# A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

Communications in Mathematics (2013)

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

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topSontz, 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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- 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
- 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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