On $q$-normal operators and quantum complex plane
  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. 
Jaka Cimpric, Yurii Savchuk, Konrad Schm\"udgen
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