Weight-partially greedy bases and weight-Property $(A)$

TL;DR

This paper introduces weighted variants of partially greedy bases and properties, providing characterizations and conditions under which these weighted properties imply the original weighted Property (A).

Contribution

It defines $w$-left and $w$-right Property (A), introduces $w$-partially greedy bases, and studies their characterizations and implications for weighted properties.

Findings

Characterizations of $w$-partially greedy and $w$-reverse partially greedy bases.

Conditions on weight sequences for $w$-left and $w$-right Property (A) to imply $w$-Property (A).

Abstract

In this paper, motivated by the notion of $w$-Property $(A)$ defined in [2], we introduce the notions of $w$-left Property $(A)$ and $w$-right Property $(A)$. We also introduce the notions of $w$-partially greedy basis (using a characterization of partially greedy basis from [4]) and $w$-reverse partially greedy basis. The main aim of this paper is to study $(i)$ some characterizations of $w$-partially greedy and $w$-reverse partially greedy basis $(ii)$ conditions on the weight sequences when $w$-left Property $(A)$ and (or) $w$-right Property $(A)$ implies $w$-Property $(A)$.