Uniform subsequential estimates on weakly null sequences

Brixey, Milena; Causey, Ryan M.; Frankart, Patrick

doi:10.2140/involve.2021.14.743

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Uniform subsequential estimates on weakly null sequences

Milena Brixey, Ryan M. Causey and Patrick Frankart

Vol. 14 (2021), No. 5, 743–774

DOI: 10.2140/involve.2021.14.743

Abstract

We provide a generalization of two results of Knaust and Odell. 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.

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Keywords

Banach spaces, Ramsey theory

Mathematical Subject Classification

Primary: 46B03, 46B15

Milestones

Received: 17 April 2020

Revised: 26 March 2021

Accepted: 26 May 2021

Published: 9 February 2022

Communicated by Kenneth S. Berenhaut

Authors

Milena Brixey
Department of Mathematics Miami University Oxford, OH United States

Ryan M. Causey
Department of Mathematics Miami University Oxford, OH United States

Patrick Frankart
Department of Mathematics Miami University Oxford, OH United States

Vol. 14, No. 5, 2021