Almost isometric constants for partial unconditionality

TL;DR

This paper investigates the optimal constants for projections on subsequences of weakly null sequences, achieving near-isometric bounds and applications to quasi-greedy subsequences under certain conditions.

Contribution

It provides new bounds close to 1 for projections in weakly null sequences and links spreading models to quasi-greedy subsequences with near-optimal constants.

Findings

Constants arbitrarily close to 1 for Schreier type projections

Constants arbitrarily close to 1 for Elton type projections under specific conditions

Existence of quasi-greedy subsequences with near-1 constants in certain weakly null sequences

Abstract

We discuss optimal constants of certain projections on subsequences of weakly null sequences. Positive results yield constants arbitrarily close to $1$ for Schreier type projections, and arbitrarily close to $1$ for Elton type projections under the assumption that the weakly null sequence admits no subsequence generating a $c_0$ spreading model. As an application, we prove that a weakly null sequence admitting a spreading model not equivalent to the $c_0$ basis has a quasi-greedy subsequence with quasi-greedy constant arbitrarily close to $1$.