Summary: A HEREDITARILY INDECOMPOSABLE ASYMPTOTIC 2 BANACH
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
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 . One of the reasons this problem has become so attractive is that by a result of N.
Aronszajn and K.T. Smith , 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 ,.
The ground breaking work  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  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  that if X is a complex