Summary: Let $B(X )$ be the algebra of all bounded linear operators on a complex Banach space X and let I * (X ) be the set of non-zero idempotent operators in $B(X )$. A surjective map $\varphi : B(X ) \rightarrow B(X )$ preserves nonzero idempotency of the Jordan products of two operators if for every pair A, B $\in B(X )$, the relation AB + BA $\in I$ * (X ) implies $\varphi (A)\varphi (B) + \varphi (B)\varphi (A) \in I$ * (X ). In this paper, the structures of linear surjective maps on $B(X )$ preserving the nonzero idempotency of Jordan products of two operators are given.

47B49

Banach space, preserver, idempotent, Jordan product