管域上全纯函数的希尔伯特空间的标准子空间
Standard subspaces of Hilbert spaces of holomorphic functions on tube domains
摘要：
本文研究了管域E + i C ^0上向量值全纯函数的希尔伯特空间的标准子空间，其中C \subeq E是在e^{Rh}下对某个自同态\in \End（E）的点生成锥不变量，可对角化，特征值为1，0，-1（推广了Lorentz boost）.该数据指定了一个楔形域W（E，C，h）\subeq E和我们的主要结果之一，表现出相应的标准子空间作为使用这些域上的检验函数生成。我们还研究了三元组（E，C，e^{πi h}）的反射正性和E上分布的支撑性质，这些性质是定义希尔伯特空间H的算子值测度的傅里叶变换。对于这些分布的虚部，我们发现相似之处，众所周知的惠更斯原理，有关楔形对偶的闵可夫斯基的背景下。有趣的例子是与欧几里德约当代数相关的Riesz分布。
英文摘要：
In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C^0, where C \subeq E is a pointed generating cone invariant under e^{R h} for some endomorphism h \in \End(E), diagonalizable with the eigenvalues 1,0,-1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E,C,h) \subeq E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E,C,e^{πi h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens' principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.
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