A Reproducing Kernel and Toeplitz Operators in the Quantum Plane

TL;DR

This paper develops Toeplitz operators with symbols from the non-commutative complex quantum plane, introducing a reproducing kernel and analyzing their algebraic properties, extending finite-dimensional results to an infinite-dimensional setting.

Contribution

It introduces a new framework for Toeplitz operators on the quantum plane using a reproducing kernel, extending prior finite-dimensional results to the infinite-dimensional case.

Findings

Construction of a reproducing kernel for the quantum plane

Analysis of commutation relations of creation and annihilation operators

Extension of finite-dimensional Toeplitz operator results to the quantum plane

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.