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A note on *-paranormal operators. (English) Zbl 1097.47022

An operator \(T\) on a complex separable Hilbert space \(H\) is said to be paranormal if \(\| Tx\|^2\leq\| T^2x\|\) for every unit vector \(x\in H\). An operator \(T\) is said to be \(^*\)-paranormal if \(\| T^*x\|^2\leq\| T^2x\|\) for every unit vector \(x\in H\). An operator \(T\) is said to be totally \(^*\)-paranormal if the operator \(T-\lambda\) is \(^*\)-paranormal for every complex number \(\lambda\).
In this paper, the authors show that if either \(T\) or \(T^*\) is totally \(^*\)-paranormal, then Weyl’s theorem holds for \(f(T)\), where \(f\) is analytic near the spectrum of \(T\), and if \(T^*\) is totally \(^*\)-paranormal, then a-Weyl’s theorem also holds for \(f(T)\). Furthermore, the authors prove that if either \(T\) or \(T^*\) is \(^*\)-paranormal, then the spectral mapping theorem holds for the Weyl spectrum of \(T\) and the essential approximate point spectrum of \(T\).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A10 Spectrum, resolvent
47A53 (Semi-) Fredholm operators; index theories
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