I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that $$Tx = \begin{pmatrix}A & B \\ C & D \end{pmatrix}\begin{pmatrix} x_1\\ x_2 \end{pmatrix}$$

In particular, what is the relation between the spectrum of $T$ and the spectrum of $A,B$, are there any inequalities with norms? Currently, I can only find some scattered references, like http://www.sciencedirect.com/science/article/pii/S0024379598102197 which says that $$\sigma(T)\subset\sigma(A)\cup\left\{\lambda\in \rho(A):\|(A - \lambda)^{-1}\|^{-1}\leq \|C\|\right\} \cup\sigma(B) \cup \left\{\lambda\in \rho(D):\|(D - \lambda)^{-1}\|^{-1}\leq \|B\|\right\}$$ Is there a systematic treatment of such things?

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    $\begingroup$ This is not correct: for example, take $A=B=C=0$. $\endgroup$ Commented Sep 13, 2014 at 14:56
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    $\begingroup$ Taking $E=F={\mathbb C}$ should impose some immediate limits on what you hope might be true $\endgroup$
    – Yemon Choi
    Commented Sep 13, 2014 at 15:55