论文标题
在两个正交投影中相关的自我辅助多项式的时刻
Relating moments of self-adjoint polynomials in two orthogonal projections
论文作者
论文摘要
给定两个正交投影$ \ {p，q \} $在非交换纹状体概率空间中，我们证明了$ p+q $的矩之间的关系，$ \ sqrt {-1}（-1}}（pq-qp）$和$ p+qPQ $和angle Angle Operator $ pqp $的矩之间。我们的证明纯粹是代数和枚举，并且不承担满足Voiculescu的Freeness财产或处于一般位置的$ P，Q $。就总和和换向器而言，从二项式类型的公式中获得的关系是由与$ p $和$ q $一起满足的二项式类型公式以及跟踪财产。在这方面，它们扩展了与自由统一操作员旋转两个投影之一或更普遍地通过自由统一布朗运动旋转的情况的情况。至于运算符$ p+qPQ $，我们得出了其矩的系数（双顺序）的自主复发关系，作为$ pqp $的线性组合，并明确确定其中的几个。这些关系是在对字母$ \ {p，qpq \} $中单词结构进行仔细分析后获得的。我们通过探索以前的结果与所谓的Kato双对双对的联系来结束论文。这样做会导致他们的时刻满足新身份。
Given two orthogonal projections $\{P,Q\}$ in a non commutative tracial probability space, we prove relations between the moments of $P+Q$, of $\sqrt{-1}(PQ-QP)$ and of $P+QPQ$ and those of the angle operator $PQP$. Our proofs are purely algebraic and enumerative and does not assume $P,Q$ satisfying Voiculescu's freeness property or being in general position. As far as the sum and the commutator are concerned, the obtained relations follow from binomial-type formulas satisfied by the orthogonal symmetries associated to $P$ and $Q$ together with the trace property. In this respect, they extend those corresponding to the cases where one of the two projections is rotated by a free Haar unitary operator or more generally by a free unitary Brownian motion. As to the operator $P+QPQ$, we derive autonomous recurrence relations for the coefficients (double sequence) of the expansion of its moments as linear combinations of those of $PQP$ and determine explicitly few of them. These relations are obtained after a careful analysis of the structure of words in the alphabet $\{P, QPQ\}$. We close the paper by exploring the connection of our previous results to the so-called Kato's dual pair. Doing so leads to new identities satisfied by their moments.
