自伴标准算子代数上强保持k-斜交换性映射的刻画

令H是维数大于2的复Hilbert空间，A是H上自伴标准算子代数.对于给定的正整数k≥1，H上算子A与B的k-斜交换子递推地定义为_*A，B]_k=_*A，_*A，B]_k-1]，其中_*A，B]₀=B，_*A，B]₁=AB-BA^*.设k≥4，φ是A上的值域包含所有一秩投影的映射.本文证明了φ满足_*φ（A），φ（B）]_k=_*A，B]_k对任意A，B∈A都成立的充分必要条件是φ（A）=A对任意A∈A都成立，或φ（A）=-A对任意A∈A都成立.当k是偶数时后一情形不出现.

Strong k-skew Commutativity Preserving Maps on Self-Adjoint Standard Operator Algebras

Let H be a complex Hilbert space with dim H>2 and A be a selfadjoint standard operator algebra on H.For given positive integer k≥1, the k-skew commutator of operators A and B on H is defined as _*A, B]_k=_*A, _*A, B]_k-1], where _*A, B]₀=B, _*A, B]₁=AB-BA^*.Assume k≥4 and Φ is a map on A with range containing all rank one projections.It is shown that, _*Φ(A), Φ(B)]_k=_*A, B]_k holds for all A, B∈A if and only if Φ(A)=A for all A∈A, or Φ(A)=-A for all A∈A.The latter case does not occur if k is even.