Toeplitz and Translation Operators on the q-Fock Spaces
Author(s)
Fethi Soltani
ABSTRACT
In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z); and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq ; and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .
In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z); and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq ; and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .
KEYWORDS
q-Fock Spaces, q-Exponential Function, q-Derivative Operator, q-Translation Operators, q-Toeplitz Operators, q-Weyl Commutation Relations
q-Fock Spaces, q-Exponential Function, q-Derivative Operator, q-Translation Operators, q-Toeplitz Operators, q-Weyl Commutation Relations
Cite this paper
nullF. Soltani, "Toeplitz and Translation Operators on the q-Fock Spaces," Advances in Pure Mathematics, Vol. 1 No. 6, 2011, pp. 325-333. doi: 10.4236/apm.2011.16059.
nullF. Soltani, "Toeplitz and Translation Operators on the q-Fock Spaces," Advances in Pure Mathematics, Vol. 1 No. 6, 2011, pp. 325-333. doi: 10.4236/apm.2011.16059.
References
[1] 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 [2] C. A. Berger and L. A. Coburn, “Toeplitz Operators on the Segal-Bargmann Space,” Transactions of the American Mathematical Society, Vol. 301, 1987, pp. 813-829. doi:10.1090/S0002-9947-1987-0882716-4 [3] F. M. Cholewinski “Generalized Fock Spaces and Associated Operators,” Society for Industrial and Applied Mathematics, Journal on Mathematical Analysis, Vol. 15, No. 1, 1984, pp. 177-202. doi:10.1137/0515015 [4] G. H. Jackson, “On a q-Definite Integrals,” Quarterly Journal of Pure and Applied Mathematics, Vol. 41, 1910, pp. 193-203. [5] T. H. Koornwinder, “Special Functions and q-Commuting Variables,” Fields Institute Communications, Vol. 14, 1997, pp. 131-166. [6] G. Andrews, R. Askey and R. Roy, “Special Functions,” Cambridge University Press, Cambridge, 1999. [7] M. Naimark, “Normed Rings,” Noordhoff, Groningen, 1959. [8] T. Hida, “Brownian Motion,” Springer-Verlag, 1980.
[1] 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 [2] C. A. Berger and L. A. Coburn, “Toeplitz Operators on the Segal-Bargmann Space,” Transactions of the American Mathematical Society, Vol. 301, 1987, pp. 813-829. doi:10.1090/S0002-9947-1987-0882716-4 [3] F. M. Cholewinski “Generalized Fock Spaces and Associated Operators,” Society for Industrial and Applied Mathematics, Journal on Mathematical Analysis, Vol. 15, No. 1, 1984, pp. 177-202. doi:10.1137/0515015 [4] G. H. Jackson, “On a q-Definite Integrals,” Quarterly Journal of Pure and Applied Mathematics, Vol. 41, 1910, pp. 193-203. [5] T. H. Koornwinder, “Special Functions and q-Commuting Variables,” Fields Institute Communications, Vol. 14, 1997, pp. 131-166. [6] G. Andrews, R. Askey and R. Roy, “Special Functions,” Cambridge University Press, Cambridge, 1999. [7] M. Naimark, “Normed Rings,” Noordhoff, Groningen, 1959. [8] T. Hida, “Brownian Motion,” Springer-Verlag, 1980.