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I am doing a research on a certain type of operator, and in the course of it I need to determine the following: Given the operator $D$ below, and identity operator $I$, $$ D=\begin{pmatrix} 0 & -\text{div}\,A \\ \nabla & 0 \\ \end{pmatrix} $$ where $A$ is a certain operator which is not important here, can I get an explicit expression for its resolvent $(\lambda I- D)^{-1}$ ?

Thank you.

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What space is everything acting on here? In particular, what is $\nabla$, what sort of operator is $A$, and how are you taking the divergence of $A$? –  Branimir Ćaćić Jan 1 '13 at 18:17

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