{"paper":{"title":"On the parallel sum of positive operators, forms, and functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Zsigmond Tarcsay","submitted_at":"2015-01-08T18:39:09Z","abstract_excerpt":"The parallel sum $A:B$ of two bounded positive linear operators $A,B$ on a Hilbert space $H$ is defined to be the positive operator having the quadratic form\n  \\begin{equation*}\n  \\inf\\{(A(x-y)\\,|\\,x-y)+(By\\,|\\,y)\\,|\\,y\\in H\\}\n  \\end{equation*} for fixed $x\\in H$. The purpose of this paper is to provide a factorization of the parallel sum of the form $J_APJ_A^*$ where $J_A$ is the embedding operator of an auxiliary Hilbert space associated with $A$ and $B$, and $P$ is an orthogonal projection onto a certain linear subspace of that Hilbert space. We give similar factorizations of the parallel s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01922","kind":"arxiv","version":1},"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"}