# zbMATH — the first resource for mathematics

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.
##### MSC:
 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:
##### References:
  Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Springer, New York (2006) · Zbl 1094.46002  Astashkin, SV, On the geometric properties of Cesàro spaces, Sb. Math., 203, 514-533, (2012) · Zbl 1253.46022  Astashkin, SV; Curbera, GP; Tikhomirov, KE, On the existence of RUC systems in rearrangement invariant spaces, Math. Nach., 289, 175-186, (2016) · Zbl 1355.46015  Astashkin, SV; Maligranda, L, Cesàro function spaces fail the fixed point property, Proc. Am. Math. Soc., 136, 4289-4294, (2008) · Zbl 1168.46014  Astashkin, SV; Maligranda, L, Structure of Cesàro function spaces, Indag. Math. (N.S.), 20, 329-379, (2009) · Zbl 1200.46027  Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988) · Zbl 0647.46057  Billard, P., Kwapién, S., Pełczyński, A., Samuel, C.h.: Biorthogonal systems of random unconditional convergence in Banach spaces, In: Texas Functional Analysis Seminar 1985-1986, Longhorn Notes, pp. 13-35 (1986) · Zbl 0424.42014  Curbera, GP; Ricker, WJ, Abstract Cesàro spaces: integral representations, J. Math. Anal. Appl., 441, 25-44, (2016) · Zbl 1355.46031  Dodds, PG; Semenov, EM; Sukochev, FA, RUC systems in rearrangement invariant spaces, Studia Math., 151, 161-173, (2002) · Zbl 1031.46017  Garling, DJH; Tomczak-Jaegermann, N, RUC-systems and Besselian systems in Banach spaces, Math. Proc. Camb. Philos. Soc., 106, 163-168, (1989) · Zbl 0693.46004  Johnson, WB; Maurey, B; Schechtman, G, Weakly null sequences in $$L_1$$, J. Am. Math. Soc., 20, 25-36, (2007) · Zbl 1125.46014  Kashin, B.S., Saakyan, A.A.: Orthogonal series. Am. Math. Soc., Providence RI (1989) · Zbl 0668.42011  Krein, S.G., Petunin, Ju.I., Semenov, E.M.: Interpolation of Linear Operators (Am. Math. Soc., Providence RI) (1982) · Zbl 1355.46015  Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol. II. Springer, Berlin (1979) · Zbl 0403.46022  López-Abad, J; Tradacete, P, Bases of random unconditional convergence in Banach spaces, Trans. Am. Math. Soc., 368, 9001-9032, (2016) · Zbl 1372.46013  Novikov, I., Semenov, E.: Haar Series and Linear Operators. Kluwer, Dordrecht (1996) · Zbl 0865.42024  Ovsepian, RI; Pełczyński, A, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space and related constructions of uniformly bounded orthogonal systems in $$L^2$$, Studia Math., 54, 149-159, (1975) · Zbl 0317.46019  Szarek, SJ, On the best constants in the Khinchin inequality, Studia Math., 58, 197-208, (1976) · Zbl 0424.42014  Wojtaszczyk, P, Existence of some special bases in Banach spaces, Studia Math., 47, 83-93, (1973) · Zbl 0261.46020  Wojtaszczyk, P.: Every separable Banach space containing $$c_0$$ has an RUC system, Texas Functional Analysis Seminar 1985-1986, pp. 37-39, Longhorn Notes. Univ. Texas, Austin (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.