On $q$-normal operators and quantum complex plane

TL;DR

This paper explores the properties of $q$-normal operators and related algebraic structures, addressing the quantum complex plane, the $q$-moment problem, and positivity conditions within a specific algebraic framework.

Contribution

It introduces and analyzes the algebra generated by a $q$-normal operator, providing new insights into the quantum complex plane and associated positivity problems.

Findings

Characterization of $q$-normal operators

Solutions to the complex $q$-moment problem

Conditions for positivity and sums of squares

Abstract

For $q>0$ let $\cA$ denote the unital $\ast$-algebra with generator $x$ and defining relation $xx^\ast=qxx^\ast$. Based on this algebra we study $q$-normal operators, the complex $q$-moment problem, positive elements and sums of squares.