The Weyl's theorem and its perturbations in semisimple Banach algebra
The classical theories of Fredholm and Riesz operators on infinite-dimensional Banach spaces were extented to general Banach algebras by B. A. Barnes, L. D. Pearlman, M. R. F. Smyth, R. E. Harte, C. Schmoeger and many others. The book of B. A. Barnes, G. J. Murphy, M. R. F. Smyth, and T. T. West [\textit{B. A. Barnes} et al., Riesz and Fredholm theory in Banach algebras. Boston-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0534.46034)] gives a nice overwiev of those extended theories. In the paper under review the authors define Fredholm, Weyl and Browder elements in a unital semisimple Banach algebra \(\mathcal{A}\) and prove some properties of them. They give equivalent conditions in order that Weyl's and Browder's theorems hold for an element \(a\in\mathcal{A}\) and for some of its perturbations, or images by holomorphic functions.
scientific article; zbMATH DE number 7375664
scientific article