-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

DOI:
https://doi.org/10.1090/S0002-9939-04-07769-X

Published electronically:
October 7, 2004

MathSciNet review:
2113941

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the canonical framework of white noise analysis over the Gel'fand triple and be the space of continuous linear operators from to . Let be a self-adjoint operator in with spectral representation . In this paper, it is proved that under appropriate conditions upon , there exists a unique linear mapping such that for each . The mapping is then naturally used to define as , where is the Dirac -function. Finally, properties of the mapping are investigated and several results are obtained.

**1.**L. Accardi, Y.G. Lu and I.V. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, Berlin, 2002. MR**1925437 (2003h:81116)****2.**T. Hida, H. H. Kuo, J. Potthoff and L. Streit, White Noise-An Infinite Dimensional Calculus, Kluwer Academic, Dordrecht, 1993. MR**1244577 (95f:60046)****3.**Z. Y. Huang, Quantum white noises--white noise approach to quantum stochastic calculus, Nagoya Math. J. 129 (1993) 23-42. MR**1210001 (94e:81153)****4.**Z. Y. Huang, C. S. Wang and X. J. Wang, Quantum cable equations in terms of generalized operators, Acta Appl. Math. 63 (2000) 151-164. MR**1831253 (2002b:81071)****5.**Z. Y. Huang, J. A. Yan, Introduction to Infinite Dimensional Calculus, Kluwer, Dordrecht, 1997.**6.**R.L. Hudson, K. R. Parthasarathy, Quantum Itô's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984) 301-323. MR**0745686 (86e:46057)****7.**H. H. Kuo, White Noise Distribution Theory, CRC, Boca Raton, 1996. MR**1387829 (97m:60056)****8.**S. L. Luo, Wick algebra of generalized operators involving quantum white noise, J. Operator Theory 38 (1997) 367-378. MR**1606956 (99b:47063)****9.**N. Obata, White Noise Calculus and Fock Space, Springer-Verlag, Berlin, 1994. MR**1301775 (96e:60061)****10.**K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992. MR**1164866 (93g:81062)****11.**J. Potthoff, L. Streit, A characterization of Hida distributions, J. Funct. Anal. 101 (1991) 212-229. MR**1132316 (93a:46078)****12.**C.S. Wang and Z.Y. Huang, A filtration of Wick algebra and its application to Quantum SDE's, Acta Math. Sinica, English Series (in press).**13.**C.S. Wang, Z.Y. Huang and X. J. Wang, Analytic characterization for Hilbert-Schmidt operators on Fock space, preprint.**14.**C.S. Wang, Z.Y. Huang and X. J. Wang, A -transform-based criterion for the existence of bounded extensions of -operators, preprint.**15.**J. A. Yan, Products and transforms of white-noise functionals (in general setting), Appl. Math. Optim., 31 (1995), 137-153. MR**1309303 (95m:60096)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
60H40

Retrieve articles in all journals with MSC (2000): 60H40

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:
https://doi.org/10.1090/S0002-9939-04-07769-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.