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Co-Toeplitz operators and their associated quantization

  • Stephen Bruce SontzEmail author
Original Paper
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Abstract

We define co-Toeplitz operators, a new class of Hilbert space operators, in order to define a co-Toeplitz quantization scheme that is dual to the Toeplitz quantization scheme introduced by the author in the setting of symbols that come from a possibly non-commutative algebra with unit. In the present dual setting the symbols come from a possibly non-co-commutative co-algebra with co-unit. However, this co-Toeplitz quantization is a usual quantization scheme in the sense that to each symbol we assign a densely defined linear operator acting in a fixed Hilbert space. Creation and annihilation operators are also introduced as certain types of co-Toeplitz operators, and then their commutation relations provide the way for introducing Planck’s constant into this theory. The domain of the co-Toeplitz quantization is then extended as well to a set of co-symbols, which are the linear functionals defined on the co-algebra. A detailed example based on the quantum group (and hence co-algebra) \(SU_q(2)\) as symbol space is presented.

Keywords

Co-Toeplitz operator Co-Toeplitz quantization Creation and annihilation operators Second quantization 

Mathematics Subject Classification

47B35 47B99 

Notes

Acknowledgements

I thank Micho Durdevich and Jean-Pierre Gazeau for providing me insights from rather complementary points of view of mathematical physics. I can not imagine how I could ever have possibly written this paper without their generosity in sharing ideas with me.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2020

Authors and Affiliations

  1. 1.Centro de Investigación en Matemáticas, A.C. (CIMAT)GuanajuatoMexico

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