A study of conditional spreading sequences

TL;DR

This paper investigates the structure of conditional spreading sequences, demonstrating their decomposition into well-understood parts and exploring their implications for the universality and generation of spreading models in Banach spaces.

Contribution

It introduces a decomposition of conditional spreading sequences into unconditional and convex block homogeneous parts, advancing understanding of their structure and universality.

Findings

Every conditional spreading sequence decomposes into unconditional and convex block homogeneous parts.

The space C(ω^ω) is universal for all spreading models, including conditional and unconditional.

Every conditional spreading sequence can be generated by a sequence in a quasi-reflexive space of order one.

Abstract

It is shown that every conditional spreading sequence can be decomposed into two well behaved parts, one being unconditional and the other being convex block homogeneous, i.e. equivalent to its convex block sequences. This decomposition is then used to prove several results concerning the structure of spaces with conditional spreading bases as well as results in the theory of conditional spreading models. Among other things, it is shown that the space $C(\omega^\omega)$ is universal for all spreading models, i.e., it admits all spreading sequences, both conditional and unconditional, as spreading models. Moreover, every conditional spreading sequence is generated as a spreading model by a sequence in a space that is quasi-reflexive of order one.