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Spaces of operators between Fréchet spaces

Published online by Cambridge University Press:  24 October 2008

José Bonet  and
Mikael Lindström
José Bonet
Affiliation:
Departamento de Matemática Aplicada, E.T.S. Arquitectura, Universidad Politécnica de Valencia, E-46071 Valencia, Spain
Mikael Lindström
Affiliation:
Department of Mathematics, Åbo Akademi, SF-20500 Åbo, Finland
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Abstract

Motivated by recent results on the space of compact operators between Banach spaces and by extensions of the Josefson–Nissenzweig theorem to Fréchet spaces, we investigate pairs of Fréchet spaces (E, F) such that every continuous linear map from E into F is Montel, i.e. it maps bounded subsets of E into relatively compact subsets of F. As a consequence of our results we characterize pairs of Köthe echelon spaces (E, F) such that the space of Montel operators from E into F is complemented in the space of all continuous linear maps from E into F.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 1 , January 1994 , pp. 133 - 144
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Bator, E. M.. Unconditionally converging and compact operators on c0. Rocky Mountain J. Math. 22 (1992), 417–422.Google Scholar
[2]Bellenot, S. F.. Basic sequences in non-Schwartz–Fréchet spaces. Trans. Amer. Math. Soc. 258 (1980), 199–216.Google Scholar
[3]Bierstedt, K. D., Meise, R. G. and Summers, W. H.. Köthe sets and Köthe sequence spaces, in Functional Analysis, Holomorphy and Approximation theory (ed. Barroso, J. A.) (North Holland Math. Studies 71 (1982)).Google Scholar
[4]Bonet, J.. On the identity L(E, F) = LB(E, F) for pairs of locally convex spaces E and F. Proc. Amer. Math. Soc. 99 (1987), 249–255.Google Scholar
[5]Bonet, J.. A question of Valdivia on quasi-normable Fréchet spaces. Canad. Math. Bull. 34 (1991), 301–304.CrossRefGoogle Scholar
[6]Bonet, J. and Lindström, M.. Convergent sequences in duals of Fréchet spaces. Proc. Essen Conference in Functional Analysis (M. Dekker, 1993), to appear.Google Scholar
[7]Bonet, J., Lindström, M. and Valdivia, M.. Two theorems of Josefson-Nissenzweig type for Fréchet spaces. Proc. Amer. Math. Soc. 117 (1993), 363–364.Google Scholar
[8]Collins, H. S. and Ruess, W.. Weak compactness in spaces of compact operators and of vector-valued functions. Pacific J. Math. 106 (1982), 45–71.CrossRefGoogle Scholar
[9]Constantinescu, C.. Spaces of Measures (de Gruyter Studies in Mathematics 4, 1984).CrossRefGoogle Scholar
[10]Wilde, M. De. Closed Graph Theorems and Webbed Spaces (Pitman, 1978).Google Scholar
[11]Diaz, J. C.. Montel subspaces in the countable projective limits of Lp(μ)-spaces. Canad. Math. Bull. 32 (1989), 169–176.CrossRefGoogle Scholar
[12]Diestel, J.. Sequences and Series in Banach Spaces (Springer, 1984).Google Scholar
[13]Dierolf, S. and Domanski, P.. Factorization of Montel operators, preprint 1992.Google Scholar
[14]Domanski, P. and Drewnowski, L.. Injectivity of spaces of operators, preprint 1992.Google Scholar
[15]Drewnowski, L.. An extension of a theorem of Rosenthal on operators acting from l∞(Γ). Studia Math. 62 (1976), 207–215.Google Scholar
[16]Emmanuele, G.. A remark on the containment of c0 in spaces of compact operators. Math. Proc. Cambridge Philos. Soc. 111 (1992), 331–335.Google Scholar
[17]Jarchow, H.. Locally Convex Spaces (Teubner, 1981).CrossRefGoogle Scholar
[18]John, K.. On the uncomplemented subspace K(X, Y). Czech. Math. J. 42 (1992), 167–173.Google Scholar
[19]Kalton, N.. Spaces of compact operators. Math. Ann. 208 (1974), 267–278.Google Scholar
[20]Kalton, N.. Exhaustive operators and vectors measures. Proc. Edinb. Math. Soc. 19 (1975). 291–300.CrossRefGoogle Scholar
[21]Köthe, G.. Topological Vector Spaces I (Springer, 1969).Google Scholar
[22]Köthe, G.. Topological Vector Spaces II (Springer, 1979).CrossRefGoogle Scholar
[23]Lindström, M. and Schlumprecht, Th.. A Josefson–Nissenzweig theorem for Fréchet spaces. Bull. London Math. Soc. 25 (1993), 55–58.CrossRefGoogle Scholar
[24]Metafune, G. and Moscatelli, V. B.. Quojections and Prequojections, pp. 235–254 in Advances in the Theory of Fréchet Spaces (ed. Terzioglu, T.) (Kluwer Acad. Publ., 1989).CrossRefGoogle Scholar
[25]Metafune, G. and Moscatelli, V. B.. On the space lp+ = ∩q>plq Math. Nachr. 147 (1990), 7–12.Google Scholar
[26]Meise, R. and Vogt, D.. Einführung in die Funktionalanalysis (Vieweg, Wiesbaden, 1992).CrossRefGoogle Scholar
[27]Önal, S. and Terzioglu, T.. Unbounded linear operators and nuclear Köthe quotients. Arch. Math. 54 (1990), 576–581.CrossRefGoogle Scholar
[28]Ramanujan, M. S. and Vogt, D.. Fréchet spaces between which all continuous linear maps are compact. Proc. Essen Conference in Functional Analysis (M. Dekker, 1993), to appear.Google Scholar
[29]Terzioglu, T.. A note on bounded linear operators. Doga Tr. Math. J. 10 (1986), 338–344.Google Scholar
[30]Terzioglu, T. and Yurdakul, M.. Restrictions of unbounded linear operators on Fréchet spaces. Arch. Math. 46 (1986), 547–550.Google Scholar
[31]Valdivia, M.. Topics in Locally Convex Spaces (North-Holland Math. Studies 67, 1982).Google Scholar
[32]Weill, L. J.. Unconditional and shrinking bases in locally convex spaces. Pacific J. Math. 29 (1969), 467–483.Google Scholar
[33]Vogt, D.. Frécheträume zwischen denen jede stetige lineare Abbildung beschränkt ist. J. Reine Angew. Math. 345 (1983), 182–200.Google Scholar