{"paper":{"title":"Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian","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":"2015-04-06T17:42:21Z","abstract_excerpt":"We compare the space of square-summable functions on an infinite graph (denoted $\\ell^2(G)$) with the space of functions of finite energy (denoted $\\mathcal{H}_{\\mathcal{E}}$). There is a notion of inclusion that allows $\\ell^2(G)$ to be embedded into $\\mathcal{H}_{\\mathcal{E}}$, but the required inclusion operator is unbounded in most interesting cases. These observations assist in the construction of the Krein extension of the Laplace operator on $\\mathcal{H}_{\\mathcal{E}}$. We investigate the Krein extension and compare it to the Friedrichs extension developed by the authors in a previous p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01332","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"}