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自伴标准算子代数上强保持k-斜交换性映射的刻画
Strong k-skew Commutativity Preserving Maps on Self-Adjoint Standard Operator Algebras
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.
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.
Hilbert空间 / k-斜交换子 / 自伴标准算子代数 / 保持映射 {{custom_keyword}} /
Hilbert spaces / k-skew commutators / self-adjoint standard operator algebras / preservers {{custom_keyword}} /
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国家自然科学基金资助项目（11271217，11671294）
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