

How to cite

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Bernués, Julio, and Pascual, Javier. "On total incomparability of mixed Tsirelson spaces." Czechoslovak Mathematical Journal 53.4 (2003): 841-859. <http://eudml.org/doc/30819>.

@article{Bernués2003,
abstract = {We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T[(\mathcal \{M\}_k,\theta _k)_\{k =1\}^\{l\}]$ with index $i(\mathcal \{M\}_k)$ finite are either $c_0$ or $\ell _p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i(\mathcal \{M\}_k)$ and the parameter $\theta _k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T[(\mathcal \{A\}_k,\theta _k)_\{k = 1\}^\infty ]$ in terms of the asymptotic behaviour of the sequence $\Bigl \Vert \sum _\{i=1\}^n e_i\Bigr \Vert $ where $(e_i)$ is the canonical basis.},
author = {Bernués, Julio, Pascual, Javier},
journal = {Czechoslovak Mathematical Journal},
keywords = {mixed Tsirelson spaces; totally incomparable spaces; mixed Tsirelson spaces; totally incomparable spaces},
language = {eng},
number = {4},
pages = {841-859},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On total incomparability of mixed Tsirelson spaces},
url = {http://eudml.org/doc/30819},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Bernués, Julio
AU - Pascual, Javier
TI - On total incomparability of mixed Tsirelson spaces
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 4
SP - 841
EP - 859
AB - We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T[(\mathcal {M}_k,\theta _k)_{k =1}^{l}]$ with index $i(\mathcal {M}_k)$ finite are either $c_0$ or $\ell _p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i(\mathcal {M}_k)$ and the parameter $\theta _k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T[(\mathcal {A}_k,\theta _k)_{k = 1}^\infty ]$ in terms of the asymptotic behaviour of the sequence $\Bigl \Vert \sum _{i=1}^n e_i\Bigr \Vert $ where $(e_i)$ is the canonical basis.
LA - eng
KW - mixed Tsirelson spaces; totally incomparable spaces; mixed Tsirelson spaces; totally incomparable spaces
UR - http://eudml.org/doc/30819
ER -

References

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  1. Banach spaces of the type of Tsirelson, Preprint (1992). (1992) 
  2. 10.1090/S0002-9947-97-01774-1, Trans. Amer. Math. Soc. 349 (1997), 973–995. (1997) MR1390965DOI10.1090/S0002-9947-97-01774-1
  3. 10.1016/0022-1236(86)90089-3, J. Funct. Anal. 69 (1986), 207–228. (1986) MR0865221DOI10.1016/0022-1236(86)90089-3
  4. 10.1002/1522-2616(200102)222:1<15::AID-MANA15>3.0.CO;2-2, Math. Nach. 222 (2001), 15–29. (2001) MR1812486DOI10.1002/1522-2616(200102)222:1<15::AID-MANA15>3.0.CO;2-2
  5. El problema de la distorsión y el problema de la base incondicional, Colloquium del departamento de análisis, Universidad Complutense, Sección 1, Vol. 33, 1995. (1995) 
  6. Tsirelson’s Space, LNM 1363, Springer-Verlag, Berlin, 1989. (1989) MR0981801
  7. A uniformly convex Banach space which contains no l p , Compositio Math. 29 (1974), 179–190. (1974) MR0355537
  8. Classical Banach Spaces I, II, Springer-Verlag, New York, 1977. (1977) MR0500056
  9. 10.1023/A:1011456204116, Positivity 5 (2001), 193–238. (2001) Zbl0988.46009MR1836747DOI10.1023/A:1011456204116
  10. 10.1090/S0002-9947-99-02425-3, Trans. Amer. Math. Soc. 352 (2000), 1859–1888. (2000) MR1637094DOI10.1090/S0002-9947-99-02425-3
  11. 10.1007/BF02782845, Israel J. Math. 76 (1991), 81–95. (1991) Zbl0796.46007MR1177333DOI10.1007/BF02782845
  12. 10.1007/BF01078599, Funct. Anal. Appl. 8 (1974), 138–141. (1974) DOI10.1007/BF01078599
  13. 10.1007/BF02761182, Israel J. Math. 32 (1979), 32–38. (1979) Zbl0402.46013MR0531598DOI10.1007/BF02761182

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