On the Structure of the Spreading Models of a Banach Space

Abstract

Abstract. We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space X. In particular we give an example of a reflexive X so that all spreading models of X contain ℓ 1 but none of them is isomorphic to ℓ 1 . We also prove that for any countable set C of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of C. In certain cases this ensures that X admits, for each α < ω 1 , a spreading model (x (α)i ) i is dominated by (and not equivalent to) (x (β) i ) i . Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.

An Extension of Schreier Unconditionality

“…COLUMN 2: Apply Remark 2.6 to (y (2,2)} to obtain a subarray (y 2 i,j ) (i,j)∈I with the property that for all a in A,…”

“…Therefore for the "multiple of the inclusion plus compact problem" we only examine HI saturated Banach spaces. In Section 3 we prove Theorems 3.2 and 3.8 which give sufficient conditions on a Banach space X so that the "multiple of the inclusion plus compact" problem to have an affirmative answer in X, and extend results of [2] and [21]. In his 2000 dissertation N. Dew [7] introduced a new HI space which we refer to as space D. In Section 4 we examine some of the basic properties of D and we apply Theorem 3.8 to prove that the"multiple of the inclusion plus compact" problem has an affirmative answer in D.…”

On spaces admitting no ℓ p or c 0 spreading model

“…Recently, in [4], it was shown that there exist hereditarily indecomposable spaces not admitting any ℓ p or c 0 as a spreading model. In [1], the authors construct a space not admitting an ℓ p , c 0 or reflexive spreading model. In paper [3] they show that a variant of the space X S does not admit any ℓ p or c 0 as a k-iterated spreading model for any k ∈ N.…”

Lattice structures and spreading models

“…Sometimes knowledge of the spreading models can be used to deduce subspace knowledge about X itself (e.g., [AOST,OS1]) but the relationship is still not completely understood. Spaces with no "nice" subspaces can have very nice spreading models (e.g., [AD]).…”