Weyl type theorems in Banach algebras and hyponormal elements in C∗ algebras

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. We prove that if the commutant of a is an element of A contains a left or right injective quasinilpotent element, then f(a) satisfies Weyl's theorem for f belongs to Hol(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 A are invariant under any commuting perturbation f such that F-n is an element of soc(A) for some n is an element of 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(& lowast;)a >= aa(& lowast;)) in a C-& lowast; algebra A and f is an element of Hol(a), then f(a) satisfies Weyl's theorem. If additionally 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 is an element of iso sigma(a), and the spectral mapping theorem of the Weyl spectrum.