A meticulous characterization of the pseudo-residual spectrum and pseudo-continuous spectrum of upper-triangular block operator matrices

For $A\in \mathcal{B}(X),B\in \mathcal{B}(Y),C\in \mathcal{B}(Y,X),$ let $M=\left(\begin{array}{ccc}A& C0& B\end{array}\right)$ denote a $2\times 2$ upper-triangular block operator matrix on $X\oplus Y$. Employing the spatial decomposition technique and stability of the closed range under small perturbations, this paper investigated the $\epsilon$-injectivity and $\epsilon$-density of upper-triangular block operator matrices under upper-triangular perturbations. Based on this, we studied meticulous characterization of the pseudo-residual spectrum and pseudo-continuous spectrum of upper-triangular block operator matrices under $2\times 2$ upper-triangular perturbations.