{"paper":{"title":"Pencils of pairs of projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Guoxing Ji, Miaomiao Cui","submitted_at":"2017-09-04T02:57:35Z","abstract_excerpt":"Let $T$ be a self-adjoint operator on a complex Hilbert space $\\mathcal{H}$. We give a sufficient and necessary condition for $T$ to be the pencil $\\lambda P+Q$ of a pair $( P, Q)$ of projections at some point $\\lambda\\in\\mathbb{R}\\backslash\\{-1, 0\\}$. Then we represent all pairs $(P, Q)$ of projections such that $T=\\lambda P+Q$ for a fixed $\\lambda$, and find that all such pairs are connected if $\\lambda\\in\\mathbb{R}\\backslash\\{-1, 0, 1\\}$. Afterwards, the von Neumann algebra generated by such pairs $(P,Q)$ is characterized. Moreover, we prove that there are at most two real numbers such that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.01418","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"}