Complemented Subspaces of Linear Bounded Operators

Abstract

AbstractWe study the complementation of the spaceW(X,Y) of weakly compact operators, the spaceK(X,Y) of compact operators, the spaceU(X,Y) of unconditionally converging operators, and the spaceCC(X,Y) of completely continuous operators in the spaceL(X,Y) of bounded linear operators fromXtoY. Feder proved that ifXis infinite-dimensional andc0↪Y, thenK(X,Y) is uncomplemented inL(X,Y). Emmanuele and John showed that ifc0↪K(X,Y), thenK(X,Y) is uncomplemented inL(X,Y). Bator and Lewis showed that ifXis not a Grothendieck space andc0↪Y, thenW(X,Y) is uncomplemented inL(X,Y). In this paper, classical results of Kalton and separably determined operator ideals with property (∗) are used to obtain complementation results that yield these theorems as corollaries.