Toeplitz and Translation Operators on the q-Fock Spaces

Author(s)
Fethi Soltani

ABSTRACT

In this work, we introduce a class of Hilbert spaces F_{q} of entire functions on the disk , , with reproducing kernel given by the q-exponential function e_{q}(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 D_{q} by and the q-Derivative operator on the Fock space F_{q} ; 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 F_{q} .

In this work, we introduce a class of Hilbert spaces F

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,"

References

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[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.