Iteration of spreading models

Abstract

If E is a Banach space and (xn),e N is a bounded sequence in E, Brunel--Sucheston [2] showed that there exists a subsequence (X')neN of (Xn)n~ N such that the limit of Ilalx'l§247 exists, when n l < n 2 < . . . < n k go to infinity, for all finite sequences of scalars al, . . . , a k. A sequence (x~),c N satisfying this property will be called a good sequence. If (e,),e N denotes the canonical basis of the space S of finite sequences of scalars, we denote by la le l+ ... +akekl the previous limit. It is a norm on this space provided that the sequence (x,)n~ N has no norm-convergent subsequence. Let F be the completion of S under this norm: F will be called a spreading model of E, built on the sequence (Xn)n~ N. The sequence (e,),c N will be called the fundamental sequence of F. This notion was first introduced and studied by A. Brtmel and L. Sucheston (see for example [2] and [3]); it has been applied to the study of the Banach--Saks properties by the first named author in [1], in which the reader may find the proofs of all the statements we give here. Our aim in this paper will be to investigate the following question, raised by H. P. Rosenthal: given a space E, a spreading model F~, built on some bounded sequence (Xn),~ N of E, a spreading model F2, built on some bounded sequence (Y,)n~N of F1, is F2 isomorphic to some spreading model F of E, built on some bounded sequence (Zn)ne N of E? The isometric version of this question was answered negatively by the authors in [1]. We shall show here that the question in its full generality has also a negative answer: we shall present a Banach space E, a spreading model F 1 of E, a spreading model F~ of FI, such that F~ is not isomorphic to any spreading model of E. The tools we use for this purpose will be the connections between the Banach-Saks properties and the isomorphism of a spreading model to /1, established by the first named author in [1]. More precisely, let us recall the following definition and theorem, given in [11: