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Grothendieck spaces with the Dunford–Pettis property

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Abstract

Banach spaces which are Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet spaces occurs in Bonet and Ricker (Positivity 11:77–93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For \({p\in \{0\}\cup[1,\infty)}\) it is shown that the classical co-echelon spaces k p (V) and \({K_p(\overline{V})}\) are GDP-spaces if and only if they are Montel. On the other hand, \({K_\infty(\overline{V})}\) is always a GDP-space and k ∞(V) is a GDP-space whenever its (Fréchet) predual, i.e., the Köthe echelon space λ 1(A), is distinguished.

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References

  1. Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. (to appear)

  2. Bierstedt K.D.: An introduction to locally convex inductive limits. In: Hogbe–Nlend, H. (eds) Functional Analysis and its Applications., pp. 35–133. World Scientific Publ. Co., Singapore (1988)

    Google Scholar 

  3. Bierstedt K.D., Meise R.G., Summers W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272, 107–160 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bierstedt, K.D., Meise, R.G., Summers, W.H.: Köthe sets and Köthe sequence spaces. In: Functional Analysis, Holomorphy and Approximation Theory (Rio de Janeiro, 1980), vol. 17, pp. 27–91. North Holland Math. Studies, Amsterdam (1982)

  5. Bierstedt, K.D., Meise, R.: Distinguished echelon spaces and the projective description of weighted inductive limits of type \({\mathcal{V}_d\mathcal{C}(X)}\) . In: Aspects of Mathematics and its Applications, pp. 169–226. North Holland Math. Library, Amsterdam (1986)

  6. Bierstedt K.D., Bonet J.: Some aspects of the modern theory of Fréchet spaces. Rev. R. Acad. Cien. Serie A Mat. RACSAM 97, 159–188 (2003)

    MATH  MathSciNet  Google Scholar 

  7. Bonet, J., Lindström, M.: Convergent sequences in duals of Fréchet spaces. In: Proc. of the Essen Conference in Functional Analysis, pp. 391–404. Marcel Dekker, New York (1993)

  8. Bonet J., Domański P., Lindström M., Ramanujan M.S.: Operator spaces containing c 0 or ℓ ∞. Results Math. 28, 250–269 (1995)

    MATH  MathSciNet  Google Scholar 

  9. Bonet J., Ricker W.J.: Spectral measures in classes of Fréchet spaces. Bull. Soc. Roy. Sci. Liège 73, 99–117 (2004)

    MATH  MathSciNet  Google Scholar 

  10. Bonet J., Ricker W.J.: The canonical spectral measure in Köthe echelon spaces. Integral Equ. Oper. Theory 53, 477–496 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bonet J., Okada S., Ricker W.J.: The canonical spectral measure and Köthe function spaces. Quaestiones Math. 29, 91–116 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Bonet J., Ricker W.J.: Schauder decompositions and the Grothendieck and Dunford–Pettis properties in Köthe echelon spaces of infinite order. Positivity 11, 77–93 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cascales B., Orihuela J.: On compactness in locally convex spaces. Math. Z. 195, 365–381 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dean D.W.: Schauder decompositions in (m). Proc. Am. Math. Soc. 18, 619–623 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  15. Díaz J.C., Miñarro M.A.: Distinguished Fréchet spaces and projective tensor product. Doǧa-Tr. J. Math. 14, 191–208 (1990)

    MATH  Google Scholar 

  16. Díaz J.C., Dierolf S.: On duals of weakly angelic (LF)-spaces. Proc. Am. Math. Soc. 125, 2897–2905 (1997)

    Article  MATH  Google Scholar 

  17. Diestel, J., Uhl J.J., Jr.: Vector Measures. Math. Surveys No. 15. Amer. Math. Soc., Providence (1977)

  18. Diestel J.: A survey of results related to the Dunford–Pettis property. Contemporary Math. 2, 15–60 (1980) (Amer. Math. Soc.)

    MathSciNet  Google Scholar 

  19. Domański P.: Fréchet injective spaces of continuous functions. Results Math. 18, 209–230 (1990)

    MATH  MathSciNet  Google Scholar 

  20. Domański P.: On the injective C(T)-spaces. Indag. Math. (N.S.) 3, 289–299 (1991)

    Article  Google Scholar 

  21. Edwards R.E.: Functional Analysis. Reinhart and Winston, New York (1965)

    MATH  Google Scholar 

  22. Gowers W.T., Maurey B.: The unconditional basic sequence problem. J. Am. Math. Soc. 6, 851–874 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kalton N.J.: Schauder decompositions in locally convex spaces. Proc. Camb. Phil. Soc. 68, 377–392 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  24. Köthe G.: Topological Vector Spaces I. 2nd Rev. edn. Springer, Berlin (1983)

    Google Scholar 

  25. Köthe G.: Topological Vector Spaces II. Springer, Berlin (1979)

    MATH  Google Scholar 

  26. Lotz H.P.: Tauberian theorems for operators on L ∞ and similar spaces. In: Bierstedt, K.D., Fuchssteiner, B. (eds) Functional Analysis: Surveys and Recent Results III., pp. 117–133. North Holland, Amsterdam (1984)

    Chapter  Google Scholar 

  27. Lotz H.P.: Uniform convergence of operators on L ∞ and similar spaces. Math. Z. 190, 207–220 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lusky W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  30. Okada S.: Spectrum of scalar-type spectral operators and Schauder decompositions. Math. Nachr. 139, 167–174 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces, vol. 131. North HollandMath. Studies, Amsterdam (1987)

    Google Scholar 

  32. Qiu J.H.: Retakh’s conditions and regularity properties of (LF)-spaces. Arch. Math. 67, 302–307 (1996)

    Article  MATH  Google Scholar 

  33. Ricker W.J.: Spectral operators of scalar-type in Grothendieck spaces with the Dunford–Pettis property. Bull. London Math. Soc. 17, 268–270 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  34. Ricker W.J.: Spectral measures, boundedly σ–complete Boolean algebras and applications to operator theory. Trans. Am. Math. Soc. 304, 819–838 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ricker W.J.: Operator algebras generated by Boolean algebras of projections in Montel spaces. Integral Equ. Oper. Theory 12, 143–145 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ricker W.J.: Well-bounded operators of type (B) in H.I. spaces. Acta Sci. Math. (Szeged) 59, 475–488 (1994)

    MATH  MathSciNet  Google Scholar 

  37. Vogt, D.: Regularity properties of (LF)-spaces. In: Progress in Functional Analysis, vol. 170, pp. 57–84. North Holland Math. Studies, Amsterdam (1992)

  38. Walsh B.: Structure of spectral measures on locally convex spaces. Trans. Am. Math. Soc. 120, 295–326 (1965)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università del Salento, C.P. 193, 73100, Lecce, Italy

    Angela A. Albanese

  2. Instituto Universitario de Matemática Pura y Aplicada IUMPA, Edificio IDI5 (8E), Cubo F, Cuarta Planta, Universidad Politécnica de Valencia, 46071, Valencia, Spain

    José Bonet

  3. Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85072, Eichstätt, Germany

    Werner J. Ricker

Authors
  1. Angela A. Albanese
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  2. José Bonet
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  3. Werner J. Ricker
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Correspondence to Angela A. Albanese.

Additional information

A. A. Albanese’s support of the Alexander von Humboldt Foundation and J. Bonet’s support of the Alexander von Humboldt Foundation, MEC and FEDER Project MTM 2007-62643 and GV Project Prometeo/2008/101 are gratefully acknowledged.

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Albanese, A.A., Bonet, J. & Ricker, W.J. Grothendieck spaces with the Dunford–Pettis property. Positivity 14, 145–164 (2010). https://doi.org/10.1007/s11117-009-0011-x

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