Bases of random unconditional convergence in Banach spaces (Q2822727)

The aim of this paper is to analyze the relations among randon unconditional convergence for a basis in a Banach space and the classical notion of unconditional basis. A series \(\sum_n x_n\) in a Banach space is randomly unconditionally convergent when \(\sum_n \varepsilon_n x_n\) converges almost surely on signs \((\varepsilon_n)_n\) with respect to the Haar probability measure on \(\{-1,1\}^\mathbb N\). A random unconditionally convergent coordinate system \((e_i)_i\) in a Banach space (RUC) has the property that every element can be written as an expansion that is randomly unconditionally convergent. This can also be characterized in terms of domination by the norm of a vector \(x\) of the sign-average integrals of the finite projections of \(x\) using the corresponding biorthogonal systems. If the converse inequality holds, then it is said that the system is random unconditionally divergent (RUD). The analysis of both kinds of coordinate systems seems to be useful in applications of functional analysis, for example in signal processing and harmonic analysis.NEWLINENEWLINEUnconditional bases have major differences with respect to RUC and RUD bases. For instance, in the second case block basis stability may fail. Also, every separable Banach space can be linearly embedded in a space with an RUC basis. More interesting differences are explained in the paper. Section 2 is devoted to explain some fundamental facts: known and new sufficient and necessary conditions for a space to have an RUC or RUD basis are given. For example, Proposition 2.5 states that a basic sequence is unconditional if and only if it is RUC and RUD. The Walsh basis in \(L^1[0,1]\) is RUD (Ex. 2.21) and the Rademacher functions in \(\mathrm{BMO}[0,1]\) define an RUC basic sequence (Ex. 2.22).NEWLINENEWLINEIn Section 3, uniqueness of RUC and RUD bases is analyzed. For example, a classical result states that every basis of \(\ell^1\) is equivalent to the unit basis of \(\ell^1\), and in fact it is a characterization of \(\ell^1\)-spaces (Corollary 3.3). In the case of RUD bases, this does not hold. Indeed, it is stated in Theorem 2.3 that every infinite-dimensional Banach space with an RUD basis has at least two non-equivalent RUD bases. Section 4 confronts the problem of the comparison of RUC, RUD and unconditional bases of Banach spaces. Not every basic sequence has RUC or RUD subsequences. The question is if every weakly null sequence has subsequences with partial random unconditionality RUC or RUD. It is shown here that the classical Maurey-Rosenthal example of a weakly null basis without unconditional subsequences does not have RUD subsequences either. RUD bases without unconditional subsequences are also studied, providing constructive tools for finding such and related sequences (Theorem 4.4). RUD sequences in rearrangement invariant spaces are studied in Section 5. For example, it is stated in Theorem 5.1 that every weakly null sequence in a separable r.i. space on \([0,1]\) has a subsequence which is basic RUD. Open questions, examples and precise explanations are also given in this interesting paper.