Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$

Consider the block matrix

$$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$

where $A_1 = \operatorname{diag}(\lambda_1,\lambda_2)$ for $\lambda_i \in \mathbb C$ and $A_2: V^2 \to V^2.$

We then consider $$K=(A-\lambda)^{-1}.$$

Question: Can we express the resolvent in the form

$$K = \begin{pmatrix} T_1(T_2-\lambda)^{-1} & * \\ * & T_3 (T_4-\lambda)^{-1}\end{pmatrix}$$

where $T_1,..,T_4$ are some matrices and $*$ elements I do not really care about.