A Class of Banach Spaces with Few Non Strictly Singular Operators

The original motivation for this paper is based on the natural question left open by the Gowers-Maurey solution of the unconditional basic sequence problem for Banach spaces ([12]). Recall that Gowers and Maurey have constructed a Banach space X with a Schauder basis (en)n but with no unconditional basic sequence. Thus, while every infinite dimensional Banach space contains a sequence (xn)n which forms a Schauder basis for its closure Y = 〈xn〉n, meaning that every vector of Y has a unique representation ∑ n anxn, one may not be able to get such (xn)n such that the sums ∑ n anxn converge unconditionally whenever they converge. The fundamental role of Schauder basis and the fact that the notion is very much dependent on the order lead to the natural variation of the notion, the definition of transfinite Schauder basis (xα)α<γ , where vectors of X have a unique representations as sums ∑ α<γ aαxα. In fact, as it will be clear from some results in this paper, considering transfinite Schauder basis, even if one knows that X has the ordinary Schauder basis, can be an advantage. Thus, the natural question which originated the research of this paper asks whether one can have Banach spaces with long (even of uncountable length) Schauder bases but with no unconditional basic sequence. There is actually a more fundamental reason for asking this question. As noticed originally by W. B. Johnson, the Gowers-Maurey space X is hereditarily indecomposable which in particular yields that the space of operators on X is very small in the sense that every bounded linear operator on X can be written as λIdX + S, where S is a strictly singular operator. On the other hand, if X has a transfinite Schauder basis (eα)α<γ of length, say, γ = ω, it could not longer have so small operator space as projections on infinite intervals 〈eα〉α∈I are all (uniformly) bounded, so one would like to find out the amount of control on the space of non strictly singular operators that is possible in this case. In fact, our solution of the transfinite variation of the unconditional basic sequence problem has led us to many other new questions of this sort, has forced us to introduce several new methods to this area, and has revealed several new phenomena that could have been perhaps difficult to discover by working only in the context of ordinary Schauder bases. To arrive at first necessity for a new method we repeat that our goal here is to construct a Banach space Xω1 with a transfinite Schauder basis (eα)α<ω1 with no unconditional basic sequence as well as to understand its separable initial segments Xγ = 〈eα〉α<γ . The original Gowers-Maurey method for preventing unconditional basic sequences is to force the unconditional constants of initial finite-dimensional subspaces, according to the fixed Schauder basis, grow to infinity. Since initial finite-dimensional subspaces according to our transfinite Schauder basis (eα)α<ω1 are far from exhausting the whole space their method will not work here. It turns out that in order to impose the conditional structure to our space(s) Xγ (γ ≤ ω1) we needed to import a tool from another area of mathematics, a rather canonical semi-distance function ̺ on the space ω1 of all countable ordinals ([26]). What ̺ does in our context here is to essentially identify the structure of finite-dimensional subspaces of various Xγ ’s which globally are of course very much different, since for example Xω is hereditarily indecomposable while, say, Xω2 has a rich space of non-strictly singular operators.