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<p>Property&#xa0;(A) is a weak symmetry condition that plays a fundamental role in the characterization of greedy-type bases in the <i>isometric</i> 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&#xa0;(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.</p>

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Conditional Bases with Property (A)

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月31日

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  • Fernando Albiac,
  • José L. Ansorena,
  • Miguel Berasategui,
  • Pablo M. Berná

摘要

Property (A) is a weak 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.