It is established that the relationships between Weyl’s theorem, Browder’s theorem, generalized Weyl’s theorem and generalized Browder’s theorem in a semiprime Banach algebra A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document}. We prove that if the commutant of a∈A\documentclass[12pt]{minimal}
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\begin{document}$$a \in {\mathcal {A}}$$\end{document} contains a left or right injective quasinilpotent element, then f(a) satisfies Weyl’s theorem for f belongs to Hol(a)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}ol(a)$$\end{document}, the set of analytic functions on a neighborhood of σ(a)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (a)$$\end{document}. It is shown that the accumulation points of the spectrum of an element in A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document} are invariant under any commuting perturbation f such that fn∈soc(A)\documentclass[12pt]{minimal}
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\begin{document}$$f^{n} \in \textrm{soc}({\mathcal {A}})$$\end{document} for some n∈N\documentclass[12pt]{minimal}
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\begin{document}$$n \in {\mathbb {N}}$$\end{document}. This result provides a positive answer to Question 2.8 in (Linear Multilinear Algebra 64(2):247–257, 2016), and it is then applied to investigate the perturbations of Weyl’s theorem and generalized Weyl’s theorem. It is also shown that if a is a hyponormal element (that is, a∗a≥aa∗\documentclass[12pt]{minimal}
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\begin{document}$$a^{*}a \ge aa^{*}$$\end{document}) in a C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document} algebra A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document} and f∈Hol(a)\documentclass[12pt]{minimal}
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\begin{document}$$f \in {\mathcal {H}}ol(a)$$\end{document}, then f(a) satisfies Weyl’s theorem. If additionally A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document} is primitive then f(a) obeys generalized Weyl’s theorem. We also consider some other interesting properties of hyponormal elements in C* algebras, including simply polaroidness, topological divisor of zero, self-adjointness of the spectral projection with respect to λ∈isoσ(a)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in \text {iso}\sigma (a)$$\end{document}, and the spectral mapping theorem of the Weyl spectrum.
