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 λ0∈C such that λ0I−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 λ0I−T is of Kato type or quasi-Fredholm.