Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) is said to satisfy property (UW Π) if σ a (T)\σ ea (T) = Π(T), where σ a (T) and σ ea (T) denote the approximate point spectrum and the essential approximate point spectrum of T respectively, Π(T) denotes the set of all poles of T. T ∈ B(H) satisfies a-Weyl's theorem if σ a (T)\σ ea (T) = π a 00 (T), where π a 00 (T) = {λ ∈ isoσ a (T): 0 < n(T − λI) < ∞}. In this paper, we give necessary and sufficient conditions for a bounded linear operator and its function calculus to satisfy both property (UW Π) and a-Weyl's theorem by topological uniform descent. In addition, the property (UW Π) and a-Weyl's theorem under perturbations are also discussed.