Complexity of weakly null sequences

TL;DR

This paper introduces an ordinal index to measure the complexity of weakly null sequences, constructs sequences with arbitrary complexity levels, and applies these to build Tsirelson-like spaces, linking the index to Baire-1 functions and refining existing results.

Contribution

It develops a new ordinal index for weakly null sequences, enabling the construction of sequences with specified complexity and connecting this to function space indices.

Findings

Constructed weakly null sequences with arbitrary complexity levels.

Linked the ordinal index to the Lavrentiev index of Baire-1 functions.

Sharpened results on averaging weakly null sequences.

Abstract

We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each alpha < omega_1, a weakly null sequence (x^{alpha}_n)_n in C(omega^{omega^{alpha}})) with complexity alpha. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are used to construct Tsirelson-like spaces of large index. We also show that this new ordinal index is related to the Lavrentiev index of a Baire-1 function and use the index to sharpen some results of Alspach and Odell on averaging weakly null sequences.