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Towards a “Matrix theory” for unbounded operator matrices. (English) Zbl 0672.47001

Consider \(2\times2\) block operator matrices \(\mathfrak A = \left[ \begin{matrix} A & B \\ C & D \end{matrix} \right]\), acting on the direct sum \(E \times F\) of Banach spaces \(E\), \(F\). The operators \(A\), \(B\), \(C\), \(D\) are linear but not necessarily bounded. Under certain assumptions (\(A\) and \(D\) have non-empty resolvent sets, \(B\) is relatively \(D\)-bounded, \(C\) is relatively \(A\)-bounded and the whole matrix \(A\) is a closed operator) the invertibility and spectrum of \(\mathfrak A\) are studied. Also, criteria are given for the operator matrix \(\left[ \begin{matrix} A & B \\ 0 & D\end{matrix} \right]\) to be generator of a semigroup, or generator of an analytic semigroup. Typical applications involve operator matrices of the form \(\left[ \begin{matrix} a & \delta_0 \\ c & d/dx \end{matrix} \right]\), where \(\delta_0\) is the Dirac measure at 0. Another application considered in the paper is the operator \[ \left[ \begin{matrix} \Delta_ n + \text{grad.div} & \text{grad} & \text{grad} \\ \text{div} & \Delta & 0 \\ \text{div} & 0 & 0 \end{matrix} \right], \]
which is the linearization of equations describing the flow of viscous compressible and heat conducting fluids.
Reviewer: L.Rodman

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47L60 Algebras of unbounded operators; partial algebras of operators
47F05 General theory of partial differential operators
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