# Article

Full entry | PDF   (0.2 MB)
Keywords:
compact operator; approximation property; reflexive Banach space; projection; separability
Summary:
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 $\lambda$.
References:
Amir D., Lindenstrauss J.: The structure of weakly compact sets in Banach spaces. Ann. of Math. 88 (1968), 35-59. MR 0228983 | Zbl 0164.14903
Arterburn D., Whitney R.: Projections in the space of bounded linear operators. Pacific J. Math. 15 (1965), 739-746. 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. 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. 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. 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. MR 1435603
Feder M.: On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 24 (1980), 196-205. 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. 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. MR 0955384 | Zbl 0631.46015
Godun B.V.: Unconditional bases and basic sequences. Izv. Vyssh. Uchebn. Zaved. Mat 24 (1980), 69-72. 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. MR 1258564
John K.: On the uncomplemented subspace ${\Cal K}(X,Y)$. Czechoslovak Math. J. 42 (1992), 167-173. MR 1152178
Johnson J.: Remarks on Banach spaces of compact operators. J. Funct. Analysis 32 (1979), 304-311. MR 0538857 | Zbl 0412.47024
Kuo T.H.: Projections in the space of bounded linear operators. Pacific. J. Math. 52 (1974), 475-480. MR 0352939
Kalton N.J.: Spaces of compact operators. Math. Ann. 208 (1974), 267-278. MR 0341154 | Zbl 0266.47038
Lindenstrauss J.: On nonseparable reflexive Banach spaces. Bull. Amer. Math. Soc. 72 (1966), 967-970. 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. 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. 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. MR 0296663
Tong A.E., Wilken D.R.: The uncomplemented subspace $K(E,F)$. Studia Math. 37 (1971), 227-236. MR 0300058 | Zbl 0212.46302
Zippin M.: Banach spaces with separable duals. Trans. Amer. Math. Soc. 310 (1988), 371-379. MR 0965758 | Zbl 0706.46015

Partner of