On the Friedrichs extension of some block operator matrices (Q1600109)

For symmetric semibounded \(2\times 2\) matrix operators of the form \(\left(\begin{smallmatrix} A&B\\ C&D\end{smallmatrix}\right)\) acting in the product of Hilbert spaces \({\mathcal H}_1 \times {\mathcal H}_2\), an explicit representation of the resolvent of the Friedrichs extension is obtained under some natural assumptions on the entries \(A,B,C,D\). The essential spectrum of such operators is also investigated.