Browder spectra of upper-triangular operator matrices

Abstract

Let M C = 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 thatthe Browder resolvent of an operator T and C∈B(K,H) σ (M C ) 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 (M C ), σ b (A) ∪ σ b (B) \ σ b (M C ) and σ w (A) ∪ σ w (B) \ σ w (M C ), where σ (A), σ b (A) and σ w (A) denote the spectrum, the Browder spectrum and the Weyl spectrum of A, respectively.

The semi-Fredholmness of 2×2 operator matrices

“…One way to study operators is to see them as entries of simpler operators. The operator matrices have been studied by numerous authors [1][2][3][4][5][6][7][8][9][10][11][12][13]. This paper is concerned with the semi-Fredholmness of 2 × 2 operator matrices.…”

Samuel multiplicities and Browder spectrum of operator matrices

“…Moreover, for T ∈ B(H), we introduce its corresponding spectra as following [19]: Moreover, it has been proved that s mul(T ), b.s. mul(T ) ∈ {0, 1, 2, .…”

“…(K, H) σ sb (M C ) and C∈B(K, H) σ b (M C ) completely, where M C = is a 2 × 2 upper triangular operator matrix defined on H ⊕ K. For the study advances of 2 × 2 upper triangular operatormatrix, see ([1][2][3][4],[13][14][15][16][17][18][19]). …”

A note on Browder spectrum of operator matrices

“…In[2], Zhang and Du found that formula (12) used in their proofs of Theorem 4 in[1] is not always true and proposed an additional condition to fill this gap. However, there is another gap in the proof of Theorem 4 in[1], that is, the claim in line 17 on p. 705 is not always true.…”

“…However, there is another gap in the proof of Theorem 4 in[1], that is, the claim in line 17 on p. 705 is not always true. It seems that this claim had not been corrected by the authors, but the additional condition in[2] can also fill this gap, in fact, if iso(∂Λ (A,B)) = φ, then W 3 (A, B, C ) = W 1(A, B, C ).…”