Advertisement

manuscripta mathematica

, Volume 59, Issue 4, pp 399–422 | Cite as

Operatoren H1→X

  • Wolfgang Hensgen
Article

Abstract

Let X be a complex Banach space and H1 the usual Hardy space. Various properties of operators L1/H 0 1 →X and, mainly, H1→X are considered, e.g. being weakly compact, Riesz representable, Dunford-Pettis. Connections with RNP resp. aRNP and with the validity of the equation\(H^1 \widehat \otimes X \cong \mathbb{H}^1 \left( X \right)\) are also studied, the latter space being an X-valued Hardy space. Whereas results for operators L1/H 0 1 →X closely resemble well-known theorems about operators L1→ X, this is not the case for operators H1→X. E.g., for “most” classical Banach spaces X it isnot true that\(H^1 \widehat \otimes X \cong \mathbb{H}^1 \left( X \right)\) (canonically).

Keywords

Banach Space Number Theory Algebraic Geometry Hardy Space Topological Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. [1]
    Bourgain, J.: Embedding L1 in L1/H1, TAMS278 (1983), 689–702Google Scholar
  2. [2]
    Bourgain, J.: Bilinear forms on H and bounded bianalytic functions, TAMS286 (1984), 313–337Google Scholar
  3. [3]
    Bukhvalov, A.V.: On an analytic representation of operators with abstract norm, Sov. Math. Dokl.14 (1973), 197–201Google Scholar
  4. [4]
    Bukhvalov, A.V.: Hardy spaces of vector-valued functions, J. Sov. Math.16 (1981), 1051–1059Google Scholar
  5. [5]
    Bukhvalov, A.V., Danilevich, A.A.: Boundary properties of analytic and harmonic functions with values in Banach space, Math. Notes Acad. Sci. USSR31 (1982), 104–110Google Scholar
  6. [6]
    Diestel, J.: A survey of results related to the Dunford-Pettis porperty, in: Proc. Conf. on Integration, Topology, and Geometry in Linear Spaces 1979, Providence (Rh.I.), AMS (1980), 15–60Google Scholar
  7. [7]
    Diestel, J., Uhl, Jr., J.J.: Vector Measures, AMS, Providence (Rh.I.) 1977 (Math. Surveys15)Google Scholar
  8. [8]
    Dowling, P.N.: Representable operators and the analytic Radon-Nikodým property in Banach spaces, Proc. R. Ir. Acad.85 A (1985), 143–150Google Scholar
  9. [9]
    Dowling, P.N.: The analytic Radon-Nikodým property in Banach spaces, Dissertation, Kent (Ohio) 1986Google Scholar
  10. [10]
    Duren, P.L.: Theory of Hp Spaces, Academic Press, New York etc. 1970 (Pure and Appl Math.38)Google Scholar
  11. [11]
    Edgar, G.A.: Analytic martingale convergence, J.F.A.69 (1986), 268–280Google Scholar
  12. [12]
    Garnett, J.B.: Bounded Analytic Functions, Academic Press, New York etc. 1981 (Pure and Appl. Math.96)Google Scholar
  13. [13]
    Grossetête, C.: Sur certaines classes de fonctions harmoniques dans le disque à valeur dans un espace vectoriel topologique localement convexe, C.R. Acad. Sci. Paris273 (1971), 1048–1051Google Scholar
  14. [14]
    Grossetête, C.: Classes de Hardy et de Nevanlinna pour les fonctions holomorphes à valeurs vectorielle-métrique, C.R. Acad. Sci. Paris274 (1972), 251–253Google Scholar
  15. [15]
    Grothendieck, A.: Une caractérisation vectorielle-metrique des espaces L1, Can.J.Math.,7 (1955), 552–561Google Scholar
  16. [16]
    Hensgen, W.: Hardy-Räume vektorwertiger Funktionen, Dissertation, München 1986Google Scholar
  17. [17]
    Hoffman, K.: Banach Spaces of Analytic Functions, Prentice Hall, Inc., Englewood Cliffs (N.J.) 1962Google Scholar
  18. [18]
    Ionescu-Tulcea, A., Ionescu-Tulcea, C.: Topics in the Theory of Lifting, Springer, Berlin etc. 1969 (Ergebnisse ...48)Google Scholar
  19. [19]
    Köthe, G.: Topological Vector Spaces II, Springer, Berlin etc. 1979 (Grundlehren ...237)Google Scholar
  20. [20]
    Kwapień, S., Pelczyński, A.: Some linear topological properties of the Hardy space H1, Comp. Math.33 (1976), 261–288Google Scholar
  21. [21]
    Lindenstrauss, J., Pelczyński, A.: Absolutely summing operators in Lp-spaces and their applications, Studia Math.29 (1968), 275–326Google Scholar
  22. [22]
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, II, Springer, Berlin etc. 1977, 1979 (Ergebnisse ...92,97)Google Scholar
  23. [23]
    Nachbin, L.: A theorem of the Hahn-Banach type for linear transformations, TAMS68 (1950), 28–64Google Scholar
  24. [24]
    Pelczyński, A.: Banach Spaces of Analytic Functions and Absolutely Summing Operators, AMS, Providence (Rh.I.) 1977 (Reg. Conf. Ser. in Math.30)Google Scholar
  25. [25]
    Rudin, W.: Real and Complex Analysis, 2nd ed., Tata McGraw-Hill Publ. Co. Ltd., New Delhi 1981Google Scholar
  26. [26.]
    Ryan, R.: Boundary values of analytic vector-valued functions, Indag. Math.65 (1962), 558–572Google Scholar
  27. [27]
    Ryan, R.: The F. and M. Riesz theorem for vector measures, Indag. Math.66 (1963), 408–412Google Scholar
  28. [28]
    Talagrand, M.: Pettis Integral and Measure Theory, AMS, Providence (Rh.I.) 1984 (MAMS307)Google Scholar
  29. [29]
    Wojtaszczyk, P.: The Banach space H1, in: Functional Analysis: Surveys and Recent Results III, North Holland Math. Studies90, 1934Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Wolfgang Hensgen
    • 1
  1. 1.NWF I-MathematikUniversität RegensburgRegensburg

Personalised recommendations