单位元算子的共轭轨道
The conjugate orbit of a unitary operator
AI总结：
本文研究了单位元算子的共轭轨道，探讨了其性质及与经典双边移位算子、单位元乘法算子等的关系。
摘要：
本文讨论了单位元算子$CUC$的各个方面，其中$U$是固定在复Hilbert空间$\mathcal{H}$上的单位元算子，而$C$则遍历$\mathcal{H}$上的所有共轭算子（反线性、等距、自轭）。我们将此类单位元算子称为$U$的共轭轨道，并记为$\mathfrak{O}_c(U)$。我们看到$U^{*}$，即$U$的Hilbert空间伴随算子，总是属于$\mathfrak{O}_c(U)$，而$U$仅当$U$与$U^{*}$单位等价时才属于$\mathfrak{O}_c(U)$，这使$U$成为$\mathfrak{O}_c(U)$的成员成为罕见事件。我们完整地描述了经典双边移位算子的共轭轨道，并讨论了经典单位圆上的Lebesgue空间上的单位元乘法算子何时属于此共轭轨道。我们还将此讨论扩展到具有更高乘数的双边移位算子，通过单位等价性，使它们与其它有趣的单位元算子如实直线上的翻译和扩张算子在Lebesgue空间上建立联系。有限单位元矩阵为我们提供了丰富的共轭单位元轨道讨论实例。特别是，我们确定哪些对角矩阵，如果有的话，属于固定单位元矩阵的共轭轨道。与有限单位元矩阵密切相关的是相对于某些，可能是无限的，正交基的可对角化单位元算子。我们给出了一大类这些单位元算子的变体，它们属于共轭轨道，并建立与经典傅里叶-普朗歇尔和Hilbert变换的联系。最后，我们开发了一个使用实Hilbert空间的单位元算子模型，并用它来描述共轭轨道，以及以另一种视角回顾我们之前的讨论。
英文摘要：
This paper discusses various aspects of the collection of unitary operators $CUC$, where $U$ is a fixed unitary operator on a complex Hilbert space $\mathcal{H}$ and $C$ varies over the set of all conjugations on $\mathcal{H}$ (antilinear, isometric, involutions). We call this class of unitary operators, the {\em conjugate orbit }of $U$ and denote it by $\mathfrak{O}_c(U)$. We will see that $U^{*}$, the Hilbert space adjoint of $U$, always belongs to $\mathfrak{O}_c(U)$, while $U$ belongs to $\mathfrak{O}_c(U)$ only when $U$ is unitarily equivalent to $U^{*}$, making $U$ a member of $\mathfrak{O}_c(U)$ an uncommon event. We completely describe the conjugate orbit of the classical bilateral shift and discuss when a unitary multiplication operator on the classical Lebesgue space of the unit circle belongs to this conjugate orbit. We also broaden this discussion to include the bilateral shifts of higher multiplicity which, via unitary equivalence, makes connections to other interesting unitary operators such as the translation and dilation operators on the Lebesgue space of the real line. Finite unitary matrices provide us with a rich source of examples of conjugate unitary orbits to discuss. In particular, we determine which diagonal matrices, if any, belong to the conjugate orbit of a fixed unitary matrix. Closely related to the finite unitary matrices are the diagonalizable unitary operators with respect to some, possibly infinite, orthonormal basis. We give a large class of variations of these unitary operators that belong to the conjugate orbit and establish a connection to the classical Fourier--Plancherel and Hilbert transforms. Finally, we develop a model for a unitary operator using real Hilbert spaces and use it to describe the conjugate orbit as well as revisit some of our previous discussions in another light.
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