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On the operator equation \(AB = zBA\). (English) Zbl 1213.47022

The authors study the operator equation \(AB = z BA\) for bounded operators on a complex Hilbert space. They extend an earlier result of J. Yang and H. K. Du in which the Fuglede-Putnam theorem is used to show that, if \(A\) and \(B\) are normal operators satisfying \(AB = z BA\), then \(|z| = 1\). The conditions on \(A\) and \(B\) are relaxed to hyponormal and paranormal, respectively, with \(A\) either invertible or having \(0\) as an isolated point of its spectrum. Additionally, if \(U|T|\) is the polar decomposition of \(T\) and \(\Delta(T) = |T|^{1/2} U|T|^{1/2}\) is the Aluthge transform, then \(T=z \Delta(T)\) for some \(z \in \mathbb C\) is shown to imply \(z=1\) provided that \(\sigma(T) \neq \{ 0 \}\).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A10 Spectrum, resolvent
47B47 Commutators, derivations, elementary operators, etc.
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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