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 \({\mathcal {A}}\) . We prove that if the commutant of \(a \in {\mathcal {A}}\) contains a left or right injective quasinilpotent element, then f(a) satisfies Weyl’s theorem for f belongs to \({\mathcal {H}}ol(a)\) , the set of analytic functions on a neighborhood of \(\sigma (a)\) . It is shown that the accumulation points of the spectrum of an element in \({\mathcal {A}}\) are invariant under any commuting perturbation f such that \(f^{n} \in \textrm{soc}({\mathcal {A}})\) for some \(n \in {\mathbb {N}}\) . 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, \(a^{*}a \ge aa^{*}\) ) in a \(C^{*}\) algebra \({\mathcal {A}}\) and \(f \in {\mathcal {H}}ol(a)\) , then f(a) satisfies Weyl’s theorem. If additionally \({\mathcal {A}}\) 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 \(\lambda \in \text {iso}\sigma (a)\) , and the spectral mapping theorem of the Weyl spectrum.
