Summary: AN EXTENSION OF SCHREIER UNCONDITIONALITY
G. ANDROULAKIS, F. SANACORY
Abstract. The main result of the paper extends the classical result of E. Odell on Schreier
unconditionality to arrays in Banach spaces. An application is given on the "multiple of the
inclusion plus compact" problem which is further applied to a hereditarily indecomposable
Banach space constructed by N. Dew.
A finite subset F of N is called a Schreier set if |F| min(F) (where |F| denotes the
cardinality of F). The important notion of Schreier unconditionality was introduced by
E. Odell  and has inspired rich literature on the subject (see for example , ). Earlier
very similar results can be found in [15, page 77] and [19, Theorem 2.1 ]. A basic sequence
(xn) in a Banach space is defined to be Schreier unconditional if there is a constant C > 0
such that for all scalars (ai) c00 and for all Schreier sets F we have
aiei C aiei .
In this case (ei) is called C-Schreier unconditional.
Theorem 1.1.  Let (xn) be a normalized weakly null sequence in a Banach space. Then
for any > 0, (xn) contains a (2 + )- Schreier unconditional subsequence.
Our main result is Theorem 1.2 where we extend Theorem 1.1 to arrays of vectors of a Banach
space such that each row is a seminormalized weakly null sequence. Then Theorem 1.2