Complemented Subspaces of Linear Bounded Operators

Bahreini, Manijeh ; Bator, Elizabeth ; Ghenciu, Ioana

Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 449-461

Résumé

We study the complementation of the space $W\left( X,Y \right)$ of weakly compact operators, the space $K\left( X,Y \right)$ of compact operators, the space $U\left( X,Y \right)$ of unconditionally converging operators, and the space $CC\left( X,Y \right)$ of completely continuous operators in the space $L\left( X,Y \right)$ of bounded linear operators from $X$ to $Y$ . Feder proved that if $X$ is infinite-dimensional and ${{c}_{0}}\,\to \,Y$ , then $K\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$ . Emmanuele and John showed that if ${{c}_{0}}\,\to \,K(X,\,Y)$ , then $K\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$ . Bator and Lewis showed that if $X$ is not a Grothendieck space and ${{c}_{0}}\,\to \,Y$ , then $W\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$ . In this paper, classical results of Kalton and separably determined operator ideals with property $\left( * \right)$ are used to obtain complementation results that yield these theorems as corollaries.

Détail

DOI : 10.4153/CMB-2011-097-2

Mots-clés : 46B20, 46B28, spaces of operators, complemented subspaces, compact operators, weakly compact operators, completely continuous operators