Abstract
Using an isometric version of the Davis, Figiel, Johnson, and Peŀczyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Amir and J. Lindenstrauss,The structure of weakly compact sets in Banach spaces, Annals of Mathematics88 (1968), 35–46.
K. Astala and H. O. Tylli,Seminorms related to weak compactness and to Tauberian operators, Mathematical Proceedings of the Cambridge Philosophical Society107 (1990), 367–375.
P. G. Casazza and H. Jarchow,Self-induced compactness in Banach spaces, Proceedings of the Edinburgh Mathematical Society. Section A126 (1996), 355–362.
C. M. Cho and W. B. Johnson, A characterization of subspaces X of ℓ p for which K(X) is an M-ideal in L(X), Proceedings of the American Mathematical Society93 (1985), 466–470.
W. J. Davis, T. Figiel, W. B. Johnson and A. Peŀczyński,Factoring weakly compact operators, Journal of Functional Analysis17 (1974), 311–327.
J. Diestel,Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics485, Springer-Verlag, Berlin-Heidelberg-New York, 1975.
J. Diestel,Sequences and Series in Banach Spaces, Graduate Texts in Mathematics92, Springer-Verlag, Berlin, 1984.
J. Diestel and J. J. Uhl, Jr.,Vector Measures, Mathematical Surveys15, American Mathematical Society, Providence, 1977.
N. Dunford and J. T. Schwartz,Linear Operators. Part 1: General Theory, Wiley Interscience, New York, 1958.
G. Emmanuele and K. John, Some remarks on the position of the spaceK(X,Y) inside the spaceW(X,Y), New Zealand Journal of Mathematics26 (1997), 183–189.
H. Fakhoury,Sélections linéaires associées au théoréme de Hahn-Banach, Journal of Functional Analysis11 (1972), 436–452.
M. Feder and P. D. Saphar,Spaces of compact operators and their dual spaces, Israel Journal of Mathematics21 (1975), 38–49.
T. Figiel,Factorization of compact operators and applications to the approximation problem, Studia Mathematica45 (1973), 191–210.
G. Godefroy, N. J. Kalton and P. D. Saphar,Unconditional ideals in Banach spaces, Studia Mathematica104 (1993), 13–59.
G. Godefroy and P. D. Saphar,Duality in spaces of operators and smooth norms on Banach spaces, Illinois Journal of Mathematics32 (1988), 672–695.
A. Grothendieck,Produits tensoriels topologiques et espaces nucléaires, Memoirs of the American Mathematical Society16 (1955).
N. Grønbæk and G. A. Willis,Approximate identities in Banach algebras of compact operators, Canadian Mathematical Bulletin36 (1993), 45–53.
P. Harmand, D. Werner and W. Werner,M-ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics1547, Springer-Verlag, Berlin, 1993.
H. Jarchow,Locally Convex Spaces, B.G. Teubner, Stuttgart, 1981.
J. Johnson,Remarks on Banach spaces of compact operators, Journal of Functional Analysis32 (1979), 304–311.
W. B. Johnson,Factoring compact operators, Israel Journal of Mathematics9 (1971), 337–345.
N. J. Kalton,Locally complemented subspaces and L p-spaces for 0<p<1, Mathematische Nachrichten115 (1984), 71–97.
Å. Lima,The metric approximation property, norm-one projections and intersection properties of balls, Israel Journal of Mathematics84 (1993), 451–475.
Å. Lima,Property (ωM*)and the unconditional metric compact approximation property, Studia Mathematica113 (1995), 249–263.
Å. Lima and E. Oja,Ideals of finite rank operators, intersection properties of balls, and the approximation property, Studia Mathematica133 (1999), 175–186.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Ergebnisse der Mathematik und ihrer Grenzgebiete92, Springer-Verlag, Berlin, 1977.
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Ergebnisse der Mathematik und ihrer Grenzgebiete97, Springer-Verlag, Berlin, 1979.
E. Oja,Geometry of Banach spaces having shrinking approximations of the identity. Transactions of the American Mathematical Society (to appear).
E. Oja,Géométrie des espaces de Banach ayant des approximations de l'identité contractantes, Comptes Rendus de l'Académie des Sciences, Paris, Série I328 (1999), 1167–1170.
O. I. Reinov,How bad can a Banach space with the approximation property be?, Matematicheskie Zametki33 (1983), 833–846 (in Russian): English translation in Mathematical Notes33 (1983), 427–434.
A. Szankowski,Subspaces without approximation property, Israel Journal of Mathematics30 (1978), 123–129.
G. Willis,The compact approximation property does not imply the approximation property, Studia Mathematica103 (1992), 99–108.
P. Wojtaszczyk,Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics25, Cambridge University Press, 1991.
Author information
Authors and Affiliations
Additional information
The third-named author wishes to acknowledge the warm hospitality provided by Åsvald Lima and his colleagues at Agder College, where a part of this work was done in May–June 1998. Her visit was supported by the Norwegian Academy of Science and Letters and by Estonian Science Foundation Grant 3055.
Rights and permissions
About this article
Cite this article
Lima, A., Nygaard, O. & Oja, E. Isometric factorization of weakly compact operators and the approximation property. Isr. J. Math. 119, 325–348 (2000). https://doi.org/10.1007/BF02810673
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02810673