On left democracy function

TL;DR

This paper investigates the properties of the left democracy function in Banach spaces, providing examples and conditions under which bases are greedy or have optimal greedy projections, extending previous research.

Contribution

It introduces a basis with a non-doubling left democracy function and establishes the link between greediness and the optimality of greedy projections in such bases.

Findings

Existence of a basis with non-doubling $h_l$

Greedy projection is not optimal for non-doubling $h_l$ bases

Basis is greedy iff the greedy projection is optimal

Abstract

We continue the study undertaken in \cite{GHN} of left democracy function $h_l(N)=\inf_{#\Lambda=N}\left\|\sum_{n\in \Lambda_N} x_n\right\| $ of an unconditional basis in a Banach space $X$. We provide an example of a basis with $h_l$ non-doubling. Then we show that for bases with non-doubling $h_l$ the greedy projection is not optimal. Together with results from \cite{GHN} and improved by C. Cabrelli, G. Garrig\'os, E. Hernandez and U. Molter we get that the basis is greedy if and only if the greedy projection is optimal.