On the Weyl spectrum: Spectral mapping theorem and Weyl's theorem
Let \({\mathcal B}({\mathcal H})\) be the set of all bounded linear operators on a Banach space \({\mathcal H}\) and \({\mathcal F}_0\subset {\mathcal B}({\mathcal H})\) be the set of all Fredholm operators of index \(0.\) The Weyl spectrum \(\sigma _w(T)\) of an operator \(T\in {\mathcal B}({\mathcal H})\) is defined by \(\sigma _w(T) =\{\lambda\in {\mathbb{C}}: T-\lambda I\notin {\mathcal F}_0\}.\) They say that \(T\) satisfies the Weyl spectral mapping theorem if \(\sigma _w(f(T))=f(\sigma _w(T))\) for any function \(f\) analytic on a neighborhood of the spectrum \(\sigma (T).\) They say that Weyl's theorem holds for \(T\) if \(\sigma _w(T)\) consists precisely of all points in \(\sigma(T)\) except the isolated eigenvalues of finite multiplicity. It is shown that if \(T\) is a dominant operator or an analytic quasi-hyponormal operator then \(T\) satisfies the Weyl spectral mapping theorem. Moreover, Weyl's theorem holds for \(f(T)+F\) if ``dominant'' is replaced by ``\(M\)-hyponormal'', where \(F\) is any finite rank operator commuting with \(T.\)
scientific article; zbMATH DE number 1218591
scientific article