A note on Browder spectrum of operator matrices

When A ∈ B(H) and B ∈ B(K) are given, we denote by MC the operator acting on the Hilbert space H ⊕ K of the form MC = A C 0 B . 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); (ii) int σp(B) = φ; (iii) σ(A) ∩ σ(B) = φ; (iv) 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) and H/R(A − λ) are not isomorphic up to a finite dimensional subspace}.