Interpolating Hereditarily Indecomposable Banach Spaces

A Banach space X is said to be Hereditarily Indecomposable (H.I.) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace. (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X .) The H.I. spaces form a new and, as we believe, fundamental class of Banach spaces. The celebrated example of a Banach space with no unconditional basic sequence, due to W. Gowers and B. Maurey ([GM]), is the first construction of a H.I. space. It is easily seen that every H.I. space does not contain any unconditional basic sequence. Actually, the concept of H.I. spaces came after W. Johnson’s observation that this was a property of the Gowers-Maurey example. To describe even further the peculiar structure of a H.I. space, we recall an alternative definition of such a space. A Banach space X is a H.I. space if and only if for every pair of subspaces Y , Z and ε > 0 there exist y ∈ Y , z ∈ Z with ||y|| = ||z|| = 1 and ||y − z|| < ε. Thus, H.I. spaces are structurally irrelevant to classical Banach spaces, in particular to Hilbert spaces. Other constructions of H.I. spaces already exist. We mention Argyros and Deliyanni’s construction of H.I. spaces which are asymptotic ` spaces ([AD2]), V. Ferenczi’s example of a uniformly convex H.I. space ([F2]) and H.I. modified asymptotic ` spaces contained in [ADKM]. Other examples of Banach spaces which are H.I. or which have a H.I. subspace are given in [G1], [H], [OS1]. The construction of such a space requires several steps and it uses two fundamental ideas. The first is Tsirelson’s recursive definition of saturated norms ([Ts]) and the second is Maurey-Rosenthal’s construction of weakly null sequences without unconditional basic subsequences ([MR]). An important ingredient in the GowersMaurey construction is the Schlumprecht space. This is a Tsirelson type Banach space which is arbitrarily distortable and has been used in the solution of important problems. Thus beyond its use in the constructions of H.I. spaces it plays a central role in the solution of the distortion problem for Hilbert spaces ([OS]). The essential difference between Schlumprecht and Tsirelson spaces became more transparent in [AD2] where the mixed Tsirelson spaces were introduced. It is natural to expect that H.I. spaces share special and interesting properties not found in the previously known Banach spaces. Indeed, the following theorem is proven in