A note on the approximate pseudospectrum of upper triangular operator matrices

Abstract

We study the approximate pseudospectrum $\sigma_{ap,\varepsilon}(\mathcal{T})$ of a $2 \times 2$ upper triangular bounded operator matrix
\begin{equation*}
\mathcal{T}=\begin{pmatrix}A & B\\ 0 & D\end{pmatrix}
: \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X},
\end{equation*}
on a complex Banach space, focusing on its relationship with the approximate pseudospectrum $\sigma_{ap,\varepsilon}(A)$ and $\sigma_{ap,\varepsilon}(D)$ of the diagonal entries. First, by constructing counterexamples, we show that in general there is no simple inclusion between $\sigma_{ap,\varepsilon}(\mathcal{T})$ and $\sigma_{ap,\varepsilon}(A)\cup\sigma_{ap,\varepsilon}(D)$. Next, we establish a sufficient condition: if $\lambda\in\sigma_{ap,\varepsilon}(D)$ and $\mathcal{R}(B)\subseteq\mathcal{R}(A-\lambda I)$, then $\lambda\in\sigma_{ap,\varepsilon}(\mathcal{T})$. Under this condition and an additional inequality constraint, we obtain the equality $\sigma_{ap,\varepsilon}(\mathcal{T})=\sigma_{ap,\varepsilon}(A)\cup\sigma_{ap,\varepsilon}(D)$. In addition, when the coupling operator $B$ can be regarded as a sufficiently small perturbation, we show that $\sigma_{ap,\varepsilon}(\mathcal{T})$ may be “sandwiched” between appropriately expanded or contracted approximate pseudospectrum of the diagonal entries $A$ and $D$.