Classification of Ideals in Banach Spaces

Let an operator T belong to an operator ideal J, then for any operators A and B which can be composed with T asBTA then BTA \(\in\) J. Indeed, J contains the class of finite rank Banach Space operators. Now given L(X; Y ). Then J(X; Y ) \(\subseteq\) L(X; Y ) such that J(X; Y ) = {T : X \(\gets\) Y : T \(\subseteq\) }. Thus an operator ideal is a subclass J of L containing every identity operator acting on a one-dimensional Banach space such that: S + T \(\in\) J(X; Y ) where S; T \(\in\) J(X; Y ). If W;Z;X; Y \(\in\) K;A \(\in\) L(W;X);B \(\in\) L(Y;Z) then BTA \(\in\) J(W;Z) whenever T \(\in\) J(X; These properties compare very well with the algebraic notion of ideals in Banach Algebras within whose classes lie compact operators, weakly compact operators, finitely strictly regular operators, completely continuous operators, strictly singular operators among others. Thus, the aim of this paper is to characterize the various classes of ideals in Banach spaces. Special attention is given to the characteristics involving the ideal properties, the metric approximation properties, the hereditary properties in relation to the ideal extensions in the Hahn-Banach space, projection and embedments in the biduals of the Banach Space.