Equivalent characterization of centralizers on B(H)

Abstract

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 Φ: B(H) → B(H) satisfies Φ(AB) = Φ(A)B = AΦ(B) for any A,B ∈ B(H) with AB = Z if and only if Φ(AB) = Φ(A)B = AΦ(B), ∀A,B ∈ B(H), that is, Φ is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Φ(A2) = AΦ(A) = Φ(A)A for any A ∈ B(H) with A2 = 0 if and only if Φ(A) = AΦ(I) = Φ(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.