We show that as soon as $c_0$ embeds complementably into the space of all weakly compact operators from $X$ to $Y$, then it must live either in $X^\ast$ or in $Y$.
@article{118664, author = {Giovanni Emmanuele}, title = {On complemented copies of $c\_0$ in spaces of operators, II}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {35}, year = {1994}, pages = {259-261}, zbl = {0862.46010}, mrnumber = {1286572}, language = {en}, url = {http://dml.mathdoc.fr/item/118664} }
Emmanuele, Giovanni. On complemented copies of $c_0$ in spaces of operators, II. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 259-261. http://gdmltest.u-ga.fr/item/118664/
On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. (1958) | MR 0115069 | Zbl 0084.09805
Sequences and Series in Banach Spaces, Graduate Text in Mathematics 97, Springer Verlag 1984. | MR 0737004
Dominated operators on $C[0,1]$ and the $(CRP)$, Collect. Math. 41 (1990), 21-25. (1990) | MR 1134442 | Zbl 0752.47006
Remarks on the uncomplemented subspace $W(E,F)$, J. Funct. Analysis 99 (1991), 125-130. (1991) | MR 1120917 | Zbl 0769.46006
On the containment of $c_0$ by spaces of compact operators, Bull. Sci. Math. 115 (1991), 177-184. (1991) | MR 1101022
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
On complemented copies of $c_0$ in spaces of operators, Comm. Math. 32 (1992), 29-32. (1992) | MR 1202755
About the position of $K_{w^\ast}(X^\ast,Y)$ in $L_{w^\ast}(X^\ast,Y)$, Atti Sem. Mat. Fisico Univ. Modena, to appear. | MR 1282327
Uncomplementability of spaces of compact operators in larger spaces of operators, to appear. | MR 1435603 | Zbl 0903.46006
On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math. 34 (1980), 196-205. (1980) | MR 0575060 | Zbl 0411.46009
Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151 (1970), 563-579. (1970) | MR 0279564 | Zbl 0216.16203
Spaces of compact operators, Math. Annalen 208 (1974), 267-278. (1974) | MR 0341154 | Zbl 0266.47038