算子代数上(m,n)-Jordan导子的刻画

Characterizations of (m,n)-Jordan Derivations on Operator Algebras

设R是一个环,M是一个R-双边模,m和n是两个非负整数满足m+n≠0,如果δ是一个从R到M的可加映射满足对任意A∈R,(m+n)δ(A^2)=2mAδ(A)+2nδ(A)A,则称δ是一个(m,n)-Jordan导子.本文证明了,如果R是一个单位环,M是一个单位R-双边模含有一个由R中幂等元代数生成的左(右)分离集,那么,当m,n>0且m≠n时,每一个从R到M的(m,n)-Jordan导子恒等于零.还证明了,如果A和B是两个单位环,M是一个忠实的单位(A,B)-双边模(N是一个忠实的单位(B,A)-双边模),m,n>0且m≠n,U=[A N M B]是一个|mn(m-n)(m+n)|-无挠的广义矩阵环,那么每一个从U到自身的(m,n)-Jordan导子恒等于零.

Let R be a ring, U be an R-bimodule, m and n be two fixed nonnegative integers with m ＋ n ≠ 0. If an additive mapping 5 from .R into U satisfies （m ＋ n）5（A2） = 2mAS（A） ＋ 2nS（A）A for every A in R, then 5 is called an （m, n）-Jordan derivation. In this paper, we prove that if .R is a unital ring and .U is a unital R- bimodule with a left （right） separating set generated algebraically by all idempotents in R, then every （m, n）-Jordan derivation from R into U is identical with zero whenever m,n 〉 0 and m ≠n; We also show that if A and B be two unitalrings, Uis a faithful unital （A,B）-bimodule N is a faithful unital （B, A）-bimodule）, m, n 〉 0 and m ≠ n, U= [A,U,N ,B] is a ｜mn（m- n）（m ＋ n）｜-torsion-free generalized matrix ring, then every （m, n）-Jordan derivation from into itself is equal to zero.