Browder spectra of upper-triangular operator matrices

Let MC=[AC0B] be a 2×22×2 upper triangular operator matrix acting on the Hilbert space H⊕KH⊕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)ρb(T)=C∖σb(T) denotes the Browder resolvent of an operator T   and ⋂C∈B(K,H)σ(MC)⋂C∈B(K,H)σ(MC) has been determined in [H.K. Du, P. Jin, Perturbation of spectrums of 2×22×2 operator matrices, Proc. Amer. Math. Soc. 121 (1994) 761–776]. Moreover, we explore the relations of σ(A)∪σ(B)∖σ(MC),σ(A)∪σ(B)∖σ(MC),σb(A)∪σb(B)∖σb(MC)σb(A)∪σb(B)∖σb(MC) and σw(A)∪σw(B)∖σw(MC)σw(A)∪σw(B)∖σw(MC), where σ(A)σ(A), σb(A)σb(A) and σw(A)σw(A) denote the spectrum, the Browder spectrum and the Weyl spectrum of A, respectively.