On 𝑞-normal operators and the quantum complex plane

Cimprič, Jaka; Savchuk, Yurii; Schmüdgen, Konrad

doi:10.1090/S0002-9947-2013-05733-9

On $q$-normal operators and the quantum complex plane
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by Jaka Cimprič, Yurii Savchuk and Konrad Schmüdgen PDF

Trans. Amer. Math. Soc. 366 (2014), 135-158 Request permission

Abstract:

For $q>0$ let $\mathcal {A}$ denote the unital $*$-algebra with generator $x$ and defining relation $xx^*=qx^*x$. Based on this algebra we study $q$-normal operators and the complex $q$-moment problem. Among other things, we prove a spectral theorem for $q$-normal operators, a variant of Haviland’s theorem and a strict Positivstellensatz for $\mathcal {A}.$ We also construct an example of a positive element of $\mathcal {A}$ which is not a sum of squares. It is used to prove the existence of a formally $q$-normal operator which is not extendable to a $q$-normal one in a larger Hilbert space and of a positive functional on $\mathcal {A}$ which is not strongly positive.

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