Operators on $C(K)$ spaces preserving copies of Schreier spaces

Abstract

Abstract. It is proved that an operator T : C(K) → X, K compact metrizable, X a separable Banach space, for which the -Szlenk index of T * (B X * ) is greater than or equal to ω ξ , ξ < ω 1 , is an isomorphism on a subspace of C(K) isomorphic to X ξ , the Schreier space of order ξ. As a corollary, one obtains that a complemented subspace of C(K) with Szlenk index equal to ω ξ+1 contains a subspace isomorphic to X ξ .

Absolutely convex sets of large Szlenk index

“…Assertion (i) of Proposition 2.7 is a trivial generalisation of Lemma 2.7 of [13], requiring little change in the proof. Assertion (ii) is Lemma 2.8 of [13]. We refer the reader to [13] for the proofs.…”

“…Blossomed trees are used in [13] to study fixing properties of operators of large Szlenk index acting on C(K) spaces. The important property of a blossomed tree T in studying the Szlenk index is that for every t ∈ T \ MAX(T ) and cofinite subset…”

“…The following example guarantees a rich supply of blossomed trees. Note that other examples of blossomed trees, namely the Schreier families of finite subsets of N, are used by Gasparis in [13]. The construction of trees in Example 2.6 below is essentially the same as that given by Bourgain on p.91 of [2].…”

Szlenk and w∗-dentability indices of the Banach spaces C([0,Brooker1 2013Journal of Mathematical Analysis and Applications

“…Since then the Szlenk index has found a variety of uses in the study of Banach space geometry, a survey of which can be found in [14]. One of the main applications of the Szlenk index is in the study of C(K) spaces and their operators, as witnessed in particular by the work of Alspach [2], Alspach and Benyamini [1], Benyamini [3], Bourgain [5] and Gasparis [8]; we refer to the survey article [17] for a detailed discussion of this topic. The purpose of the current paper is to enlarge the class of C(K) spaces for which the Szlenk index of C(K) is known.…”

Operators whose dual has non-separable range

“…Typically, one has an operator T : X → Y which is "large" in a suitable sense and tries to find a concrete object that the operator T preserves. Various versions of this problem have been studied in the literature and several satisfactory answers have been obtained; see, for instance, [1,4,5,13,14,15,16,23,24]. Among them, there are two fundamental results that deserve special attention.…”