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A Reproducing Kernel and Toeplitz Operators in the Quantum Plane
Communications in Mathematics, Tome 21 (2013) no. 2, pp. 137-160
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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.
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.
Classification : 46E22, 47B32, 47B35, 81S99
Keywords: Reproducing kernel; Toeplitz operator; quantum plane; second quantization; creation and annihilation operators 
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Sontz, Stephen Bruce. A Reproducing Kernel and Toeplitz Operators in the Quantum Plane. Communications in Mathematics, Tome 21 (2013) no. 2, pp. 137-160. http://geodesic.mathdoc.fr/item/COMIM_2013_21_2_a3/

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