{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:344VNPU3JXEKSERFZQOXMDTWRF","short_pith_number":"pith:344VNPU3","schema_version":"1.0","canonical_sha256":"df3956be9b4dc8a91225cc1d760e76897a10bd848062771288c8147371eab3e3","source":{"kind":"arxiv","id":"1903.01746","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1903.01746","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-03-05T09:38:24Z","cross_cats_sorted":[],"title_canon_sha256":"1ba58b15332ed093324bfc5832c412fc9ff4e9bba2e61542da0e846917979a5e","abstract_canon_sha256":"e0bfb787ae46990dabae136ff92ebc695bda15dd656a599945ded981fbc378c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:59.869743Z","signature_b64":"bakKFct5dRUjysNjdfEik2ROPmbrdUqGp+KSasoNQTCFJm5kwz5zVpt4tVFwCT2lBG8w1C2m4/w/au9GUOsBAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df3956be9b4dc8a91225cc1d760e76897a10bd848062771288c8147371eab3e3","last_reissued_at":"2026-05-17T23:51:59.869367Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:59.869367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.01746","created_at":"2026-05-17T23:51:59.869425+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.01746v1","created_at":"2026-05-17T23:51:59.869425+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.01746","created_at":"2026-05-17T23:51:59.869425+00:00"},{"alias_kind":"pith_short_12","alias_value":"344VNPU3JXEK","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"344VNPU3JXEKSERF","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"344VNPU3","created_at":"2026-05-18T12:33:07.085635+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF","json":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF.json","graph_json":"https://pith.science/api/pith-number/344VNPU3JXEKSERFZQOXMDTWRF/graph.json","events_json":"https://pith.science/api/pith-number/344VNPU3JXEKSERFZQOXMDTWRF/events.json","paper":"https://pith.science/paper/344VNPU3"},"agent_actions":{"view_html":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF","download_json":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF.json","view_paper":"https://pith.science/paper/344VNPU3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.01746&json=true","fetch_graph":"https://pith.science/api/pith-number/344VNPU3JXEKSERFZQOXMDTWRF/graph.json","fetch_events":"https://pith.science/api/pith-number/344VNPU3JXEKSERFZQOXMDTWRF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF/action/storage_attestation","attest_author":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF/action/author_attestation","sign_citation":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF/action/citation_signature","submit_replication":"https://pith.science/pith/344VNPU3JXEKSERFZQOXMDTWRF/action/replication_record"}},"created_at":"2026-05-17T23:51:59.869425+00:00","updated_at":"2026-05-17T23:51:59.869425+00:00"}