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• W3104668839 abstract "For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>q>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the unital <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=asterisk> <mml:semantics> <mml:mo>∗<!-- ∗ --></mml:mo> <mml:annotation encoding=application/x-tex>*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra with generator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and defining relation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x x Superscript asterisk Baseline equals q x Superscript asterisk Baseline x> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>xx^*=qx^*x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Based on this algebra we study <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-normal operators and the complex <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-moment problem. Among other things, we prove a spectral theorem for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-normal operators, a variant of Haviland’s theorem and a strict Positivstellensatz for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper A period> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {A}.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We also construct an example of a positive element of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not a sum of squares. It is used to prove the existence of a formally <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-normal operator which is not extendable to a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=application/x-tex>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-normal one in a larger Hilbert space and of a positive functional on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not strongly positive." @default.
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• W3104668839 date "2013-09-04" @default.
• W3104668839 modified "2023-09-26" @default.
• W3104668839 title "On 𝑞-normal operators and the quantum complex plane" @default.
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• W3104668839 doi "https://doi.org/10.1090/s0002-9947-2013-05733-9" @default.
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