A spectral study of operator matrices

Consider two infinite dimensional complex Banach spaces $X$ and $Y$. The operator matrix $\mathcal{M}_{N}$ is defined on $X \oplus Y$ by $\mathcal{M}_{N} = \begin{pmatrix} M_1 & N \\ 0 & M_2 \end{pmatrix}$, where $M_1 \in \mathcal{L}(X)$, $M_2 \in \mathcal{L}(Y)$, and $N \in \mathcal{L}(Y,X)$. The central focus of this manuscript revolves around the examination of the relation between certain spectral properties of the operator matrix $\mathcal{M}_{N}$ and those of their diagonal terms $M_1$ and $M_2$.