Abstract:

An operator T is said to satisfy a-Browder's theorem if σ_a(T)\σ_ea(T)???π₀₀^a(T), where σ_a(T) and σ_ea(T) denote the approximate point spectrum and the essential approximate point spectrum, respectively, and π₀₀^a(T)={λ∈isoσ_a(T),0< dimN(T-λI)<∞}. If σ_a(T)\σ_ea(T)=π₀₀^a(T), we say that T satisfies a-Weyl's theorem. In this note, by using the characteristics of semi-Fredholm domain of the diagonal of the upper triangular operator matrix, we investigate the stability of a-Browder's theorem and a-Weyl's theorem for the upper triangular operator matrices under compact perturbations.

Key words: upper triangular operator matrices, a-Browder’s theorem, compact perturbations, asymptotic intertwining operator, semi-Fredholm domain