Time stopping for Tsirelson’s norm

Beanland, Kevin; Duncan, Noah; Holt, Michael

doi:10.2140/involve.2018.11.857

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Time stopping for Tsirelson's norm

Kevin Beanland, Noah Duncan and Michael Holt

Vol. 11 (2018), No. 5, 857–866

DOI: 10.2140/involve.2018.11.857

Abstract

Tsirelson’s norm $∥ \cdot ∥_{T}$ on $c_{00}$ is defined as the limit of an increasing sequence of norms ${(∥ \cdot ∥_{n})}_{n = 1}^{\infty}$ . For each $n \in ℕ$ let $j (n)$ be the smallest integer satisfying $∥ x ∥_{j (n)} = ∥ x ∥_{T}$ for all $x$ with $max supp x = n$ . We show that $j (n)$ is $O (n^{1 ∕ 2})$ . This is an improvement of the upper bound of $O (n)$ given by P. Casazza and T. Shura in their 1989 monograph on Tsirelson’s space.

Keywords

Tsirelson's space, Banach space

Mathematical Subject Classification 2010

Primary: 46B03

Milestones

Received: 18 April 2017

Revised: 21 July 2017

Accepted: 14 August 2017

Published: 2 April 2018

Communicated by Stephan Garcia

Authors

Kevin Beanland
Department of Mathematics Washington and Lee University Lexington, VA United States

Noah Duncan
Department of Mathematics Washington and Lee University Lexington, VA United States

Michael Holt
Department of Mathematics Washington and Lee University Lexington, VA United States

Vol. 11, No. 5, 2018