Upper triangular operator matrices, asymptotic intertwining and Browder, Weyl theorems

Given a Banach space X, let MC∈B(X⊕X) denote the upper triangular operator matrix MC=(AC0B), and let δAB∈B(B(X)) denote the generalized derivation δAB(X)=AX−XB. If limn→∞∥δABn(C)∥1n=0, then σx(MC)=σx(M0), where σx stands for the spectrum or a distinguished part thereof (but not the point spectrum); furthermore, if R=R1⊕R2∈B(X⊕X) is a Riesz operator which commutes with MC, then σx(MC+R)=σx(MC), where σx 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 M0, MC, M0+R and MC+R. Sufficient conditions for the equivalence of Weyl’s (a-Weyl’s) theorem are also considered.