Complemented subspaces of bounded linear operators

For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y) be a Banach space of all bounded linear operator between Banach spaces X and Y, K(X, Y) be the space of all compact operators, and W(X, Y) be the space of all weakly compact operators. We denote space all operator ideals by O.; Work by Bator, Lewis, and Kalton on the space of compact and weakly compact operators motivates much of this paper. Conditions that make K( X, Y) and W(X, Y) complemented in L(X, Y) are generalized to the space of operator ideals. Also an extension will be given from the previous results that yield copies of c_o as well as l ∞ in L(X, Y) to the space O(X, Y). For instance, if X is an infinite-dimensional space and c_o embeds in L(X, Y), then l ∞ embeds in L(X, Y). This notion is generalized to the case when X is separable, Y contain a copy of c_o, and O(X, Y) is complemented in L( X, Y). Then l∞ embeds isomorphically in O(X, Y) if and only if c _o embeds isomorphically in O(X, Y ).; The results concerning operator ideals are applied to the study of CC(X, Y), the Banach space of completely continuous operators. For example, suppose O is a separably determined closed operator ideal that has property (*). Then O( X, c_o) = L(X, c_o) if and only if O(X, c_o) is complemented in L(X, c_o). It is immediate that if X be a Banach space, then CC( X, c_o) = L(X, c_o) if and only if CC(X c_o) is complemented in L(X, c_o).