Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras
Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras
复制标题
白噪声平方和有限差分代数的希尔伯特模实现
DOI：
10.1023/a:1026644229489
复制
发表时间：
2000
影响因子：
0.6
通讯作者：
Michael Skeide
中科院分区：
文献类型：
--
作者：
L. Accardi;Michael Skeide
本文基于Hilbert模的构造方法，给出了白色噪声平方代数的表示理论。我们找到了唯一的Fock表示，并证明了表示空间是通常的对称Fock空间。虽然我们从一个自由度开始，但最终会有无数个自由度。令人惊讶的是，我们的代表原来有一个密切的关系Feinsilver的有限差分代数。实际上，在白色噪声的平方代数中存在有限差分代数的全纯像。我们的表示限于这个图像是有限差分Fock空间上的Boukas表示。因此，我们将Boukas表示扩展到一个更大的代数，它是由创造者，零化子和数运算符生成的。
We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.