Projections from $L(X,Y)$ onto $K(X,Y)$
John, Kamil
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000), p. 765-771 / Harvested from Czech Digital Mathematics Library
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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^*$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^*$ has the approximation property). Suppose that $L(X,Y)\neq K(X,Y)$ and let $1<\lambda<2$. Then $X$ (respectively $Y$) can be equivalently renormed so that any projection $P$ of $L(X,Y)$ onto $K(X,Y)$ has the sup-norm greater or equal to $\lambda $.

Publié le : 2000-01-01
Classification:  46B28 
@article{119208,
     author = {Kamil John},
     title = {Projections from $L(X,Y)$ onto $K(X,Y)$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {41},
     year = {2000},
     pages = {765-771},
     zbl = {1050.46016},
     mrnumber = {1800167},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119208}
}

John, Kamil. Projections from $L(X,Y)$ onto $K(X,Y)$. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 765-771. http://gdmltest.u-ga.fr/item/119208/

Amir D.; Lindenstrauss J. The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35-59. (1968) | MR 0228983 | Zbl 0164.14903

Arterburn D.; Whitney R. Projections in the space of bounded linear operators, Pacific J. Math. 15 (1965), 739-746. (1965) | MR 0187052

Casazza P.G.; Kalton N.J. Notes on approximation properties on separable Banach spaces, in: Geometry of Banach spaces, London Math. Soc. Lecture Note Ser. 158, Cambridge Univ. Press, 1991, pp.49-63. | MR 1110185

Diestel J.; Morrison T.J. The Radon-Nikodym property for the space of operators, Math. Nachr. 92 (1979), 7-12. (1979) | MR 0563569 | Zbl 0444.46021

Emmanuele G. On the containment of $c_0$ by spaces of compact operators, Bull. Sci. Mat. 115 (1991), 177-184. (1991) | MR 1101022

Emmanuele G. A remark on the containment of $c_0$ in spaces of compact operators, Math. Proc. Cambridge Phil. Soc. 111 (1992), 331-335. (1992) | MR 1142753

Emmanuele G.; John K. Uncomplementability of spaces of compact operators in lager spaces of operators, Czechoslovak J. Math. 47 (122) (1997), 19-32. (1997) | MR 1435603

Feder M. On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 24 (1980), 196-205. (1980) | MR 0575060 | Zbl 0411.46009

Feder M. On the non-existence of a projection onto the spaces of compact operators, Canad. Math. Bull. 25 (1982), 78-81. (1982) | MR 0657655

Godefroy G.; Saphar P. Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32 (4) (1988), 672-695. (1988) | MR 0955384 | Zbl 0631.46015

Godun B.V. Unconditional bases and basic sequences, Izv. Vyssh. Uchebn. Zaved. Mat 24 (1980), 69-72. (1980) | MR 0603941

John K. On the space $K(P,P^\ast)$ of compact operators on Pisier space P, Note di Matematica 72 (1992), 69-75. (1992) | MR 1258564

John K. On the uncomplemented subspace ${\Cal K}(X,Y)$, Czechoslovak Math. J. 42 (1992), 167-173. (1992) | MR 1152178

Johnson J. Remarks on Banach spaces of compact operators, J. Funct. Analysis 32 (1979), 304-311. (1979) | MR 0538857 | Zbl 0412.47024

Kuo T.H. Projections in the space of bounded linear operators, Pacific. J. Math. 52 (1974), 475-480. (1974) | MR 0352939

Kalton N.J. Spaces of compact operators, Math. Ann. 208 (1974), 267-278. (1974) | MR 0341154 | Zbl 0266.47038

Lindenstrauss J. On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967-970. (1966) | MR 0205040 | Zbl 0156.36403

Ruess W. Duality and geometry of spaces of compact operators, in Functional Analysis: Surveys and Recent Results III, Math. Studies 90, North Holland, 1984. | MR 0761373 | Zbl 0573.46007

Saphar P.D. Projections from $L(E,F)$ onto $K(E,F)$, Proc. Amer. Math. Soc. 127 (4) (1999), 1127-1131. (1999) | MR 1473679 | Zbl 0912.46011

Singer I. Bases in Banach spaces. Vol. II, Berlin-Heidelberg-New York, Springer, 1981. | MR 0610799

Thorp E. Projections onto the space of compact operators, Pacific J. Math. 10 (1960), 693-696. (1960) | MR 0114128

Tong A.E. On the existence of non-compact bounded linear operators between certain Banach spaces, Israel J. Math. 10 (1971), 451-456. (1971) | MR 0296663

Tong A.E.; Wilken D.R. The uncomplemented subspace $K(E,F)$, Studia Math. 37 (1971), 227-236. (1971) | MR 0300058 | Zbl 0212.46302

Zippin M. Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), 371-379. (1988) | MR 0965758 | Zbl 0706.46015