{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CUEUP55AENRUPZ5ZFRPWOKMP4R","short_pith_number":"pith:CUEUP55A","canonical_record":{"source":{"id":"1806.10757","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-28T03:36:53Z","cross_cats_sorted":["math.CV","math.OA"],"title_canon_sha256":"15a9dbea8ef4906856e45b8fa9a238d6d0ffc311854ec6486e5958bc849e9182","abstract_canon_sha256":"0a267bc05296994cd645effa85434d31a1c30837a438a2379be8fd5a24cd1eda"},"schema_version":"1.0"},"canonical_sha256":"150947f7a0236347e7b92c5f67298fe44698feff2d286f808262702fd241f116","source":{"kind":"arxiv","id":"1806.10757","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.10757","created_at":"2026-05-18T00:12:08Z"},{"alias_kind":"arxiv_version","alias_value":"1806.10757v1","created_at":"2026-05-18T00:12:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10757","created_at":"2026-05-18T00:12:08Z"},{"alias_kind":"pith_short_12","alias_value":"CUEUP55AENRU","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"CUEUP55AENRUPZ5Z","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"CUEUP55A","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CUEUP55AENRUPZ5ZFRPWOKMP4R","target":"record","payload":{"canonical_record":{"source":{"id":"1806.10757","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-28T03:36:53Z","cross_cats_sorted":["math.CV","math.OA"],"title_canon_sha256":"15a9dbea8ef4906856e45b8fa9a238d6d0ffc311854ec6486e5958bc849e9182","abstract_canon_sha256":"0a267bc05296994cd645effa85434d31a1c30837a438a2379be8fd5a24cd1eda"},"schema_version":"1.0"},"canonical_sha256":"150947f7a0236347e7b92c5f67298fe44698feff2d286f808262702fd241f116","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:08.017858Z","signature_b64":"WR0jgVIVKMpghEcAAY5Z1wbmtOidiIYF2yclLf3OuOUL/4m9sFUIN2YlRet5Ud5bGW2kzMm+ZeqEVlHxjr2MBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"150947f7a0236347e7b92c5f67298fe44698feff2d286f808262702fd241f116","last_reissued_at":"2026-05-18T00:12:08.017341Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:08.017341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.10757","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GHEMZgFtbPzQ/u/k4eMHqOzLkHbYyxkH0/UwUZzroDbLGu5fJVOeEXNT8CJrDRGKzpzX1EkKOS8fqW4tz+iLBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T22:21:01.247670Z"},"content_sha256":"5be73e780f968c10185808518875176a22571f6ab99f0b0f21c152c97bc2f887","schema_version":"1.0","event_id":"sha256:5be73e780f968c10185808518875176a22571f6ab99f0b0f21c152c97bc2f887"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CUEUP55AENRUPZ5ZFRPWOKMP4R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.OA"],"primary_cat":"math.FA","authors_text":"Caixing Gu, Jie Xiao, Shuaibing Luo","submitted_at":"2018-06-28T03:36:53Z","abstract_excerpt":"This paper is devoted to the study of reducing subspaces for multiplication operator $M_\\phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_\\phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_\\phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $\\phi^{-1}\\circ\\phi$. By this means, we determine the reducing subspaces of $M_\\phi$ on the Dirichlet space when the order of $\\phi$ is $5$; $6$; $7$ and answer some quest"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10757","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0wcHHEN0pM1z2MAY3u2xGytWh/SAy4v53ExJP1zliQebF7tonq+3WpUPFkbQH8sIPD2EBGV8WMPUhIhaFjE2BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T22:21:01.248019Z"},"content_sha256":"fd82406468a19ec229dfc5c4f1f373fcc2724bf7036b77590898b3a5309a6fc1","schema_version":"1.0","event_id":"sha256:fd82406468a19ec229dfc5c4f1f373fcc2724bf7036b77590898b3a5309a6fc1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R/bundle.json","state_url":"https://pith.science/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-19T22:21:01Z","links":{"resolver":"https://pith.science/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R","bundle":"https://pith.science/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R/bundle.json","state":"https://pith.science/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CUEUP55AENRUPZ5ZFRPWOKMP4R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CUEUP55AENRUPZ5ZFRPWOKMP4R","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0a267bc05296994cd645effa85434d31a1c30837a438a2379be8fd5a24cd1eda","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-28T03:36:53Z","title_canon_sha256":"15a9dbea8ef4906856e45b8fa9a238d6d0ffc311854ec6486e5958bc849e9182"},"schema_version":"1.0","source":{"id":"1806.10757","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.10757","created_at":"2026-05-18T00:12:08Z"},{"alias_kind":"arxiv_version","alias_value":"1806.10757v1","created_at":"2026-05-18T00:12:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10757","created_at":"2026-05-18T00:12:08Z"},{"alias_kind":"pith_short_12","alias_value":"CUEUP55AENRU","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"CUEUP55AENRUPZ5Z","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"CUEUP55A","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:fd82406468a19ec229dfc5c4f1f373fcc2724bf7036b77590898b3a5309a6fc1","target":"graph","created_at":"2026-05-18T00:12:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper is devoted to the study of reducing subspaces for multiplication operator $M_\\phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_\\phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_\\phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $\\phi^{-1}\\circ\\phi$. By this means, we determine the reducing subspaces of $M_\\phi$ on the Dirichlet space when the order of $\\phi$ is $5$; $6$; $7$ and answer some quest","authors_text":"Caixing Gu, Jie Xiao, Shuaibing Luo","cross_cats":["math.CV","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-28T03:36:53Z","title":"Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10757","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5be73e780f968c10185808518875176a22571f6ab99f0b0f21c152c97bc2f887","target":"record","created_at":"2026-05-18T00:12:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0a267bc05296994cd645effa85434d31a1c30837a438a2379be8fd5a24cd1eda","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-28T03:36:53Z","title_canon_sha256":"15a9dbea8ef4906856e45b8fa9a238d6d0ffc311854ec6486e5958bc849e9182"},"schema_version":"1.0","source":{"id":"1806.10757","kind":"arxiv","version":1}},"canonical_sha256":"150947f7a0236347e7b92c5f67298fe44698feff2d286f808262702fd241f116","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"150947f7a0236347e7b92c5f67298fe44698feff2d286f808262702fd241f116","first_computed_at":"2026-05-18T00:12:08.017341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:08.017341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WR0jgVIVKMpghEcAAY5Z1wbmtOidiIYF2yclLf3OuOUL/4m9sFUIN2YlRet5Ud5bGW2kzMm+ZeqEVlHxjr2MBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:08.017858Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.10757","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5be73e780f968c10185808518875176a22571f6ab99f0b0f21c152c97bc2f887","sha256:fd82406468a19ec229dfc5c4f1f373fcc2724bf7036b77590898b3a5309a6fc1"],"state_sha256":"62fbdbc43db423444f53a77e6b15b2c6ed248f0c64c4fbcc576b5ee928a42c90"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EHRqdU33JiImaug50/4kjPSPF6sbnGM3w9nJCoa70mUc4N0UzLaGzr215HLuxxMl5sQJaYgysW60rAHcQ+JrBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-19T22:21:01.250054Z","bundle_sha256":"ca2dbed44f55a15a52c0359d4df812c1bbd7925ee1e99c47ca334ac6669b2fb1"}}