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 Σᵇi, ε (M0) =Σbi, ε (A) ∪Σbi, ε (B), i∈1, 2, 4, where Σbi, ε (·) and Σᵇi, 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 Σᵇ4, ε (·) of a 2 × 2 bounded upper triangular operator matrix MC=AC0B on a separable Hilbert space. We obtain that ⋂C∈B (X) Σᵇ4, 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) =Σᵇi, ε (MC) ∪W, i∈1, 2, 4, where W is the union of certain holes in (Σbi, ε (A) ∪Σbi, ε (B) ) \Σᵇi, ε (MC).