First order differential operators in white noise analysis
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- by Dong Myung Chung and Tae Su Chung PDF
- Proc. Amer. Math. Soc. 126 (1998), 2369-2376 Request permission
Abstract:
Let $(E)$ be the space of test white noise functionals. We first introduce a family $\{\mdwhtdiamond _\gamma ;\gamma \in \mathbb {C}\}$ of products on $(E)$ including Wiener and Wick products, and then show that with each product $\mdwhtdiamond _\gamma$, we can associate a first order differential operator, called a first order $\gamma$-differential operator. We next show that a first order $\gamma$-differential operator is indeed a continuous derivation under the product $\mdwhtdiamond _\gamma$. We finally characterize $\gamma \Delta _G+N$ by means of rotation-invariance and continuous derivation under the product $\mdwhtdiamond _\gamma$. Here $\Delta _G$ and $N$ are the Gross Laplacian and the number operator on $(E)$, respectively.References
- D. M. Chung and T. S. Chung, Wick derivations on white noise functionals, J. Korean Math. Soc. 33(1996), No. 4, 993–1008.
- D. M. Chung, T. S. Chung and U. C. Ji, A characterization theorem for operators on white noise functionals, preprint.
- D. M. Chung and U. C. Ji, Transformation groups on white noise functionals and their applications, to appear in J. Appl. Math. Optim.
- S. W. He, J. G. Wang and R. Q. Yao, The characterizations of Laplacians in white noise analysis, Nagoya Math. J. 143(1996), 93–109.
- Takeyuki Hida, Hui-Hsiung Kuo, and Nobuaki Obata, Transformations for white noise functionals, J. Funct. Anal. 111 (1993), no. 2, 259–277. MR 1203453, DOI 10.1006/jfan.1993.1012
- Takeyuki Hida, Hui-Hsiung Kuo, Jürgen Potthoff, and Ludwig Streit, White noise, Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. An infinite-dimensional calculus. MR 1244577, DOI 10.1007/978-94-017-3680-0
- H.-H. Kuo, “White Noise Distribution Theory,” CRC Press, 1996.
- Nobuaki Obata, Rotation-invariant operators on white noise functionals, Math. Z. 210 (1992), no. 1, 69–89. MR 1161170, DOI 10.1007/BF02571783
- Nobuaki Obata, White noise calculus and Fock space, Lecture Notes in Mathematics, vol. 1577, Springer-Verlag, Berlin, 1994. MR 1301775, DOI 10.1007/BFb0073952
- Nobuaki Obata, Derivations on white noise functionals, Nagoya Math. J. 139 (1995), 21–36. MR 1355267, DOI 10.1017/S0027763000005286
- J. Potthoff and L. Streit, A characterization of Hida distributions, J. Funct. Anal. 101 (1991), no. 1, 212–229. MR 1132316, DOI 10.1016/0022-1236(91)90156-Y
Additional Information
- Dong Myung Chung
- Affiliation: Department of Mathematics, Sogang University Seoul, 121-742, Korea
- Email: dmchung@ccs.sogang.ac.kr
- Tae Su Chung
- Affiliation: Department of Mathematics, Sogang University Seoul, 121-742, Korea
- Received by editor(s): January 21, 1997
- Additional Notes: Research supported by KOSEF 996-0100-00102 and BSRI 97-1412.
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2369-2376
- MSC (1991): Primary 46F25
- DOI: https://doi.org/10.1090/S0002-9939-98-04323-8
- MathSciNet review: 1451792