The Perturbation of the Sub-Noncommutative Pseudo-Browder Essential Spectrum of Bounded Upper Triangular Operator Matrices

Let ε>0 and TB(X×X) be the Banach algebra of all 2×2 bounded upper triangular operator matrices on a separable Hilbert space X×X. In this paper, we first establish the spectrum equalities for special cases of upper triangular operator matrices—diagonal block operator matrix M0=A00B. We obtain that Σ^bi,ε(M0)=Σbi,ε(A)∪Σbi,ε(B), i∈{1,2,4}, where Σbi,ε(·) and Σ^bi,ε(·) denote the noncommutative pseudo-upper (resp. lower) semi-Browder essential spectrum, noncommutative pseudo-Browder essential spectrum, sub-noncommutative pseudo-upper (resp. lower) semi-Browder essential spectrum, and sub-noncommutative pseudo-Browder essential spectrum. Secondly, based on Cao and Bai’s works, we study the perturbation of the sub-noncommutative pseudo-Browder essential spectrum Σ^b4,ε(·) of a 2 × 2 bounded upper triangular operator matrix MC=AC0B on a separable Hilbert space. We obtain that ⋂C∈B(X)Σ^b4,ε(MC)=Σb1,ε(A)∪Σb2,ε(B)∪Δ, where Δ={λ∈C: there exist Pi∈B(X) with ∥Pi∥<ε,i∈{1,2}, such that α(A+P1−λI)+α(B+P2−λI)≠β(A+P1−λI)+β(B+P2−λI)}. Finally, we obtain Σbi,ε(A)∪Σbi,ε(B)=Σ^bi,ε(MC)∪W,i∈{1,2,4}, where W is the union of certain holes in (Σbi,ε(A)∪Σbi,ε(B))\Σ^bi,ε(MC).