Uniform subsequential estimates on weakly null sequences

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.

References in corpus (1)

• The $\xi ,\zeta$-Dunford Pettis property