Abstract:

Let I(X) be the set of all idempotent operators on a complex Banach space X.  The purpose of this note is to give  the concrete form of either that every surjective map φ on I(X) such that AB is a nonzero idempotent if and only if φ(A)φ(B) is for all A,B∈I(X) or that every surjective map φ on I(X) such that ABA is a nonzero idempotent if and only if φ(A)φ(B)φ(A) is for all A,B∈I(X), when the dimension of X is at least 3.

Key words: preserver map; idempotent operator; products of idempotent operators