Linear versus nonlinear forms of partial unconditionality of bases (Q6592089)

Let \(\mathbb{X}\) be a Banach (or more generally a quasi-Banach) space over the real or complex field. As basis in \(\mathbb{X}\) be understood a norm-bounded sequence \(\mathfrak{B}=(x_n)_{n=1}^\infty\) in \(\mathbb{X}\) whose linear span is dense in \(\mathbb{X}\) and for which there is a (unique) norm bounded sequence of linear functionals \(\mathfrak{B}=(x^*_n)\) in \(\mathbb{X}^*\) biorthogonal to \(\mathbb{X}\). For the study various problems in the theory of greedy algorithms in space \(\mathbb{X}\), some bases (for example: bounded-oscillation unconditional basis; \((D,d)\) bounded-oscillation unconditional basis, where \(1\leq d\leq D <\infty\); nearly unconditional basis; quasi-greedy basis; truncation quasi-greedy basis) were introduced, the identification of connections between which is of great importance.\N\NNaturally, the problem arises of establishing a connection between these concepts. The solution of this problem contribute to bringing to the new connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of Banach spaces. It is proved that for a given basis \(\mathfrak{B}\) of a quasi-Banach space \(\mathbb{X}\), the following are equivalent: (i) \(\mathfrak{B}\) is bounded-oscillation unconditional; (ii) \(\mathfrak{B}\) is \((1,1)\) bounded-oscillation unconditional; (iii) \(\mathfrak{B}\) is truncation quasi-greedy. It is also proved that there is a Schauder basis \(\mathfrak{B}\) of a Banach space \(\mathbb{X}\) that is nearly unconditional but is not bounded-oscillation unconditional. Let us also note that new bases have been introduced, which are named as unconditional for constant coefficients (UCC for short) and as flattening quasi-greedy. It is proved that there is a monotone Schauder basis of a Banach space that is flattening quasi-greede but is not UCC.