The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view. If (xn, x * n ) ∞ n=1 is a biorthogonal system in X then for each x ∈ X we have a formal expansionxn. The thresholding greedy algorithm (with threshold ε > 0) applied to x is formally defined asxn. The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.
Abstract. This article closes the cycle of characterizations of greedy-like bases in the "isometric" case initiated in [1] with the characterization of 1-greedy bases and continued in [2] with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight characterization of almost greedy bases.
It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k m [B]) ∞ m=1 of its conditionality constants verifies the estimate k m [B] = O(log m) and that if the reverse inequality log m = O(k m [B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k m [B] = O(log m) 1−ǫ for some ǫ > 0. However, in the existing literature one finds very few instances of spaces possessing quasigreedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [16] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k m [B] = O(log m) and superreflexive classical Banach spaces having for every ǫ > 0 quasi-greedy bases B with k m [B] = O(log m) 1−ǫ . Moreover, in most cases those bases will be almost greedy. 2010 Mathematics Subject Classification. 46B15, 41A65.
Abstract. For a conditional quasi-greedy basis B in a Banach space the associated conditionality constants k m [B] verify the estimate k m [B] = O(log m). Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies k m [B] = (log m) 1−ǫ for some 0 < ǫ < 1, and this is optimal. Our first goal in this paper will be to fill the gap in between the general case and the superreflexive case and investigate the growth of the conditionality constants in non-superreflexive spaces. Roughly speaking, the moral will be that we can guarantee optimal bounds only for quasigreedy bases in superreflexive spaces. We prove that if a Banach space X is not superreflexive then there is a quasi-greedy basis B in a Banach space Y finitely representable in X with k m [B] ≈ log m. As a consequence we obtain that for every 2 < q < ∞ there is a Banach space X of type 2 and cotype q possessing a quasi-greedy basis B with k m [B] ≈ log m. We also tackle the corresponding problem for Schauder bases and show that if a space is non-superreflexive then it possesses a basic sequence B with k m [B] ≈ m.
This paper initiates the study of the structure of a new class of p-Banach spaces, 0 < p < 1, namely the Lipschitz free p-spaces (alternatively called Arens-Eells p-spaces) F p (M) over pmetric spaces. We systematically develop the theory and show that some results hold as in the case of p = 1, while some new interesting phenomena appear in the case 0 < p < 1 which have no analogue in the classical setting. For the former, we, e.g., show that the Lipschitz free p-space over a separable ultrametric space is isomorphic to p for all 0 < p ≤ 1. On the other hand, solving a problem by the first author and N. Kalton, there are metric spaces N ⊂ M such that the natural embedding from F p (N ) to F p (M) is not an isometry.
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