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Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 6, Pages 111–136 (Mi izv461)  

This article is cited in 35 scientific papers (total in 35 papers)

Singular symmetric functionals and Banach limits with additional invariance properties

P. G. Dodds, B. de Pagter, A. A. Sedaev, E. M. Semenov, F. A. Sukochev


Abstract: For symmetric spaces of measurable functions on the real half-line, we study the problem of existence of positive linear functionals monotone with respect to the Hardy–Littlewood semi-ordering, the so-called symmetric functionals. Two new wide classes of symmetric spaces are constructed which are distinct from Marcinkiewicz spaces and for which the set of symmetric functionals is non-empty. We consider a new construction of singular symmetric functionals based on the translation-invariance of Banach limits defined on the space of bounded sequences. We prove the existence of Banach limits invariant under the action of the Hardy operator and all dilation operators. This result is used to establish the stability of the new construction of singular symmetric functionals for an important class of generating sequences.

DOI: https://doi.org/10.4213/im461

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English version:
Izvestiya: Mathematics, 2003, 67:6, 1187–1212

Bibliographic databases:

UDC: 517.51
MSC: 30H05, 32E25, 46A40, 46B10, 46B42, 46E30, 46J10, 46L50
Received: 11.10.2002

Citation: P. G. Dodds, B. De Pagter, A. A. Sedaev, E. M. Semenov, F. A. Sukochev, “Singular symmetric functionals and Banach limits with additional invariance properties”, Izv. RAN. Ser. Mat., 67:6 (2003), 111–136; Izv. Math., 67:6 (2003), 1187–1212

Citation in format AMSBIB
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\pages 111--136
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Sedaev, F. A. Sukochev, S. Lord, “Connes–Dixmier Traces, Singular Symmetric Functionals, and Measurable Elements in the Sense of Connes”, Math. Notes, 76:6 (2004), 884–889  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Lord S., Sedaev A., Sukochev F., “Dixmier traces as singular symmetric functionals and applications to measurable operators”, J. Funct. Anal., 224:1 (2005), 72–106  crossref  mathscinet  zmath  isi  elib  scopus
    3. S. Lord, A. A. Sedaev, F. A. Sukochev, “Connes-Dixmier Traces, Singular Symmetric Functionals, and the Notion of Connes Measurable Element”, Math. Notes, 77:5 (2005), 671–676  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. L. Carey, F. A. Sukochev, “Dixmier traces and some applications in non-commutative geometry”, Russian Math. Surveys, 61:6 (2006), 1039–1099  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Semenov E.M., Usachev A.S., Khorpyakov O.O., “The space of almost convergent sequences”, Dokl. Math., 74:1 (2006), 587–588  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. Georgievskii P.Y., Ershov A.P., Levin V.A., Timofeev I.B., Shibkov V.M., “Transverse electric discharges in supersonic air flows: Simulation of gasdynamic effects in the discharge channel”, High Temperature, 44:1 (2006), 1–11  mathnet  crossref  isi  elib  scopus
    7. Gogatishvili A., Ovchinnikov V.I., “Interpolation orbits and optimal Sobolev's embeddings”, J. Funct. Anal., 253:1 (2007), 1–17  crossref  mathscinet  zmath  isi  elib  scopus
    8. Astashkin S.V., Sukochev F.A., “Banach-Saks property in Marcinkiewicz spaces”, J. Math. Anal. Appl., 336:2 (2007), 1231–1258  crossref  mathscinet  zmath  isi  elib  scopus
    9. Carey A.L., Rennie A., Sedaev A., Sukochev F., “The dixmier trace and asymptotics of zeta functions”, J. Funct. Anal., 249:2 (2007), 253–283  crossref  mathscinet  zmath  isi  elib  scopus
    10. Guvernyuk S.V., Savinov K.G., “Isobaric separation structures in supersonic flows with a localized inhomogeneity”, Dokl. Phys., 52:3 (2007), 151–155  mathnet  crossref  adsnasa  isi  elib  elib  scopus
    11. A. S. Usachev, “Transformations in the space of almost convergent sequences”, Siberian Math. J., 49:6 (2008), 1136–1137  mathnet  crossref  mathscinet  isi  elib
    12. Kalton N., Sukochev F., “Rearrangement-invariant functionals with applications to traces on symmetrically normed ideals”, Canad. Math. Bull., 51:1 (2008), 67–80  crossref  mathscinet  zmath  isi  elib  scopus
    13. A. A. Sedaev, “Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces”, Russian Math. (Iz. VUZ), 53:12 (2009), 77–80  mathnet  crossref  mathscinet  zmath
    14. E. M. Semenov, F. A. Sukochev, “Characteristic functions of Banach limits”, Siberian Math. J., 51:4 (2010), 723–727  mathnet  crossref  mathscinet  isi  elib  elib
    15. Steven Lord, Fedor Sukochev, “Measure Theory in Noncommutative Spaces”, SIGMA, 6 (2010), 072, 36 pp.  mathnet  crossref  mathscinet
    16. Sedaev A.A., Sukochev F.A., Zanin D.V., “Lidskii-Type Formulae for Dixmier Traces”, Integral Equations and Operator Theory, 68:4 (2010), 551–572  crossref  mathscinet  zmath  isi  scopus
    17. Semenov E.M., Sukochev F.A., “Invariant Banach limits and applications”, Journal of Functional Analysis, 259:6 (2010), 1517–1541  crossref  mathscinet  zmath  isi  elib  scopus
    18. Sukochev F., Zanin D., “zeta-function and heat kernel formulae”, J Funct Anal, 260:8 (2011), 2451–2482  crossref  mathscinet  zmath  isi  elib  scopus
    19. Kalton N.J., Sedaev A.A., Sukochev F.A., “Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces”, Adv Math, 226:4 (2011), 3540–3549  crossref  mathscinet  zmath  isi  elib  scopus
    20. Semenov E.M., Sukochev F.A., Usachev A.S., “Strukturnye svoistva mnozhestva banakhovykh predelov”, Doklady Akademii nauk, 441:2 (2011), 177–178  zmath  elib
    21. E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Structure properties of the Banach limit set”, Dokl. Math, 84:3 (2012), 802  crossref  mathscinet  isi  scopus
    22. E. Semenov, F. Sukochev, “Extreme points of the set of Banach limits”, Positivity, 17:1 (2013), 163  crossref  mathscinet  zmath  isi  scopus
    23. Sedaev A.A., Sukochev F.A., “Dixmier Measurability in Marcinkiewicz Spaces and Applications”, J. Funct. Anal., 265:12 (2013), 3053–3066  crossref  mathscinet  zmath  isi  elib  scopus
    24. Sukochev F., Usachev A., Zanin D., “Generalized Limits with Additional Invariance Properties and their Applications to Noncommutative Geometry”, Adv. Math., 239 (2013), 164–189  crossref  mathscinet  zmath  isi  elib  scopus
    25. Sukochev F., Zanin D., “Traces on Symmetrically Normed Operator Ideals”, J. Reine Angew. Math., 678 (2013), 163–200  crossref  mathscinet  zmath  isi  elib  scopus
    26. Dykema K., Skripka A., “Perturbation Formulas for Traces on Normed Ideals”, Commun. Math. Phys., 325:3 (2014), 1107–1138  crossref  mathscinet  zmath  adsnasa  isi  scopus
    27. E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Geometric properties of the set of Banach limits”, Izv. Math., 78:3 (2014), 596–620  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. E. A. Alekhno, E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Order properties of the set of Banach limits”, Dokl. Math, 91:1 (2015), 20  crossref  mathscinet  zmath  scopus
    29. E. A. Alekhno, E. M. Semenov, F. A. Sukochev, A. S. Usachev, “Order and geometric properties of the set of Banach limits”, St. Petersburg Math. J., 28:3 (2017), 299–321  mathnet  crossref  mathscinet  isi  elib
    30. Alekhno E., Semenov E., Sukochev F., Usachev A., “On the structure of invariant Banach limits”, C. R. Math., 354:12 (2016), 1195–1199  crossref  mathscinet  zmath  isi  elib  scopus
    31. Sukochev F., Usachev A., “Dixmier traces and non-commutative analysis”, J. Geom. Phys., 105 (2016), 102–122  crossref  mathscinet  zmath  isi  scopus
    32. Alekhno E.A., Semenov E.M., Sukochev F.A., Usachev A.S., “Banach Limits: Invariance and Functional Characteristics”, Dokl. Math., 96:1 (2017), 305–307  crossref  mathscinet  zmath  isi  scopus
    33. Alekhno E., Semenov E., Sukochev F., Usachev A., “Invariant Banach Limits and Their Extreme Points”, Studia Math., 242:1 (2018), 79–107  crossref  mathscinet  zmath  isi  scopus
    34. Sukochev F., Veksler A., “The Mean Ergodic Theorem in Symmetric Spaces”, Studia Math., 245:3 (2019), 229–253  crossref  mathscinet  zmath  isi
    35. E. M. Semenov, F. A. Sukochev, A. Usachev, “The main classes of invariant Banach limits”, Izv. Math., 83:1 (2019), 124–150  mathnet  crossref  crossref  adsnasa  isi  elib
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