Let H be a Hilbert space, dim(H) 3, and B(H) the algebra of all bounded linear operators on H. We study bijective linear operators : B(H) B(H) that maps preserves some fixed sets of elements. In particular we show that if preserves the set of all normal operators on H then there exist scalar c, a unitary operator U B(H) and a linear functional f on B(H) such that either (T)=c TU+f(T)I for all T B(H), or (T)=c U+f(T)I for all T B(H), where I is the identity operator. We also describe bijective linear operators that preserves idempotents. It turns out that such operators must be an automorphism or an antiautomorphism. This result is inspired by the following theorem of Herstein. Let R be a prime ring of characteristic not 2 and : R R be a bijective map such that ( )= for all x R. Then is either an automorphism or an antiautomorphism of R. |