On the reducible M-ideals in Banach spaces

The object of the investigation is to study reducible M -ideals in Banach spaces. It is shown that if the number of M - ideals in a Banach space X is n( ∞), then the number of reducible M -ideals does not exceed of (n−2)(n−3)/2. Moreover, given a compact metric space X, we obtain a general form of a reducible M -ideal in the space C(X) of continuous functions on X. The intersection of two M -ideals is not necessarily reducible. We construct a sub- set of the set of all M -ideals in a Banach space X such that the intersection of any pair of it’s elements is reducible. Also, some Banach spaces X and Y for which K(X, Y ) is not a reducible M - ideal in L(X, Y ), are presented. Finally, a weak version of reducible M -ideal called semi reducible M -ideal is introduced.