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On the uncomplemented subspace \(K(X,Y)\). (English) Zbl 0776.46016

Let \(X\) and \(Y\) be Banach spaces.It is a long-standing conjecture that either every operator from \(X\) to \(Y\) is compact or the subspace of compact operators is uncomplemented. It is proved in this paper that the above conjecture holds if \(K(X,Y)\) contains a copy of \(c_ 0\). [The same result has been obtained by G. Emmanuele, Math. Proc. Cambridge Philos. Soc. 111, 331-335 (1992)].
Reviewer: D.Werner (Berlin)

MSC:

46B28 Spaces of operators; tensor products; approximation properties
47L05 Linear spaces of operators
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References:

[1] D. Arterburn, R. Whitney: Projections in the space of bounded linenar operators. Pacific J. Math. 15 (1965), 739-746. · Zbl 0138.38602
[2] C. Bessaga, A. Pelczynski: On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958), 151-164. · Zbl 0084.09805
[3] J. Diestel, T. J. Morrison: The Radon-Nikodym property for the space of operators. Math. Nachrichten 92 (1979), 7-12. · Zbl 0444.46021
[4] J. Diestel: Sequences and series in Banach spaces. Graduate Texts in Mathematics 92, Springer, 1984.
[5] L. Drewnowski: An extension of a theorem of Rosenthal on operators acting from \(l_\infty (M)\). Studia Math. 57 (1976), 209-215. · Zbl 0351.46008
[6] G. Emmanuele: On the containment of \(c_0\) by spaces of compact operators. Bull. sci. mat. 115 (1991), 177-184. · Zbl 0749.46013
[7] M. Feder: On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 24 (1980), 196-205. · Zbl 0411.46009
[8] K. John: On the space \(K(P,P^*)\) of compact operators on Pisier space \(P\). submitted for publ. · Zbl 0802.46015
[9] J. Johnson: Remarks on Banach spaces of compact operators. J. funct. anal. 32 (1979), 304-311. · Zbl 0412.47024
[10] N. J. Kalton: Exhaustive operators and vector measures. Proc. Edinburgh, Math. Soc. 19 (1974), 291-300. · Zbl 0302.47020
[11] N. J. Kalton: Spaces of compact operators. Math. Ann. 208 (1974), 267-278. · Zbl 0266.47038
[12] P. Kissel, E. Schock: Lucid operators on Banach spaces. Comment. Math. Univ. Carolinae 31 (1990), 489-499. · Zbl 0731.47046
[13] T. H. Kuo: Projections in the space of bounded linear operators. Pacific J. Math. 52 (1974), 475-480. · Zbl 0287.47030
[14] J. Pisier: Counterexamples to a conjecture of Grothendieck. Acta Math. 151 (1983), 180-208. · Zbl 0542.46038
[15] H. P. Rosenthal: On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13-36. · Zbl 0227.46027
[16] W. Ruess: Duality and geometry of spaces of compact operators. Functional Analysis: Surveys and Recent Results III, North Holland, Amsterdam, 1984 (Mathematics Studies, 90), pp. 59-78.
[17] E. Thorp: Projections onto the space of compact operators. Pacific J. Math. 10 (1960), 693-696. · Zbl 0119.31904
[18] A. E. Tong: On the existence of non-compact bounded linear operators between certain Banach spaces. Israel J. Math. 10 (1971), 451-456. · Zbl 0247.47036
[19] A. E. Tong, D. R. Wilken: The uncomplemented subspace \(K(E,F)\). Studia Math. 37 (1971), 227-236. · Zbl 0212.46302
[20] H. O. Tylli: Weak compactness of multiplication operators on spaces of bounded linear operators. · Zbl 0760.47019
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