Positivity 12 (2008), 313–340
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020313-28, published online January 11, 2008
An Extension of Schreier Unconditionality
George Androulakis and Frank 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 fur-
ther applied to a hereditarily indecomposable Banach space constructed by
Mathematics Subject Classiﬁcation (2000). 46A32, 47B07.
Keywords. Schreier unconditionality, compact operators, hereditarily
indecomposable Banach space.
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 intro-
duced 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,
]. A basic sequence (x
) in a Banach space is deﬁned to be Schreier
unconditional if there is a constant C>0 such that for all scalars (a
) ∈ c
for all Schreier sets F we have
In this case (e
) is called C-Schreier unconditional.
Theorem 1.1.  Let (x
) be a normalized weakly null sequence in a Banach space.
Then for any ε>0, (x
) contains a (2 + ε)-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 guarantees the existence of a subarray which preserves
The present paper is part of the Ph.D thesis of the second author who is partially supported
under Prof. Girardi’s NSF grant DMS-0306750.