Geodesic
Representation Theorems and Variational Principles for Self-Adjoint Operator Matrices
Matematičeskie zametki, Tome 101 (2017) no. 4, pp. 516-530.

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We use the notion of triples $\mathfrak{D}^+\hookrightarrow \mathfrak{H}\hookrightarrow\mathfrak{D}^-$ of Hilbert spaces to develop an analog of the Friedrichs extension procedure for a class of nonsemibounded operator matrices. In addition, we suggest a general approach (stated in the same terms) to the construction of variational principles for the eigenvalues of such matrices.
Keywords: rigged space, operator matrix, self-adjoint extension, variational principle. 
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A. A. Vladimirov. Representation Theorems and Variational Principles for Self-Adjoint Operator Matrices. Matematičeskie zametki, Tome 101 (2017) no. 4, pp. 516-530. http://geodesic.mathdoc.fr/item/MZM_2017_101_4_a2/

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