Conditional bases with Property~(A)

TL;DR

This paper constructs examples of Banach spaces with Schauder bases that possess Property~(A) but are not unconditional, resolving a longstanding open problem and advancing the understanding of greedy bases in Banach spaces.

Contribution

It provides the first known examples of bases with Property~(A) that are not unconditional, clarifying the relationship between these properties in Banach spaces.

Findings

Existence of bases with Property~(A) that are not unconditional

Counterexamples that address open questions in greedy basis theory

Refinement of the construction methods for Banach space bases

Abstract

Property~(A) is a week symmetry condition that plays a fundamental role in the characterization of greedy-type bases in the isometric case, i.e., when the constants involved in the study of the efficiency of the thresholding greedy algorithm in Banach spaces are sharp. In this note we build examples of Banach spaces with Schauder bases that have Property~(A) but fail to be unconditional, thus settling a long standing problem in the area. As a by-product of our work we hone our construction to produce counterexamples that solve other open questions in the isometric theory of greedy bases.