On the complemented subspaces of the Schreier spaces

I. Gasparis; D. Leung

Studia Mathematica (2000)

  • Volume: 141, Issue: 3, page 273-300
  • ISSN: 0039-3223

Abstract

top
It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space X ξ generated by subsequences ( e l n ξ ) and ( e m n ξ ) , respectively, of the natural Schauder basis ( e n ξ ) of X ξ are isomorphic if and only if ( e l n ξ ) and ( e m n ξ ) are equivalent. Further, X ξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ( e n ξ ) . It is also shown that there exists a complemented subspace spanned by a block basis of ( e n ξ ) , which is not isomorphic to a subspace generated by a subsequence of ( e n ζ ) , for every 0 ≤ ζ ≤ ξ . Finally, an example is given of an uncomplemented subspace of X ξ which is spanned by a block basis of ( e n ξ ) .

How to cite

top

Gasparis, I., and Leung, D.. "On the complemented subspaces of the Schreier spaces." Studia Mathematica 141.3 (2000): 273-300. <http://eudml.org/doc/216785>.

@article{Gasparis2000,
abstract = {It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^ξ$ generated by subsequences $(e_\{l_n\}^\{ξ\})$ and $(e_\{m_n\}^\{ξ\})$, respectively, of the natural Schauder basis $(e_\{n\}^\{ξ\})$ of $X^ξ$ are isomorphic if and only if $(e_\{l_n\}^\{ξ\})$ and $(e_\{m_n\}^\{ξ\})$ are equivalent. Further, $X^ξ$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_\{n\}^\{ξ\})$. It is also shown that there exists a complemented subspace spanned by a block basis of $(e_\{n\}^\{ξ\})$, which is not isomorphic to a subspace generated by a subsequence of $(e_n^ζ)$, for every $0 ≤ ζ ≤ ξ$. Finally, an example is given of an uncomplemented subspace of $X^ξ$ which is spanned by a block basis of $(e_\{n\}^\{ξ\})$.},
author = {Gasparis, I., Leung, D.},
journal = {Studia Mathematica},
keywords = {Schreier sets; complemented subspace},
language = {eng},
number = {3},
pages = {273-300},
title = {On the complemented subspaces of the Schreier spaces},
url = {http://eudml.org/doc/216785},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Gasparis, I.
AU - Leung, D.
TI - On the complemented subspaces of the Schreier spaces
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 3
SP - 273
EP - 300
AB - It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^ξ$ generated by subsequences $(e_{l_n}^{ξ})$ and $(e_{m_n}^{ξ})$, respectively, of the natural Schauder basis $(e_{n}^{ξ})$ of $X^ξ$ are isomorphic if and only if $(e_{l_n}^{ξ})$ and $(e_{m_n}^{ξ})$ are equivalent. Further, $X^ξ$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_{n}^{ξ})$. It is also shown that there exists a complemented subspace spanned by a block basis of $(e_{n}^{ξ})$, which is not isomorphic to a subspace generated by a subsequence of $(e_n^ζ)$, for every $0 ≤ ζ ≤ ξ$. Finally, an example is given of an uncomplemented subspace of $X^ξ$ which is spanned by a block basis of $(e_{n}^{ξ})$.
LA - eng
KW - Schreier sets; complemented subspace
UR - http://eudml.org/doc/216785
ER -

References

top
  1. [1] D. E. Alspach and S. A. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992). Zbl0787.46009
  2. [2] G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125-149. Zbl0940.46008
  3. [3] S. A. Argyros and I. Deliyanni, Examples of asymptotic l 1 Banach spaces, Trans. Amer. Math. Soc. 349 (1997), 973-995. 
  4. [4] S. A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), 243-294. 
  5. [5] S. A. Argyros and I. Gasparis, Unconditional structures of weakly null sequences, Trans. Amer. Math. Soc., to appear. 
  6. [6] S. A. Argyros, S. Mercourakis and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157-193. Zbl0942.46007
  7. [7] P. Cembranos, The hereditary Dunford-Pettis property on C(K,E), Illinois J. Math. 31 (1987), 365-373. Zbl0618.46042
  8. [8] I. Gasparis, A dichotomy theorem for subsets of the power set of the natural numbers, Proc. Amer. Math. Soc., to appear. Zbl0962.46006
  9. [9] A. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, New York, 1994. 
  10. [10] K. Kuratowski, Applications of the Baire-category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72. Zbl0311.54036
  11. [11] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, II, Ergeb. Math. Grenzgeb. 92, 97, Springer, Berlin, 1977, 1979. Zbl0362.46013
  12. [12] J. Mycielski, Almost every function is independent, Fund. Math. 81 (1973), 43-48. Zbl0311.54018
  13. [13] E. Odell, On quotients of Banach spaces having shrinking unconditional bases, Illinois J. Math. 36 (1992), 681-695. Zbl0834.46008
  14. [14] E. Odell, N. Tomczak-Jaegermann and R. Wagner, Proximity to l 1 and distortion in asymptotic l 1 spaces, J. Funct. Anal. 150 (1997), 101-145. Zbl0890.46015
  15. [15] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2 (1930), 58-62. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.