A note on Browder spectrum of operator matrices

When A∈B(H)A∈B(H) and B∈B(K)B∈B(K) are given, we denote by MCMC the operator acting on the Hilbert space H⊕KH⊕K of the form MC=(AC0B). In this note, it is shown that the following results in [Hai-Yan Zhang, Hong-Ke Du, Browder spectra of upper-triangular operator matrices, J. Math. Anal. Appl. 323 (2006) 700–707]W3(A,B,C)=W1(A,B,C)(in line 17 on p. 705) and⋂C∈B(K,H)σb(MC)=(⋂C∈B(K,H)σ(MC))∖[ρb(A)∩ρb(B)] are not always true, although the authors tried to fill the gap in their proofs by proposing an additional condition in [H.-Y. Zhang, H.-K Du, Corrigendum to “Browder spectra of upper-triangular operator matrices” [J. Math. Anal. Appl. 323 (2006) 700–707], J. Math. Anal. Appl. 337 (2007) 751–752]. A counterexample is given and then we show that under one of the following conditions:(i)σsu(B)=σ(B)σsu(B)=σ(B);(ii)intσp(B)=ϕintσp(B)=ϕ;(iii)σ(A)∩σ(B)=ϕσ(A)∩σ(B)=ϕ;(iv)σa(A)=σ(A)σa(A)=σ(A), we have⋂C∈B(K,H)σb(MC)=σle(A)∪σre(B)∪W(A,B)∪σD(A)∪σD(B), where W(A,B)={λ∈C:N(B−λ)andH/R(A−λ)¯are not isomorphic up to a finitedimensional subspace}.