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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$\delta$-function of an operator: A white noise approach


Authors: Caishi Wang, Zhiyuan Huang and Xiangjun Wang
Journal: Proc. Amer. Math. Soc. 133 (2005), 891-898
MSC (2000): Primary 60H40
Published electronically: October 7, 2004
MathSciNet review: 2113941
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Abstract: Let $(E) \subset (L^2) \subset (E)^*$ be the canonical framework of white noise analysis over the Gel'fand triple $S({\mathbb R}) \subset L^2({\mathbb R}) \subset S^*({\mathbb R})$ and ${\mathcal L} \equiv {\mathcal L}[(E),(E)^*]$ be the space of continuous linear operators from $(E)$ to $(E)^*$. Let $Q$ be a self-adjoint operator in $(L^2)$with spectral representation $Q = \int_{\mathbb R}\lambda\,P_Q(d\lambda)$. In this paper, it is proved that under appropriate conditions upon $Q$, there exists a unique linear mapping $Z:S^*({\mathbb R}) \longmapsto {\mathcal L}$ such that $Z(f)=\int_{\mathbb R}f(\lambda)\,P_Q(d\lambda)$ for each $f \in S({\mathbb R})$. The mapping is then naturally used to define $\delta(Q)$ as $Z(\delta)$, where $\delta$ is the Dirac $\delta$-function. Finally, properties of the mapping $Z$ are investigated and several results are obtained.


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Additional Information

Caishi Wang
Affiliation: Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China
Email: wangcs@nwnu.edu.cn

Zhiyuan Huang
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China
Email: zyhuang@hust.edu.cn

Xiangjun Wang
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China
Email: x.j.wang@yeah.net

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07769-X
PII: S 0002-9939(04)07769-X
Keywords: White noise analysis, self-adjoint operator, Schwartz generalized function
Received by editor(s): December 10, 2002
Received by editor(s) in revised form: September 16, 2003
Published electronically: October 7, 2004
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.