Co-Toeplitz operators and their associated quantization
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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.
KeywordsCo-Toeplitz operator Co-Toeplitz quantization Creation and annihilation operators Second quantization
Mathematics Subject Classification47B35 47B99
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.
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Conflict of interest
The author declares that he has no conflict of interest.
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