On certain equivalent norms on Tsirelson’s space

It remains open an important question as to whether or not there exists a distortable Banach space which is not arbitrarily distortable. The primary candidate for such a space is Tsirelson’s space T . While it is not difficult to directly define, for every 1 < λ < 2, an equivalent norm on T which is a λ-distortion, T does not belong to any general class of Banach spaces known to be arbitrarily distortable. In fact (see below) if there does exists a distortable not arbitrarily distortable Banach space X then X must contain a subspace which is very Tsirelson-like in appearance. It is thus of interest to examine in particular all known equivalent norms on T to see if they can arbitrarily distort T (or a subspace of T ). We do so in this paper for a previously unstudied fascinating class of renormings. The renormings we consider here are “natural” in that pertain to the deep combinatorial nature of the norm of T . Namely, for each n by ‖·‖n we denote the norm of the Tsirelson space T (Sn, 2 −n), which can be easily seen to be equivalent to the original norm on T . Our main result (Theorem 2.1) is that this family of equivalent norms does not arbitrarily distort T or even any subspace of T . The proof actually introduces a larger family of equivalent norms (‖ · ‖j )j,n and (| · | n j )j,n which are shown to not arbitrarily distort any subspace of T . Quantitative estimates for the stabilizations of these norms are given in Theorem 2.5. It is shown that (up to absolute constants) one has that for all n and subspaces X ⊆ T there is a subspace Y ⊆ X such that ‖y‖n ∼ 1 n if y ∈ Y with ‖y|| = 1. Some stabilization results for more general norms on T of various classes are also given in Section 3. In Section 4 we raise some problems. Section 1 contains the relevant terminology and background material. Otherwise our notation is standard as may be found in [LT]. More detailed information about Tsirelson’s space and Tsirelson type spaces can be found in [CS], [OTW], [AD], [AO] and the references therein.