The construction ofthe spectral theory on quantum deformed operators and its application to quantum physics

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A kind of spectral theory of q-deformedoperators in a Hilbert space isneeded so that unbounded representations of quantum algebras are analyzed. This work is devoted toinvestigating aq-normal operator in a Hilbert space, compared with a standard spectral measure and the spectral decomposition for a normal operator in a Hilbertspace. It is shown that, if a positive number q is smaller than 1, every q-normal operator in a Hilbert space is extended to a standard normal operator in a possibly larger Hilbert space.