Extensions and new characterizations of some greedy-type bases

TL;DR

This paper explores strengthened notions of partial greediness in Banach space bases, establishing equivalences and new properties that deepen understanding of greedy-type bases.

Contribution

It introduces the consecutive almost greedy property and a stronger PG property, expanding the theoretical framework of greedy bases in Banach spaces.

Findings

Consecutive almost greedy property is equivalent to almost greedy property.

A new stronger PG property is identified for general bases.

The study clarifies relationships between various greedy-type properties.

Abstract

Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.