{"paper":{"title":"Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Erin P. J. Pearse, Palle E. T. Jorgensen","submitted_at":"2015-12-10T22:07:09Z","abstract_excerpt":"We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator $A$ on a Hilbert space $\\mathcal{H}$, by means of a symmetric pair of operators. A \\emph{symmetric pair} is comprised of densely defined operators $J: \\mathcal{H}_1 \\to \\mathcal{H}_2$ and $K: \\mathcal{H}_2 \\to \\mathcal{H}_1$ which are compatible in a certain sense. With the appropriate definitions of $\\mathcal{H}_1$ and $J$ in terms of $A$ and $\\mathcal{H}$, we show that $(JJ^\\star)^{-1}$ is the Friedrichs extension of $A$. Furthermore, we use related ideas (including th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03463","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}