Non-complemented Spaces of Operators, Vector Measures, andco

Abstract

AbstractThe Banach spacesL(X,Y),K(X,Y),Lw*(X*,Y), andKw*(X*,Y) are studied to determine when they contain the classical Banach spacescoor ℓ∞. The complementation of the Banach spaceK(X,Y) inL(X,Y) is discussed as well as what impact this complementation has on the embedding ofcoor ℓ∞inK(X,Y) orL(X,Y). Results of Kalton, Feder, and Emmanuele concerning the complementation ofK(X,Y) inL(X,Y) are generalized. Results concerning the complementation of the Banach spaceKw*(X*,Y) inLw*(X*,Y) are also explored as well as how that complementation affects the embedding ofcoor ℓ∞inKw*(X*,Y) orLw*(X*,Y). The ℓpspaces for 1 =p< ∞ are studied to determine when the space of compact operators from one ℓpspace to another containsco. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.