APM  Vol.2 No.3 , May 2012
Fock Spaces for the q-Dunkl Kernel
Abstract: In this work, we introduce a class of Hilbert spaces of entire functions on the disk , 0<q<1 , with reproducing kernel given by the q-Dunkl kernel . The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Q by z and the q-Dunkl operator on the Fock space ; and we prove that these operators are adjoint-operators and continuous from this space into itself.
Cite this paper: F. Soltani, "Fock Spaces for the q-Dunkl Kernel," Advances in Pure Mathematics, Vol. 2 No. 3, 2012, pp. 169-176. doi: 10.4236/apm.2012.23023.

[1]   C. A. Berger and L. A. Coburn, “Toeplitz Operators on the Segal-Bargmann Space,” Transactions of the Ameri- can Mathematical Society, Vol. 301, No. 2, 1987, pp. 813- 829. doi:10.1090/S0002-9947-1987-0882716-4

[2]   V. Bargmann, “On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Part I,” Communications on Pure and Applied Mathematics, Vol. 14, No. 3, 1961, pp. 187-214. doi:10.1002/cpa.3160140303

[3]   M. Sifi and F. Soltani, “Generalized Fock Spaces and Weyl Relations for the Dunkl Kernel on the Real Line,” Journal of Mathematical Analysis and Applications, Vol. 270, No. 1, 2002, pp. 92-106. doi:10.1016/S0022-247X(02)00052-5

[4]   F. M. Cholewinski, “Generalized Fock Spaces and Associated Operators,” SIAM Journal of Mathematical Analysis, Vol. 15, No. 1, 1984, pp. 177-202. doi:10.1137/0515015

[5]   G. H. Jackson, “On a q-Definite Integrals,” The Quarterly Journal of Pure and Applied Mathematics, Vol. 41, No. 2, 1910, pp. 193-203.

[6]   T. H. Koornwinder, “Special Functions and q-Commuting Variables,” Fields Institute Communications, Vol. 14, 1997, pp. 131-166.

[7]   A. Fitouhi, M. M. Hamza and F. Bouzeffour, “The q-jα Bessel Function,” Journal of Approximation Theory, Vol. 115, No. 1, 2002, pp. 144-166. doi:10.1006/jath.2001.3645

[8]   F. Soltani, “Multiplication and Translation Operators on the Fock Spaces for the q-Modified Bessel Function,” The Advances in Pure Mathematics (APM), Vol. 1, No. 2, 2011, pp. 221-227. doi:10.4236/apm.2011.14039

[9]   J. J. Betancor, M. Sifi and K. Trimèche, “Hypercyclic and Chaotic Convolution Operators Associated with the Dunkl Operator on ,” Acta Mathematica Hungarica, Vol. 106, No. 1-2, 2005, pp. 101-116. doi:10.1007/s10474-005-0009-1