Some remarks on the spectral theory and commutativity of unbounded normal operators
In this paper, some standard results from the spectral theory and commutativity of unbounded normal operators are proven, for instance, the spectral theorem, the characterization of the spectrum and the spectral mapping theorem. Nonetheless, it seems that some parts of the proofs are new. As an implication of the proven results, a theorem concerning commuting powers of normal operators is presented. More precisely, the following is proven. Theorem. If \(A\) is an unbounded normal operator and \(B\) is a bounded normal operator such that the spectrum \(\sigma(A)\) is generalized irrotational \((\mathrm{mod}(\frac{2\pi}{n}))\) -- for the definition see the paper -- and \(\sigma(B)\) is generalized irrotational \((\mathrm{mod}(\frac{2\pi}{m}))\), then \(B^m A^n \subseteq A^n B^m\) gives \(BA\subseteq AB\).
scientific article; zbMATH DE number 6381415