Topological uniform descent and localized SVEP

A localized version of the single-valued extension property is studied, for a bounded linear operator T acting on a Banach space and its adjoint T ⁎ , at the points λ 0 ∈ C such that λ 0 I − T has topological uniform descent (TUD for brevity). We characterize the single-valued extension property at these points for T and T ⁎ . We also give some applications of these results. As we give a counterexample to show that the adjoint of an operator with TUD is not necessarily with TUD, it is worth to mention that the characterizations of SVEP at these points for T ⁎ cannot be obtained dually from the characterizations of SVEP at the same points for T. It is quite different from the case that λ 0 I − T is of Kato type or quasi-Fredholm.