Han, Young Min; Kim, An-Hyun A note on *-paranormal operators. (English) Zbl 1097.47022 Integral Equations Oper. Theory 49, No. 4, 435-444 (2004). 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 \(T\). Reviewer: Shanli Sun (Beijing) Cited in 16 Documents MSC: 47B20 Subnormal operators, hyponormal operators, etc. 47A10 Spectrum, resolvent 47A53 (Semi-) Fredholm operators; index theories Keywords:*-paranormal operator; Weyl spectrum; essential spectrum; spectral mapping theorem PDFBibTeX XMLCite \textit{Y. M. Han} and \textit{A.-H. Kim}, Integral Equations Oper. Theory 49, No. 4, 435--444 (2004; Zbl 1097.47022) Full Text: DOI