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Post Closed as "Needs more focus" by Yemon Choi, Ricardo Andrade, Steven Sam, Stefan Kohl♦, Chris Godsil
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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 $$\sigma(T)\subset\sigma(A)\cup\left\{\lambda\in \rho(A):\|(A - \lambda)^{-1}\|^{-1}\leq \|C\|\right\} \cup\sigma(B) \cup \left\{\lambda\in \rho(B):\|(B - \lambda)^{-1}\|^{-1}\leq \|B\|\right\}$$$$\sigma(T)\subset\sigma(A)\cup\left\{\lambda\in \rho(A):\|(A - \lambda)^{-1}\|^{-1}\leq \|C\|\right\} \cup\sigma(B) \cup \left\{\lambda\in \rho(D):\|(D - \lambda)^{-1}\|^{-1}\leq \|B\|\right\}$$ Is there a systematic treatment of such things?

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 $$\sigma(T)\subset\sigma(A)\cup\left\{\lambda\in \rho(A):\|(A - \lambda)^{-1}\|^{-1}\leq \|C\|\right\} \cup\sigma(B) \cup \left\{\lambda\in \rho(B):\|(B - \lambda)^{-1}\|^{-1}\leq \|B\|\right\}$$ Is there a systematic treatment of such things?

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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Aad
Aad
  • 11
  • 2

Reference request on operator matrices

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 $$\sigma(T)\subset\sigma(A)\cup\left\{\lambda\in \rho(A):\|(A - \lambda)^{-1}\|^{-1}\leq \|C\|\right\} \cup\sigma(B) \cup \left\{\lambda\in \rho(B):\|(B - \lambda)^{-1}\|^{-1}\leq \|B\|\right\}$$ Is there a systematic treatment of such things?