Components of topological uniform descent resolvent set and local spectral theory (Q1931726)

Let \(T\) be a bounded linear operator acting on an infinite-dimensional complex Banach space \(X\). \(T\) is said to have topological uniform descent (TUD) for order \(d\in\mathbb{N}\) if, for \(n\geq d\), \(R(T)+N(T^n)=R(T)+N(T^d)\) and \(R(T^n)\) is closed in the operator range topology of \(R(T^d)\), where \(R(T)\) and \(N(T)\) denote the range and null space of \(T\), respectively. The topological uniform descent spectrum is defined by \(\sigma_{ud}(T)=\{\lambda\in\mathbb{C}\,:\,\lambda I-T\text{ does not have TUD}\}\). In this paper, the authors prove that the topological uniform descent spectrum \(\sigma_{ud}(T)\) is a compact subset of the usual spectrum and satisfies the spectral mapping theorem. They investigate the component of \(\rho_{ud}(T)=\mathbb{C}\setminus\sigma_{ud}(T)\) using tools from local spectral theory. Some applications to Drazin invertibility are given.