反射正性与Hankel算子--无重数情形
Reflection positivity and Hankel operators -- the multiplicity free case
摘要：
我们分析反射正表示的积极汉克尔算子。这是由于正汉克尔算子是用它们的Carleson测度来描述的，而表示和反射正希尔伯特空间之间的相容条件是相当复杂的。这导致我们的概念的汉克尔正表示的三元组$(G,S,τ)$，其中$G$是一个群体，$τ$对合自同构的$G$和$S \subseteq G$一个子半群与$τ(S) = S^{-1}$。对于对应于反射正算子的三元组$(\mathbb Z,\mathbb N,-id_{\mathbb Z})$和对应于反射正单参数群的三元组$(\mathbb R,\mathbb R_+,-id_{\mathbb R})$，我们证明了每一个Hankel正表示都可以通过标量积的微小变化而成为反射正的.一个关键的方法在于使用由$H^2(\mathbb C_+)$上的正汉克尔算子$H$定义的$\mathbb R_+$上的测度$μ_H$来定义Pick函数，该函数的虚部被限制在虚轴上，为$H$提供算子符号。
英文摘要：
We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,τ)$, where $G$ is a group, $τ$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $τ(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method consists in using the measure $μ_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.
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