Holomorphic Functions and the Heat Kernel Measure on an Infinite Dimensional Complex Orthogonal Group
Holomorphic Functions and the Heat Kernel Measure on an Infinite Dimensional Complex Orthogonal Group
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无限维复正交群上的全纯函数和热核测度
DOI：
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发表时间：
1998
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通讯作者：
M. Gordina
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文献类型：
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作者：
M. Gordina
利用希尔伯特空间中的扩散，构造了无限维复群上的热核测度μt。然后证明了群上的全纯多项式关于热核测度是平方可积的。这些多项式的闭包H_L2(S·OH·S，μt)是我们所考虑的两个全纯函数空间之一。第二个空间H_L2(S O(∞))包含的函数在群的卡梅隆-马丁空间的模拟上是全纯的。证明了存在从第一个空间到第二个空间的等距，其主要定理是：无穷维泰勒展开式的无穷维非线性模拟定义了从HL2(S O(∞))到与无限维群的李代数相关的希尔伯特空间的等距.这是B.DRIVER和L.Gross关于复李群的一个无穷维等距的推广.本文的所有结果都是对一个具体群--Hilbert-Schmidt复正交群的表示，尽管我们的方法可以应用于更一般的情况.
The heat kernel measure μt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, H L2(S OH S, μt), is one of two spaces of holomorphic functions we consider. The second space, H L2(S O(∞)), consists of functions which are holomorphic on an analog of the Cameron–Martin subspace for the group. It is proved that there is an isometry from the first space to the second one.The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from H L2(S O(∞)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups.All the results of this paper are formulated for one concrete group, the Hilbert–Schmidt complex orthogonal group, though our methods can be applied in more general situations.