Equivalent characterization of centralizers on B(H)

Let H be a Hilbert space with dimH . 2 and Z . B(H) be an arbitrary but fixed operator. In this paper we show that an additive map .: B(H) . B(H) satisfies .(AB) = .(A)B = A.(B) for any A,B . B(H) with AB = Z if and only if .(AB) = .(A)B = A.(B), .A,B . B(H), that is, . is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that .(A 2) = A.(A) = .(A)A for any A . B(H) with A 2 = 0 if and only if .(A) = A.(I) = .(I)A, .A . B(H), and generalize main results in Linear Algebra and its Application, 450, 243.249 (2014) to infinite dimensional case. New equivalent characterization of centralizers on B(H) is obtained.