Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Stabilization of Tsirelson-type norms on $ \ell_p$ spaces


Author: Anna Maria Pelczar
Journal: Proc. Amer. Math. Soc. 135 (2007), 1365-1375
MSC (2000): Primary 46B03
DOI: https://doi.org/10.1090/S0002-9939-06-08599-6
Published electronically: October 27, 2006
MathSciNet review: 2276645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider classical Tsirelson-type norms of $ T[\mathcal{A}_n,\theta]$ and their modified versions on $ \ell_p$ spaces, $ 1<p<\infty$. We show that the modified Tsirelson-type norms do not distort any of the subspaces of the $ \ell_p$ spaces. We prove that Tsirelson-type norms, being equivalent to their modified versions, may at most 2-distort $ \ell_p$ spaces.


References [Enhancements On Off] (What's this?)

  • 1. G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125-149. MR 1679593 (2000f:46012)
  • 2. S. Argyros and I. Deliyanni, Banach Spaces of the type of Tsirelson, preprint arXiv:math. FA/9207206.
  • 3. S. Argyros and I. Deliyanni, Examples of asymptotic $ \ell_1$ Banach Spaces, Trans. Amer. Math. Soc. 349 (1997), 973-995. MR 1390965 (97f:46021)
  • 4. S. Argyros, I. Deliyanni, D. Kutzarova and A. Manoussakis, Modified mixed Tsirelson spaces, J. Funct. Anal. 159 (1998), 43-109. MR 1654174 (2000j:46031)
  • 5. S. Argyros, I. Deliyanni and A. Manoussakis, Distortion and spreading models in modified mixed Tsirelson spaces, Studia Math. 157 (3) (2003), 199-236. MR 1980299 (2005f:46021)
  • 6. S. Bellenot, Tsirelson superspaces and $ \ell_p$, J. Funct. Anal. 69 (1986), 207-228. MR 0865221 (88f:46033)
  • 7. P. Casazza and E. Odell, Tsirelson's space and minimal subspaces, Texas Functional Analysis Seminar 1982/3, Longhorn Notes, University of Texas, 61-72. MR 0832217
  • 8. W.B. Johnson, A reflexive Banach space which is not sufficiently Euclidean, Studia Math. 55 (1976), 201-205. MR 0430756 (55:3761)
  • 9. A. Manoussakis, A note on certain equivalent norms on Tsirelson's space, Glas. Math. J. 46 (2004), 379-390. MR 2062620 (2005d:46016)
  • 10. E. Odell and T. Schlumprecht, The distortion problem. Acta Math. 173 (1994), no. 2, 259-281. MR 1301394 (96a:46031)
  • 11. E. Odell and N. Tomczak-Jaegermann, On certain equivalent norms on Tsirelson's space, Illinois J. Math. 44 (2000), 51-71. MR 1731381 (2001g:46034)
  • 12. L. Pieniazek and J. Tabor, On the strange addition, preprint.
  • 13. T. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), 81-85. MR 1177333 (93h:46023)
  • 14. B.S. Tsirelson, Not every Banach space contains $ \ell_p$ or $ c_0$, Funct. Anal. Appl. 8 (1974), 138-141.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B03

Retrieve articles in all journals with MSC (2000): 46B03


Additional Information

Anna Maria Pelczar
Affiliation: Institute of Mathematics, Jagiellonian University, Kraków, Poland
Email: anna.pelczar@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-06-08599-6
Keywords: Tsirelson-type spaces, distortion
Received by editor(s): September 6, 2005
Received by editor(s) in revised form: November 16, 2005, and November 22, 2005
Published electronically: October 27, 2006
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society