Browder spectra of upper-triangular operator matrices

Let MC = A C 0 B be a 2 × 2 upper triangular operator matrix acting on the Hilbert space H⊕K. In this paper, for given operators A and B, we prove that C∈B(K,H) σb(MC) = C∈B(K,H) σ(MC) ρb(A) ∩ ρb(B) , where ρb(T ) = C \ σb(T ) denotes the Browder resolvent of an operator T and C∈B(K,H) σ(MC) has been determined in [H.K. Du, P. Jin, Perturbation of spectrums of 2 × 2 operator matrices, Proc. Amer. Math. Soc. 121 (1994) 761–776]. Moreover, we explore the relations of σ(A) ∪ σ(B) \ σ(MC), σb(A) ∪ σb(B) \ σb(MC) and σw(A) ∪ σw(B) \ σw(MC), where σ(A), σb(A) and σw(A) denote the spectrum, the Browder spectrum and the Weyl spectrum of A, respectively. © 2005 Elsevier Inc. All rights reserved.