A note on the relationship between the Szlenk and $$w^*$$ w ∗ -dentability indices of arbitrary $$w^*$$ w ∗ -compact sets

Abstract

We prove the optimal estimate between the Szlenk and w * -dentability indices of an arbitrary w * -compact subset of the dual of a Banach space. For a given w * -compact, convex subset K of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of K. We give applications to the w *dentability index of a Banach space and of an operator.

The Szlenk index of $L_p(X)$ and $A_p$

“…We first note that if Z ∈ codim(X), L p (X)/L p (Z) is either the zero vector space or isomorphic to L p , and therefore has Szlenk index not exceeding ω. As explained in [5], this means that for any B-tree T with o(T ) ω, any δ > 0, and any normally weakly null (f t ) t∈T.Lp(X) ⊂ B Lp(X) , there exist t ∈ T.L p (X), g ∈ co(f s : ∅ ≺ s t), and h ∈ B Lp(Z) such that g − h Lp(X) < δ. Moreover, by the density of simple functions, we may assume this h is simple.…”

“…We recall that K is w * -dentable if for any non-empty subset L of K and any ε > 0, there exists a w * -open slice S of X * such that L ∩ U = ∅ and diam(L ∩ S) < ε. We recall that a w * -open slice is a subset of X * of the form {x * ∈ X * : Re x * (x) > a} for some x ∈ X and a ∈ R. As mentioned in [5], a consequence of Corollary 2 is that if Sz(K) ω ξ , then Sz(K) Dz(K) ω 1+ξ , where Dz(K) denotes the w * -dentability index of K. Thus Corollary 2 implies that K is w * -dentable if and only if it is w * -fragmentable.…”