{"paper":{"title":"The structures and decompositions of symmetries involving idempotents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jiaxin Zhang, Nana Wei, Yuan Li","submitted_at":"2019-03-05T09:38:24Z","abstract_excerpt":"Let $\\mathcal{H}$ be a separable Hilbert space and $P$ be an idempotent on $\\mathcal{H}.$ We denote by $$\\Gamma_{P}=\\{J: J=J^{\\ast}=J^{-1} \\hbox{ }\\hbox{ and }\\hbox{ } JPJ=I-P\\}$$ and $$\\Delta_{P}=\\{J: J=J^{\\ast}=J^{-1} \\hbox{ }\\hbox{ and }\\hbox{ } JPJ=I-P^*\\}.$$ In this paper, we first get that symmetries $(2P-I)|2P-I|^{-1}$ and $(P+P^{*}-I)|P+P^{*}-I|^{-1}$ are the same. Then we show that $\\Gamma_{P}\\neq\\emptyset$ if and only if $\\Delta_{P}\\neq\\emptyset.$ Also, the specific structures of all symmetries $J\\in\\Gamma_{P}$ and $J\\in\\Delta_{P} $ are established, respectively. Moreover,\n  we prove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.01746","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"}