On the complemented subspaces of the Schreier spaces

Gasparis, I.; Leung, D.

Gasparis, I. ; Leung, D.

Studia Mathematica, Tome 141 (2000), p. 273-300 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $0 \leq ζ \leq ξ$ . Finally, an example is given of an uncomplemented subspace of $X^{ξ}$ which is spanned by a block basis of $(e_{n}^{ξ})$ .

Publié le : 2000-01-01

Zbl 1001.46003

EUDML-ID : urn:eudml:doc:216785

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     author = {I. Gasparis and D. Leung},
     title = {On the complemented subspaces of the Schreier spaces},
     journal = {Studia Mathematica},
     volume = {141},
     year = {2000},
     pages = {273-300},
     zbl = {1001.46003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv141i3p273bwm}
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Gasparis, I.; Leung, D. On the complemented subspaces of the Schreier spaces. Studia Mathematica, Tome 141 (2000) pp. 273-300. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv141i3p273bwm/

Bibliographie

[000] [1] D. E. Alspach and S. A. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992). | Zbl 0787.46009

[001] [2] G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125-149. | Zbl 0940.46008

[002] [3] S. A. Argyros and I. Deliyanni, Examples of asymptotic $l_{1}$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), 973-995.

[003] [4] S. A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), 243-294.

[004] [5] S. A. Argyros and I. Gasparis, Unconditional structures of weakly null sequences, Trans. Amer. Math. Soc., to appear.

[005] [6] S. A. Argyros, S. Mercourakis and A. Tsarpalias, Convex unconditionality and summability of weakly null sequences, Israel J. Math. 107 (1998), 157-193. | Zbl 0942.46007

[006] [7] P. Cembranos, The hereditary Dunford-Pettis property on C(K,E), Illinois J. Math. 31 (1987), 365-373. | Zbl 0618.46042

[007] [8] I. Gasparis, A dichotomy theorem for subsets of the power set of the natural numbers, Proc. Amer. Math. Soc., to appear. | Zbl 0962.46006

[008] [9] A. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, New York, 1994.

[009] [10] K. Kuratowski, Applications of the Baire-category method to the problem of independent sets, Fund. Math. 81 (1973), 65-72. | Zbl 0311.54036

[010] [11] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, II, Ergeb. Math. Grenzgeb. 92, 97, Springer, Berlin, 1977, 1979. | Zbl 0362.46013

[011] [12] J. Mycielski, Almost every function is independent, Fund. Math. 81 (1973), 43-48. | Zbl 0311.54018

[012] [13] E. Odell, On quotients of Banach spaces having shrinking unconditional bases, Illinois J. Math. 36 (1992), 681-695. | Zbl 0834.46008

[013] [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. | Zbl 0890.46015

[014] [15] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Math. 2 (1930), 58-62.