量子“对角测度”的纠缠遍历定理和一个遍历定理
The entangled ergodic theorem and an ergodic theorem for quantum "diagonal measures"
摘要：
设$U$是作用在希尔伯特空间$H$上的酉算子，$\a:\{1,..., 2k\}\mapsto\{1,..., k\}$是对--划分，最后是$A_{1},...,A_{2k-1}\in B(H)$。证明了当$H$由$U$的特征向量生成时，即当$H$上的酉$U$诱导的动力学是概周期时，遍历平均$$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$在强算子拓扑中收敛.这一结果相对于纠缠遍历定理改进了已知的结果。我们还证明了H的遍历结果的非交换形式。Furstenberg相对于对角测度。这意味着，${\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} U^{n}AU^{n}}$收敛于强算子拓扑的其他有趣的情况下，所涉及的酉算子不产生几乎周期动力学，和操作$A$是非紧的。
英文摘要：
Let $U$ be a unitary operator acting on the Hilbert space $H$, $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair--partition, and finally $A_{1},...,A_{2k-1}\in B(H)$. We show that the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$ converges in the strong operator topology when $H$ is generated by the eigenvectors of $U$, that is when the dynamics induced by the unitary $U$ on $H$ is almost periodic. This result improves the known ones relative to the entangled ergodic theorem. We also prove the noncommutative version of the ergodic result of H. Furstenberg relative to diagonal measures. This implies that ${\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} U^{n}AU^{n}}$ converges in the strong operator topology for other interesting situations where the involved unitary operator does not generate an almost periodic dynamics, and the operator $A$ is noncompact.
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