2018/03/21 by Fernando Albiac, Albiac, Fernando, José L. Ansorena +5 · 1 citation
Computer Science · Mathematics · #41A65 #46B15 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1803.08351
openalex publication_date 2018/03/21 · openalex created_date 2022/09/29 · openalex updated_date 2026/07/28
It is known that for a conditional quasi-greedy basis \B in a\nBanach space mathbbX, the associated sequence\n(km[\B])m=1\∞ of its conditionality constants verifies\nthe estimate km[\B]=\O(\log m) and that if the reverse\ninequality \log m =\O(km[\B]) holds then mathbbX is\nnon-superreflexive. Indeed, it is known that a quasi-greedy basis in a\nsuperreflexive quasi-Banach space fulfils the estimate\nkm[\B]=\O(\log m)1-\ε for some \ε>0.\nHowever, in the existing literature one finds very few instances of spaces\npossessing quasi-greedy basis with conditionality constants "as large as\npossible." Our goal in this article is to fill this gap. To that end we enhance\nand exploit a technique developed in [S. J. Dilworth, N. J. Kalton, and D.\nKutzarova, On the existence of almost greedy bases in Banach spaces, Studia\nMath. 159 (2003), no. 1, 67-101] and craft a wealth of new examples of both\nnon-superreflexive classical Banach spaces having quasi-greedy bases\n\B with km[\B]=\O(\log m) and superreflexive\nclassical Banach spaces having for every \ε>0 quasi-greedy bases\n\B with km[\B]=\O(\log m)1-\ε.\nMoreover, in most cases those bases will be almost greedy.\n