Uniform subsequential estimates on weakly null sequences

TL;DR

This paper generalizes results on domination of weakly null sequences in Banach spaces, establishing uniform subsequential estimates related to a specific Schauder basis, and introduces an ordinal-quantified unification of these results.

Contribution

It provides a unified framework for subsequential domination estimates in Banach spaces, extending previous results to include uniform bounds and an ordinal-based interpolation.

Findings

Existence of a universal constant C for subsequential domination

Domination of spreading models by subsequences of a basis

Unified ordinal-quantified result interpolating previous theorems

Abstract

We provide a generalization of two results of Knaust and Odell from \cite{KO2} and \cite{KO}. We prove that if $X$ is a Banach space and $(g_n)_{n=1}^\infty$ is a right dominant Schauder basis such that every normalized, weakly null sequence in $X$ admits a subsequence dominated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists a constant $C$ such that every normalized, weakly null sequence in $X$ admits a subsequence $C$-dominated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove that if every spreading model generated by a normalized, weakly null sequence in $X$ is dominated by some spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$, then there exists $C$ such that every spreading model generated by a normalized, weakly null sequence in $X$ is $C$-dominated by every spreading model generated by a subsequence of $(g_n)_{n=1}^\infty$. We also prove a single, ordinal-quantified result which unifies and interpolates between these two results.