function of an operator: A white noise approach
Authors:
Caishi Wang, Zhiyuan Huang and Xiangjun Wang
Journal:
Proc. Amer. Math. Soc. 133 (2005), 891898
MSC (2000):
Primary 60H40
Published electronically:
October 7, 2004
MathSciNet review:
2113941
Fulltext PDF Free Access
Abstract 
References 
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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 selfadjoint 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.
Luigi
Accardi, Yun
Gang Lu, and Igor
Volovich, Quantum theory and its stochastic limit,
SpringerVerlag, Berlin, 2002. MR 1925437
(2003h:81116)
 2.
Takeyuki
Hida, HuiHsiung
Kuo, Jürgen
Potthoff, and Ludwig
Streit, White noise, Mathematics and its Applications,
vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. An
infinitedimensional calculus. MR 1244577
(95f:60046)
 3.
Zhi
Yuan Huang, Quantum white noises—white noise approach to
quantum stochastic calculus, Nagoya Math. J. 129
(1993), 23–42. MR 1210001
(94e:81153)
 4.
Zhiyuan
Huang, Caishi
Wang, and Xiangjun
Wang, Quantum cable equations in terms of generalized
operators, Acta Appl. Math. 63 (2000), no. 13,
151–164. Recent developments in infinitedimensional analysis and
quantum probability. MR 1831253
(2002b:81071), http://dx.doi.org/10.1023/A:1010776120045
 5.
Z. Y. Huang, J. A. Yan, Introduction to Infinite Dimensional Calculus, Kluwer, Dordrecht, 1997.
 6.
R.
L. Hudson and K.
R. Parthasarathy, Quantum Ito’s formula and stochastic
evolutions, Comm. Math. Phys. 93 (1984), no. 3,
301–323. MR
745686 (86e:46057)
 7.
HuiHsiung
Kuo, White noise distribution theory, Probability and
Stochastics Series, CRC Press, Boca Raton, FL, 1996. MR 1387829
(97m:60056)
 8.
Shunlong
Luo, Wick algebra of generalized operators involving quantum white
noise, J. Operator Theory 38 (1997), no. 2,
367–378. MR 1606956
(99b:47063)
 9.
Nobuaki
Obata, White noise calculus and Fock space, Lecture Notes in
Mathematics, vol. 1577, SpringerVerlag, Berlin, 1994. MR 1301775
(96e:60061)
 10.
K.
R. Parthasarathy, An introduction to quantum stochastic
calculus, Monographs in Mathematics, vol. 85, Birkhäuser
Verlag, Basel, 1992. MR 1164866
(93g:81062)
 11.
J.
Potthoff and L.
Streit, A characterization of Hida distributions, J. Funct.
Anal. 101 (1991), no. 1, 212–229. MR 1132316
(93a:46078), http://dx.doi.org/10.1016/00221236(91)90156Y
 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 HilbertSchmidt operators on Fock space, preprint.
 14.
C.S. Wang, Z.Y. Huang and X. J. Wang, A transformbased criterion for the existence of bounded extensions of operators, preprint.
 15.
J.
A. Yan, Products and transforms of whitenoise functionals (in
general setting), Appl. Math. Optim. 31 (1995),
no. 2, 137–153. MR 1309303
(95m:60096), http://dx.doi.org/10.1007/BF01182785
 1.
 L. Accardi, Y.G. Lu and I.V. Volovich, Quantum Theory and Its Stochastic Limit, SpringerVerlag, Berlin, 2002. MR 1925437 (2003h:81116)
 2.
 T. Hida, H. H. Kuo, J. Potthoff and L. Streit, White NoiseAn Infinite Dimensional Calculus, Kluwer Academic, Dordrecht, 1993. MR 1244577 (95f:60046)
 3.
 Z. Y. Huang, Quantum white noiseswhite noise approach to quantum stochastic calculus, Nagoya Math. J. 129 (1993) 2342. 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) 151164. 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) 301323. 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) 367378. MR 1606956 (99b:47063)
 9.
 N. Obata, White Noise Calculus and Fock Space, SpringerVerlag, 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) 212229. 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 HilbertSchmidt operators on Fock space, preprint.
 14.
 C.S. Wang, Z.Y. Huang and X. J. Wang, A transformbased criterion for the existence of bounded extensions of operators, preprint.
 15.
 J. A. Yan, Products and transforms of whitenoise functionals (in general setting), Appl. Math. Optim., 31 (1995), 137153. MR 1309303 (95m:60096)
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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/S000299390407769X
PII:
S 00029939(04)07769X
Keywords:
White noise analysis,
selfadjoint 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.
