Given a Banach space X , let M C ∈ B ( X ⊕ X ) denote the upper triangular operator matrix M C = ( A C 0 B ) , and let δ A B ∈ B ( B ( X ) ) denote the generalized derivation δ A B ( X ) = A X ? X B . If lim n → ∞ ∥ δ A B n ( C ) ∥ 1 n = 0 , then σ x ( M C ) = σ x ( M 0 ) , where σ x stands for the spectrum or a distinguished part thereof (but not the point spectrum); furthermore, if R = R 1 ⊕ R 2 ∈ B ( X ⊕ X ) is a Riesz operator which commutes with M C , then σ x ( M C + R ) = σ x ( M C ) , where σ x stands for the Fredholm essential spectrum or a distinguished part thereof. These results are applied to prove the equivalence of Browder’s (a-Browder’s) theorem for M 0 , M C , M 0 + R and M C + R . Sufficient conditions for the equivalence of Weyl’s (a-Weyl’s) theorem are also considered. MSC:47B40, 47A10, 47B47, 47A11.
展开▼
