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Summary: A HEREDITARILY INDECOMPOSABLE ASYMPTOTIC 2 BANACH
SPACE
G. ANDROULAKIS, K. BEANLAND
Abstract. A Hereditarily Indecomposable asymptotic 2 Banach space is constructed.
The existence of such a space answers a question of B. Maurey and verifies a conjecture of
W.T. Gowers.
1. Introduction
A famous open problem in functional analysis is whether there exists a Banach space X
such that every (bounded linear) operator on X has the form +K where is a scalar and K
denotes a compact operator. This problem is usually called the "scalar-plus-compact" prob-
lem [14]. One of the reasons this problem has become so attractive is that by a result of N.
Aronszajn and K.T. Smith [7], if a Banach space X is a solution to the scalar-plus-compact
problem then every operator on X has a non-trivial invariant subspace and hence X provides
a solution to the famous invariant subspace problem. An important advancement in the con-
struction of spaces with "few" operators was made by W.T. Gowers and B. Maurey [16],[17].
The ground breaking work [16] provides a construction of a space without any unconditional
basic sequence thus solving, in the negative, the long standing unconditional basic sequence
problem. The Banach space constructed in [16] is Hereditarily Indecomposable (HI), which
means that no (closed) infinite dimensional subspace can be decomposed into a direct sum
of two further infinite dimensional subspaces. It is proved in [16] that if X is a complex
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