Bases of random unconditional convergence in Banach spaces
HTML articles powered by AMS MathViewer
- by J. Lopez-Abad and P. Tradacete PDF
- Trans. Amer. Math. Soc. 368 (2016), 9001-9032 Request permission
Abstract:
We study random unconditional convergence for a basis in a Banach space. The connections between this notion and classical unconditionality are explored. In particular, we analyze duality relations, reflexivity, uniqueness of these bases and existence of unconditional subsequences.References
- Spiros A. Argyros and Richard G. Haydon, A hereditarily indecomposable $\scr L_\infty$-space that solves the scalar-plus-compact problem, Acta Math. 206 (2011), no. 1, 1–54. MR 2784662, DOI 10.1007/s11511-011-0058-y
- Sergey V. Astashkin, Nigel Kalton, and Fyodor A. Sukochev, Cesaro mean convergence of martingale differences in rearrangement invariant spaces, Positivity 12 (2008), no. 3, 387–406. MR 2421142, DOI 10.1007/s11117-007-2146-y
- Sergey V. Astashkin, Mikhail Leibov, and Lech Maligranda, Rademacher functions in BMO, Studia Math. 205 (2011), no. 1, 83–100. MR 2822505, DOI 10.4064/sm205-1-6
- Jürgen Batt and Wolfgang Hiermeyer, On compactness in $L_{p}(\mu ,\,X)$ in the weak topology and in the topology $\sigma (L_{p}(\mu ,\,X),\,L_{q}(\mu ,\,X^{\prime } ))$, Math. Z. 182 (1983), no. 3, 409–423. MR 696537, DOI 10.1007/BF01179760
- P. Billard, S. Kwapień, A. Pełczyński, and Ch. Samuel, Biorthogonal systems of random unconditional convergence in Banach spaces, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986) Longhorn Notes, Univ. Texas, Austin, TX, 1986, pp. 13–35. MR 1017039
- J. Bourgain, An averaging result for $c_{0}$-sequences, Bull. Soc. Math. Belg. 30 (1978), no. 1, 83–87. MR 549653
- J. Bourgain, An averaging result for $l^{1}$-sequences and applications to weakly conditionally compact sets in $L^{1}_{X}$, Israel J. Math. 32 (1979), no. 4, 289–298. MR 571083, DOI 10.1007/BF02760458
- Jean Bourgain, New classes of ${\cal L}^{p}$-spaces, Lecture Notes in Mathematics, vol. 889, Springer-Verlag, Berlin-New York, 1981. MR 639014
- J. Bourgain and F. Delbaen, A class of special ${\cal L}_{\infty }$ spaces, Acta Math. 145 (1980), no. 3-4, 155–176. MR 590288, DOI 10.1007/BF02414188
- Pilar Cembranos and José Mendoza, Banach spaces of vector-valued functions, Lecture Notes in Mathematics, vol. 1676, Springer-Verlag, Berlin, 1997. MR 1489231, DOI 10.1007/BFb0096765
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- P. G. Dodds and F. A. Sukochev, RUC-decompositions in symmetric operator spaces, Integral Equations Operator Theory 29 (1997), no. 3, 269–287. MR 1477320, DOI 10.1007/BF01320701
- P. G. Dodds, E. M. Semenov, and F. A. Sukochev, RUC systems in rearrangement invariant spaces, Studia Math. 151 (2002), no. 2, 161–173. MR 1917951, DOI 10.4064/sm151-2-4
- John Hancock Elton, WEAKLY NULL NORMALIZED SEQUENCES IN BANACH SPACES, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–Yale University. MR 2628434
- D. J. H. Garling and N. Tomczak-Jaegermann, RUC-systems and Besselian systems in Banach spaces, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 163–168. MR 994087, DOI 10.1017/S0305004100068055
- W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
- J. Hájek and A. Rényi, Generalization of an inequality of Kolmogorov, Acta Math. Acad. Sci. Hungar. 6 (1955), 281–283 (English, with Russian summary). MR 76207, DOI 10.1007/BF02024392
- Richard Haydon, Subspaces of the Bourgain-Delbaen space, Studia Math. 139 (2000), no. 3, 275–293. MR 1762585, DOI 10.4064/sm-139-3-275-293
- Robert C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518–527. MR 39915, DOI 10.2307/1969430
- William B. Johnson, Bernard Maurey, and Gideon Schechtman, Weakly null sequences in $L_1$, J. Amer. Math. Soc. 20 (2007), no. 1, 25–36. MR 2257395, DOI 10.1090/S0894-0347-06-00548-0
- W. B. Johnson and G. Schechtman, Martingale inequalities in rearrangement invariant function spaces, Israel J. Math. 64 (1988), no. 3, 267–275 (1989). MR 995572, DOI 10.1007/BF02882423
- Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
- N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30. MR 542869, DOI 10.1090/S0002-9947-1979-0542869-X
- K. S. Kazaryan and E. M. Semenov, RUC-bases and the Olevskiĭ system, Mat. Zametki 75 (2004), no. 4, 566–579 (Russian, with Russian summary); English transl., Math. Notes 75 (2004), no. 3-4, 530–541. MR 2068766, DOI 10.1023/B:MATN.0000023334.07767.e7
- S. Kwapień, On Banach spaces containing $c_{0}$, Studia Math. 52 (1974), 187–188. MR 356156
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2011. Isoperimetry and processes; Reprint of the 1991 edition. MR 2814399
- Pascal Lefèvre, Topological dichotomy and unconditional convergence, Serdica Math. J. 25 (1999), no. 4, 297–310. MR 1742770
- J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- Michel Loève, Probability theory, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-New York-London, 1960. 2nd ed. MR 0123342
- B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), no. 1, 77–98. MR 438091, DOI 10.4064/sm-61-1-77-98
- E. Odell, On Schreier unconditional sequences, Banach spaces (Mérida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993, pp. 197–201. MR 1209461, DOI 10.1090/conm/144/1209461
- H. Witvliet, Unconditional Schauder decompositions and multiplier theorems. Ph.D. thesis, Delft University of Technology, 2000.
- P. Wojtaszczyk, Every separable Banach space containing $c_0$ has an RUC system, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986) Longhorn Notes, Univ. Texas, Austin, TX, 1986, pp. 37–39. MR 1017040
Additional Information
- J. Lopez-Abad
- Affiliation: Instituto de Ciencias Matemáticas (ICMAT), CSIC-UAM-UC3M-UCM, C/Nicolás Cabrera 13-15, Campus Cantoblanco, UAM 28049 Madrid, Spain; Instituto de Matemática e Estatística - IME/USP, Rua do Matão, 1010 - Cidade Universitária, São Paulo - SP, 05508-090, Brasil
- MR Author ID: 680200
- Email: abad@icmat.es
- P. Tradacete
- Affiliation: Department of Mathematics, Universidad Carlos III de Madrid, 28911, Leganés, Madrid, Spain
- MR Author ID: 840453
- Email: ptradace@math.uc3m.es
- Received by editor(s): September 23, 2014
- Received by editor(s) in revised form: December 19, 2014
- Published electronically: March 18, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 9001-9032
- MSC (2010): Primary 46B09, 46B15
- DOI: https://doi.org/10.1090/tran/6636
- MathSciNet review: 3551596