 
Summary: ON THE SUBSYMMETRIC SEQUENCES IN S
G. ANDROULAKIS AND TH. SCHLUMPRECHT
Abstract We establish a sufficient condition for a block sequence in the Schlumprecht
space S to have a subsymmetric subsequence, and prove the existence of an uncountable
family of subsymmetric and mutually nonisomorphic block sequences in S.
1. Introduction
In [S1] the second named author constructed the first known example of an arbitrarily
distortable Banach space. In the literature this space is denoted by S. In [S2] (see [AS1])
the space S was proved to be complementably minimal. This means that the space S embeds
complementably in every infinite dimensional subspace of itself. We do not know whether S
is a prime space. In [AS1] it is shown that every complemented block sequence in S has a
subsequence which spans a space isomorphic to S. These results suggest that in the space S
there are "no many isomorphically different structures". In the present paper we show that
this is not correct. In fact our main theorem is the following surprising result:
Theorem 1.1. There exist uncountably many nonisomorphic seminormalized subsymmet
ric block sequences in S.
The sequences that substantiate Theorem 1.1 are "stabilizing sequences" (see section 2
for the definition). Stabilizing sequences were used in [AS2] to construct strictly singular
noncompact operators on the space S as well as on the GowersMaurey space [GM1]. Recall
that a sequence (xn) in S is called subsymmetric if there exists a constant C 1 such that for
