Vol. 14, No. 5, 2021

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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
Abstract

We provide a generalization of two results of Knaust and Odell. We prove that if X is a Banach space and (gn)n=1 is a right dominant Schauder basis such that every normalized, weakly null sequence in X admits a subsequence dominated by a subsequence of (gn)n=1, then there exists a constant C such that every normalized, weakly null sequence in X admits a subsequence C-dominated by a subsequence of (gn)n=1. 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 (gn)n=1, 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 (gn)n=1. 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