The Judgment of Weyl's Theorem and Property (R)

The perturbation invariance of semi-Fredholm operators are used to study Weyl type theorems for boun-ded linear operators and upper triangular operator matrices. First, the necessary and sufficient conditions for which both the property(R₁) and Browder's theorem hold or both the property (R) and Weyl's theorem hold for bounded linear operators are given. In addition, the conditions for which both the property (R) and Weyl's theorem hold for upper triangular operator matrices are explored.