三角代数上的n阶导子系

Derivation systems of order n on triangular algebra

设U=Tri(A,M,B)是三角代数,Dn={δ0,δ1,…,δn}为U上的一组可加映射且δ0=I.若A,B∈U有δm(AB)=∑mk=0Cmkδk(A)δm-k(B)(m=0,1,2,…,n),则称Dn为U上的一个n阶导子系,若A∈U有δm(A2)=∑mk=0Cmkδk(A)δm-k(A)(m=0,1,2,…,n),则称Dn为U上的一个n阶Jordan导子系.利用算子论的方法讨论了三角代数上的n阶导子系,证明了三角代数上的每个n阶Jordan导子系都是n阶导子系.

Let U=Tri（A,M,B） be a triangular algebra and Dn={δk}nk=0 be a system of additive mappings on U with δ0=I.If δm（AB）=∑k=0^mCm^kδk（A）δm-k（B）（m=0,1,2,…,n）,arbitary A,B∈U, then Dn is said to be a derivation system of order n on U.If δm（A^2）=∑k=0^mCm^kδk（A）δm-k（A）（m=0,1,2,…,n）,arbitary A∈U, then Dn is said to be a Jordan derivation system of order n on U.By using operator theory methods,it is proved that every Jordan derivation system of order n on a triangle algebra U is a derivation system of order n on a triangle algebra U.