A Problem on Spreading Models

✍ Scribed by E. Odell; Th. Schlumprecht

• Book ID: 102589519
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 253 KB
• Volume: 153
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3191

✦ Synopsis

It is proved that if a Banach space X has a basis (e n ) satisfying every spreading model of a normalized block basis of (e n ) is 1-equivalent to the unit vector basis of l 1 (respectively, c 0 ) then X contains l 1 (respectively, c 0 ). Furthermore, Tsirelson's space T is shown to have the property that every infinite dimensional subspace contains a sequence having spreading model 1-equivalent to the unit vector basis of l 1 . An equivalent norm is constructed on T so that &s 1 +s 2 &<2 whenever (s n ) is a spreading model of a normalized basic sequence in T.

1998 Academic Press

Theorem A. Let (e i ) be a basis for X (a) If &s 1 +s 2 &=2 whenever (s n ) is any spreading model of a normalized block basis of (e i ), then X contains a subspace isomorphic to l 1 .

(b) If &s 1 +s 2 &=1 whenever (s n ) is any spreading model of a normalized block basis of (e i ), then X contains a subspace isomorphic to c 0 .

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