A note on the approximate pseudospectrum of upper triangular operator matrices

We study the approximate pseudospectrum σap,ε(τ) of a 2× 2 upper triangular bounded operator matrix τ = {pmatrix} A & B \\ 0 & D {pmatrix} : X × X → X × X, on a complex Banach space, focusing on its relationship with the approximate pseudospectrum σap,ε(A) and σap,ε(D) of the diagonal entries. First, by constructing counterexamples, we show that in general there is no simple inclusion between σap,ε(τ) and σap,ε(A) ∪ σap,ε(D) . Next, we establish a sufficient condition: if λ ∈ σap,ε(D) and R(B) ⊆ R(A - λ I) , then λ ∈ σap,ε(τ) . Under this condition and an additional inequality constraint, we obtain the equality σap,ε(τ) = σap,ε(A) ∪ σap,ε(D) . In addition, when the coupling operator B can be regarded as a sufficiently small perturbation, we show that σap,ε(τ) may be ?sandwiched? between appropriately expanded or contracted approximate pseudospectrum of the diagonal entries A and D .