A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

Stephen Bruce Sontz

Communications in Mathematics (2013)

  • Volume: 21, Issue: 2, page 137-160
  • ISSN: 1804-1388

Abstract

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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.

How to cite

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Sontz, Stephen Bruce. "A Reproducing Kernel and Toeplitz Operators in the Quantum Plane." Communications in Mathematics 21.2 (2013): 137-160. <http://eudml.org/doc/260780>.

@article{Sontz2013,
abstract = {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.},
author = {Sontz, Stephen Bruce},
journal = {Communications in Mathematics},
keywords = {Reproducing kernel; Toeplitz operator; quantum plane; second quantization; creation and annihilation operators; reproducing kernel; Toeplitz operator; quantum plane; second quantization; creation and annihilation operators},
language = {eng},
number = {2},
pages = {137-160},
publisher = {University of Ostrava},
title = {A Reproducing Kernel and Toeplitz Operators in the Quantum Plane},
url = {http://eudml.org/doc/260780},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Sontz, Stephen Bruce
TI - A Reproducing Kernel and Toeplitz Operators in the Quantum Plane
JO - Communications in Mathematics
PY - 2013
PB - University of Ostrava
VL - 21
IS - 2
SP - 137
EP - 160
AB - 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.
LA - eng
KW - Reproducing kernel; Toeplitz operator; quantum plane; second quantization; creation and annihilation operators; reproducing kernel; Toeplitz operator; quantum plane; second quantization; creation and annihilation operators
UR - http://eudml.org/doc/260780
ER -

References

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  2. Bargmann, V., 10.1002/cpa.3160140303, Commun. Pure Appl. Math., 14, 1961, 187-214. (1961) MR0157250DOI10.1002/cpa.3160140303
  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. (2010) Zbl1198.81124MR2718322
  4. Gazeau, J-P., Coherent States in Quantum Physics, 2009, Wiley-VCH. (2009) 
  5. Kassel, C., Quantum Groups, 1995, Springer. (1995) Zbl0808.17003MR1321145
  6. Khalkhali, M., Basic Noncommutative Geometry, 2009, European Math. Soc.. (2009) Zbl1210.58006MR2567651
  7. Reed, M., Simon, B., Mathematical Methods of Modern Physics, Vol. I, Functional Analysis, 1972, Academic Press. (1972) 
  8. Saitoh, S., Theory of reproducing kernels and its applications, Pitman Research Notes, Vol. 189, 1988, Longman Scientific & Technical, Essex. (1988) MR0983117
  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. (2013) MR3159286
  10. Sontz, S.B., Paragrassmann Algebras as Quantum Spaces, Part II: Toeplitz Operators, Journal of Operator Theory. To appear. arXiv:1205.5493. 

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