Co-Toeplitz Operators and their Associated Quantization

TL;DR

This paper introduces co-Toeplitz operators and a dual quantization scheme based on co-algebras, extending the Toeplitz quantization framework to non-commutative and co-commutative settings, with applications to quantum groups.

Contribution

It defines co-Toeplitz operators and a dual quantization scheme using co-algebras, expanding the mathematical framework of quantization methods.

Findings

Defined co-Toeplitz operators in Hilbert spaces

Established a dual quantization scheme to Toeplitz quantization

Provided a detailed example with quantum group $SU_q(2)$

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.