Components of topological uniform descent resolvent set and local spectral theory

In this paper, we shall study the components of topological uniform descent resolvent set ρud(T) for a bounded linear operator T acting on a Banach space. We first show the constancy of certain subspace valued mappings on the components of ρud(T). Then using these results and, the equivalences of SVEP at a point λ for T and T* in the case that λI-T has topological uniform descent, we obtain a classification of these components. Finally, we give some applications of the classification. In particular, we give a sufficient condition on an operator T such that its topological uniform descent spectrum σud(T) coincides with some distinguished parts of its spectrum σ(T).