Functiones et Approximatio Commentarii Mathematici

Some permanence results of the Dunford-Pettis and Grothendieck properties in lcHs

Angela A. Albanese and Elisabetta M. Mangino

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Abstract

We prove some permanence results with respect to quotient spaces and to projective and injective tensor products of the Dunford-Pettis and Grothendieck properties in the setting of locally convex Hausdorff spaces.

Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 243-258.

Dates
First available in Project Euclid: 22 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1308749127

Digital Object Identifier
doi:10.7169/facm/1308749127

Mathematical Reviews number (MathSciNet)
MR2841182

Zentralblatt MATH identifier
1226.46002

Subjects
Primary: 46A20: Duality theory 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45] 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Secondary: 46A04: Locally convex Fréchet spaces and (DF)-spaces 46A08: Barrelled spaces, bornological spaces 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40]

Keywords
Dunford-Petty property Grothendieck property Schur property quotient space tensor product

Citation

Albanese, Angela A.; Mangino, Elisabetta M. Some permanence results of the Dunford-Pettis and Grothendieck properties in lcHs. Funct. Approx. Comment. Math. 44 (2011), no. 2, 243--258. doi:10.7169/facm/1308749127. https://projecteuclid.org/euclid.facm/1308749127


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References

  • A.A. Albanese, J. Bonet and W. Ricker, Grothendieck spaces with the Dunford-Pettis property, Positivity 14 (2010), 145–164.
  • A.A. Albanese, J. Bonet and W. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces, J. Math. Anal. Appl. 365 (2010), 142–157.
  • A.A. Albanese, J. Bonet and W. Ricker, Mean ergodic semigroups of operators, preprint, 2010.
  • K.D. Bierstedt and J. Bonet, A question of D. Vogt on $(\rm LF)$-spaces, Arch. Math. 61(2) (1993), 170–172.
  • K.D. Bierstedt and J. Bonet, Some aspects of the modern theory of Fréchet spaces, RACSAM, 97(2) (2003), 159–188.
  • K.D. Bierstedt, R.G. Meise and W.H. Summers, Köthe sets and Köthe sequence spaces, In: Functional Analysis, Holomorphy and Approximation Theory (Rio de Janeiro, 1980), North-Holland Math. Stud. 17, Amsterdam, 1982, 27–91.
  • F. Bombal and I. Villanueva, On the Dunford-Pettis property of the tensor product of $C(K)$ spaces, Proc. Amer. Math. Soc. 129(5) (2001), 1359–1363.
  • J. Bonet and J.C. Díaz, The problem of topologies of Grothendieck and the class of Fréchet T-spaces. Math. Nachr. 150 (1991), 109–118.
  • J. Bonet and S. Dierolf, On the lifting of bounded sets, Proc. Edin. Math. Soc. 36 (1993), 277–281.
  • J. Bonet, J.C. Dí az and J. Taskinen, Tensor stable Fréchet and (DF) spaces, Collect. Math. 42 (1991), 83–120.
  • J. Bonet and W. Ricker, Schauder decompositions and the Grothendieck and Dunford-Pettis properties in Köthe echelon spaces of infinite order, Positivity 11 (2007), 77–93.
  • G. Botelho and P. Rueda, The Schur property on projective and injective tensor products, Proc. Amer. Math. Soc. 137(1) (2009), 219–225.
  • J. Bourgain, New classes of $\mathcal L^p$-spaces, Lecture Notes in Math. 889, Springer, 1981.
  • J. Diestel, A survey of results related to the Dunford–Pettis property, Contemporary Math. 2 (1980), 15–60 (Amer. Math. Soc.).
  • J. Diestel and J.J. Jr. Uhl, Vector Measures, Math. Surveys No. 15, Amer. Math. Soc. Providence, 1977.
  • J.C. Dí az, Montel subspaces in the countable projective limits of $L^1(\mu)$-spaces, Canad. Math. Bull. 32(2) (1989), 169–176.
  • J.C. Dí az and C. Fernández, On quotients of sequence spaces of infinite order, Arch. Math. 66 (1996), 207–213.
  • J.C. Dí az and M.A. Mi$\tilde\rm n$arro, Distinguished Fréchet spaces and projective tensor product, Doga Mat. 14 (1990), 191–208.
  • J.C. Dí az and M.A. Mi$\tilde\rm n$arro, On Fréchet Montel spaces and their projective tensor product, Math. Proc. Camb. Phil. Soc. 113 (1993), 335–341.
  • P. Domański, M. Lindström and G. Schlüchtermann, Grothendieck spaces and duals of injective tensor products, Bull. London Math. Soc. 28 (1996), 617–626.
  • P. Domański and M. Lindström, Grothendieck operators on tensor products, Proc. Amer. Math. Soc. 126(8) (1997), 2285-2291.
  • G. Emmanuele, Remarks on weak compactness of operators defined on certain injective tensor products, Proc. Amer. Math. Soc. 116 (1992), 473–476.
  • G. Emmanuele, Some remarks on lifting of isomorphic properties to injective and projective tensor products, Portugal. Math. 53 (1996), 253–255.
  • G. Emmanuele, Some permanence results of properties of Banach spaces, Comment. Math. Univ. Carolinae, 45(3) (2004), 491–497.
  • R.E. Edwards, Functional Analysis, Reinhart and Winston, New York, 1965.
  • F.J. Freniche, Grothendieck locally convex spaces of continuous vector valued functions, Pacif. Journal Math. 120(2) (1984), 345–355.
  • M. González and M. Gutierrez, The Dunford-Pettis property on tensor products, Math. Proc. Cambridge Philos. Soc. 131 (2001), 185–192.
  • H. Jarchow, Locally convex spaces, Teubner Stuttgart (1981).
  • M. Lindström, A note on Fréchet Montel spaces, Proc. Amer. Math. Soc. 108(1) (1990), 191–196.
  • R.H. Lohman, A note on Banach spaces containing $\ell^1$, Canad. Math. Bull. 19 (1976), 365–367.
  • R.A. Ryan, The Dunford-Pettis property and projective tensor products Bull. Polish Acad. Sci. Math. 35 (1987), 785–792.
  • M. Talagrand, La proprieté de Dunford-Pettis dans $C(K,E)$ et $L^1(E)$, Israel J. Math. 44(4) (1983), 317–321.
  • J. Taskinen, Counterexamples to “problème des topologies” of Grothendieck, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 63, 1986.
  • D. Vogt, Regularity properties of (LF)-spaces, In: Progress in functional analysis (Peñí scola, 1990), North-Holland Math. Stud. 170, Amsterdam, 1992, p.57–84.