B(X)上的(m,n)-Lie中心化子

(m,n)-Lie Centralizers on B(X)

设X是实数域或复数域F上维数大于1的Banach空间,Ф:B(X)→B(X)是一个可加映射。证明了如果存在正整数m,n使得(m+n)Ф([A,B])=m[Ф(A),B]+n[A,Ф(B)]对任意A,B∈B(X)且AB=P(其中P∈B(X)是一个固定的非平凡幂等元)成立,则存在λ∈F及在AB=P的换位子上为零的可加映射h:B(X)→F使得对任意A∈B(X),有Ф(A)=λA+h(A)I.

Let X be a Banach Space over the real or complex number field F with dim X>1 andϕ:B(X)→B(X)be an additive map.This paper prove that if there are positive integers m,n such that(m+n)ϕ([A,B])=m[ϕ(A),B]+n[A,φ(B)]holds for all A,B∈B(X)and AB=P(where P is a fixed nontrivial idempotent),then there isλ∈F and an additive map h:B(X)→F vanishing at commutators[A,B]with AB=P,such thatϕ(A)=λA+h(A)I for all A∈B(X).