A Hereditarily Indecomposable Asymptotic `2 Banach Space

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 λ+K where λ is a scalar and K denotes a compact operator. This problem is usually called the “scalar-plus-compact” problem [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 construction 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 HI space then every operator on X can be written as λ + S where λ is a scalar and S is strictly singular (i.e. the restriction of S on any infinite dimensional subspace of X is not an isomorphism). It is also shown in [16] that the same property remains true for the real HI space constructed in [16]. V. Ferenczi [10] proved that if X is a complex HI space and Y is an infinite dimensional subspace of X then every operator from Y to X can be written as λiY + S where iY : Y → X is the inclusion map and S is strictly singular. It was proved in [17] that, roughly speaking, given an algebra of operators satisfying certain conditions, there exists a Banach space X such that for every infinite dimensional subspace Y , every operator from Y to X can be written as a strictly singular perturbation of a restriction to Y of some element of the algebra. The construction of the first HI space prompted researchers to construct HI spaces having additional nice properties. In other words people tried to “marry” the exotic structure of the HI spaces to the nice structure of classical Banach spaces. The reasons behind these efforts were twofold! Firstly, by producing more examples of HI spaces having additional well understood properties we can better understand how the HI property effects other behaviors of the space. Secondly, there is hope that endowing an HI space with additional