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Random unconditional convergence and divergence in Banach spaces close to \(L^1\). (English) Zbl 1405.46012
A biorthogonal system \((x_i,x_i^*)\) in a Banach space is random unconditionally convergent (RUC) if there is some constant \(C>0\) such that, for any \(n\in\mathbb N\) and any sequence of scalars \((a_i)_{i=1}^n\), \[ \mathbb E \bigg\| \sum_{i=1}^n \varepsilon_i a_i x_i\bigg\|\leq C \bigg\|\sum_{i=1}^n a_i x_i\bigg\|, \] where \(\mathbb E\) denotes the expectation with respect to Rademacher random variables \((\varepsilon_i)_{i=1}^n\). This is equivalent to the fact that the series \(\sum_{i=1}^\infty \varepsilon_i a_i x_i\) converges for almost every choice of signs (with respect to the product measure of uniform probability on \(\{-1,1\}^{\mathbb N}\)), whenever \(\sum_{i=1}^\infty a_i x_i\) converges. Analogously, one can define the notion of a random unconditionally divergent (RUD) system by requiring the converse of the above inequality. Note that a system is unconditional if and only if it is both RUC and RUD. It is an open problem to determine whether every Banach space contains a subspace with either an RUC or RUD basis.
In [Functional analysis, Proc. 4th Annu. Semin., Austin/TX (USA) 1985–86, 37–39 (1986; Zbl 0749.46012)], P. Wojtaszczk has shown that every separable Banach space containing a subspace isomorphic to \(c_0\) has a fundamental and total RUC system, in fact, this can be taken to be a basis if the original space is known to have a basis. In the paper under review, the authors provide the dual counterpart of this result: If a Banach space \(X\) contains a complemented subspace isomorphic to \(\ell_1\), then \(X\) contains a fundamental and total RUD system; if, moreover, \(X\) has a basis, then it has an RUD basis.
In addition, several interesting results concerning RUC and RUD systems in Cesàro-type spaces are also given.
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B09 Probabilistic methods in Banach space theory
Full Text: DOI
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