Previous |  Up |  Next


Reproducing kernel; Toeplitz operator; quantum plane; second quantization; creation and annihilation operators
We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined as Toeplitz operators. This paper extends results of the author for the finite dimensional case.
[1] Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc., 108, 1950, 337-404. DOI 10.1090/S0002-9947-1950-0051437-7 | MR 0051437 | Zbl 0037.20701
[2] Bargmann, V.: On a Hilbert space of analytic functions and its associated integral transform. I. Commun. Pure Appl. Math., 14, 1961, 187-214. DOI 10.1002/cpa.3160140303 | MR 0157250
[3] Baz, M. El, Fresneda, R., Gazeau, J-P., Hassouni, Y.: Coherent state quantization of paragrassmann algebras. J. Phys. A: Math. Theor., 43, 2010, 385202 (15pp). Also see the Erratum for this article in arXiv:1004.4706v3. MR 2718322 | Zbl 1198.81124
[4] Gazeau, J-P.: Coherent States in Quantum Physics. 2009, Wiley-VCH.
[5] Kassel, C.: Quantum Groups. 1995, Springer. MR 1321145 | Zbl 0808.17003
[6] Khalkhali, M.: Basic Noncommutative Geometry. 2009, European Math. Soc.. MR 2567651 | Zbl 1210.58006
[7] Reed, M., Simon, B.: Mathematical Methods of Modern Physics, Vol. I, Functional Analysis. 1972, Academic Press.
[8] Saitoh, S.: Theory of reproducing kernels and its applications, Pitman Research Notes, Vol. 189. 1988, Longman Scientific & Technical, Essex. MR 0983117
[9] Sontz, S.B.: Paragrassmann Algebras as Quantum Spaces, Part I: Reproducing Kernels, Geometric Methods in Physics. XXXI Workshop 2012. Trends in Mathematics, 2013, 47-63, arXiv:1204.1033v3. MR 3159286
[10] Sontz, S.B.: Paragrassmann Algebras as Quantum Spaces, Part II: Toeplitz Operators. Journal of Operator Theory. To appear. arXiv:1205.5493.
Partner of
EuDML logo