Elton's near unconditionality of bases as a threshold-free form of greediness

TL;DR

This paper reveals the equivalence between Elton's near unconditionality and quasi-greediness for largest coefficients, providing new insights into threshold functions and characterizing 1-quasi-greedy bases in the isometric theory of greedy bases.

Contribution

It establishes the equivalence of Elton's near unconditionality and quasi-greediness for largest coefficients, and characterizes 1-quasi-greedy bases in the isometric setting.

Findings

Near unconditionality and quasi-greediness are equivalent properties.

New description of the threshold function associated with near unconditionality.

Characterization of 1-quasi-greedy bases in the isometric theory.

Abstract

Elton's near unconditionality and quasi-greediness for largest coefficients are two properties of bases that made their appearance in functional analysis from very different areas of research. One of our aims in this note is to show that, oddly enough, they are connected to the extent that they are equivalent notions. We take advantage of this new description of the former property to further the study of the threshold function associated with near unconditionality. Finally, we made a contribution to the isometric theory of greedy bases by characterizing those bases that are $1$-quasi-greedy for largest coefficients.