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