{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7JG7SHRXLX4FM6LIM3WNNK66XC","short_pith_number":"pith:7JG7SHRX","canonical_record":{"source":{"id":"1306.0724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-04T10:27:24Z","cross_cats_sorted":[],"title_canon_sha256":"4fcd566c4cbb3ff792234ab7cf0346b38de6ad9061907283ba0af530be728075","abstract_canon_sha256":"0ce5dbf3ca8f74f03b6d69890e685d6c8ff53b687d5f051ec9dda3f22eab0cce"},"schema_version":"1.0"},"canonical_sha256":"fa4df91e375df856796866ecd6abdeb8a672f35574a640c023525af2ab1eb4b8","source":{"kind":"arxiv","id":"1306.0724","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.0724","created_at":"2026-05-18T03:21:53Z"},{"alias_kind":"arxiv_version","alias_value":"1306.0724v1","created_at":"2026-05-18T03:21:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0724","created_at":"2026-05-18T03:21:53Z"},{"alias_kind":"pith_short_12","alias_value":"7JG7SHRXLX4F","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7JG7SHRXLX4FM6LI","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7JG7SHRX","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7JG7SHRXLX4FM6LIM3WNNK66XC","target":"record","payload":{"canonical_record":{"source":{"id":"1306.0724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-04T10:27:24Z","cross_cats_sorted":[],"title_canon_sha256":"4fcd566c4cbb3ff792234ab7cf0346b38de6ad9061907283ba0af530be728075","abstract_canon_sha256":"0ce5dbf3ca8f74f03b6d69890e685d6c8ff53b687d5f051ec9dda3f22eab0cce"},"schema_version":"1.0"},"canonical_sha256":"fa4df91e375df856796866ecd6abdeb8a672f35574a640c023525af2ab1eb4b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:53.409507Z","signature_b64":"k/k+H6nBgZRcARCEEjNwE+Du1F4t9l/wyVdy/GiziSgOzjwBOSGu78apJUmIjJEHrYzr3b4cU282VkWef45sAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa4df91e375df856796866ecd6abdeb8a672f35574a640c023525af2ab1eb4b8","last_reissued_at":"2026-05-18T03:21:53.408920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:53.408920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.0724","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":"2026-05-18T03:21:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GD8vhHqm05yN4Du78WuLyH2Y4biSMpc6nFF6y+EUH1MuWN+woeo48C0B3ywj0JF9gs5PMdoebkRlMowymw6XAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T00:43:42.612005Z"},"content_sha256":"a286c4506844bbfdb80bcbca36ba6fbe4e9697d0f5a4be81f0b094768c03b2dd","schema_version":"1.0","event_id":"sha256:a286c4506844bbfdb80bcbca36ba6fbe4e9697d0f5a4be81f0b094768c03b2dd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7JG7SHRXLX4FM6LIM3WNNK66XC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Wandering subspaces of the Bergman space and the Dirichlet space over polydisc","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"A. Chattopadhyay, B. Krishna Das, Jaydeb Sarkar, S. Sarkar","submitted_at":"2013-06-04T10:27:24Z","abstract_excerpt":"Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\\mathbb{D}^n$ (with $ n \\geq 2$) is investigated. We show that for any non-empty subset $\\alpha=\\{\\alpha_1,\\dots,\\alpha_k\\}$ of $\\{1,\\dots,n\\}$ and doubly commuting invariant subspace $\\s$ of the Bergman space or the Dirichlet space over $\\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_{\\alpha}|_\\s:=(M_{z_{\\alpha_1}}|_\\s,\\dots, M_{z_{\\alpha_k}}|_\\s)$ always possesses wandering subspace of the form \\[\\bigcap_{i=1}^k(\\s\\ominus z_{\\alpha_i}\\s). \\]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0724","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":"2026-05-18T03:21:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GYf0l7ccCaRm3fclNli40b0+O4QKjndodmpC5tMS4ibQgSHO0JWYXKIkRzXS6E1ItzPl47/7jfq9QcslhUI9Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T00:43:42.612855Z"},"content_sha256":"0a72037d490d29ab1f11a6aa86d7705b32c46916efda1d3617fda3e332f4a280","schema_version":"1.0","event_id":"sha256:0a72037d490d29ab1f11a6aa86d7705b32c46916efda1d3617fda3e332f4a280"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7JG7SHRXLX4FM6LIM3WNNK66XC/bundle.json","state_url":"https://pith.science/pith/7JG7SHRXLX4FM6LIM3WNNK66XC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7JG7SHRXLX4FM6LIM3WNNK66XC/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":"2026-06-01T00:43:42Z","links":{"resolver":"https://pith.science/pith/7JG7SHRXLX4FM6LIM3WNNK66XC","bundle":"https://pith.science/pith/7JG7SHRXLX4FM6LIM3WNNK66XC/bundle.json","state":"https://pith.science/pith/7JG7SHRXLX4FM6LIM3WNNK66XC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7JG7SHRXLX4FM6LIM3WNNK66XC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7JG7SHRXLX4FM6LIM3WNNK66XC","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":"0ce5dbf3ca8f74f03b6d69890e685d6c8ff53b687d5f051ec9dda3f22eab0cce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-04T10:27:24Z","title_canon_sha256":"4fcd566c4cbb3ff792234ab7cf0346b38de6ad9061907283ba0af530be728075"},"schema_version":"1.0","source":{"id":"1306.0724","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.0724","created_at":"2026-05-18T03:21:53Z"},{"alias_kind":"arxiv_version","alias_value":"1306.0724v1","created_at":"2026-05-18T03:21:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0724","created_at":"2026-05-18T03:21:53Z"},{"alias_kind":"pith_short_12","alias_value":"7JG7SHRXLX4F","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7JG7SHRXLX4FM6LI","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7JG7SHRX","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:0a72037d490d29ab1f11a6aa86d7705b32c46916efda1d3617fda3e332f4a280","target":"graph","created_at":"2026-05-18T03:21:53Z","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":"Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\\mathbb{D}^n$ (with $ n \\geq 2$) is investigated. We show that for any non-empty subset $\\alpha=\\{\\alpha_1,\\dots,\\alpha_k\\}$ of $\\{1,\\dots,n\\}$ and doubly commuting invariant subspace $\\s$ of the Bergman space or the Dirichlet space over $\\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_{\\alpha}|_\\s:=(M_{z_{\\alpha_1}}|_\\s,\\dots, M_{z_{\\alpha_k}}|_\\s)$ always possesses wandering subspace of the form \\[\\bigcap_{i=1}^k(\\s\\ominus z_{\\alpha_i}\\s). \\]","authors_text":"A. Chattopadhyay, B. Krishna Das, Jaydeb Sarkar, S. Sarkar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-04T10:27:24Z","title":"Wandering subspaces of the Bergman space and the Dirichlet space over polydisc"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0724","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:a286c4506844bbfdb80bcbca36ba6fbe4e9697d0f5a4be81f0b094768c03b2dd","target":"record","created_at":"2026-05-18T03:21:53Z","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":"0ce5dbf3ca8f74f03b6d69890e685d6c8ff53b687d5f051ec9dda3f22eab0cce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-04T10:27:24Z","title_canon_sha256":"4fcd566c4cbb3ff792234ab7cf0346b38de6ad9061907283ba0af530be728075"},"schema_version":"1.0","source":{"id":"1306.0724","kind":"arxiv","version":1}},"canonical_sha256":"fa4df91e375df856796866ecd6abdeb8a672f35574a640c023525af2ab1eb4b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa4df91e375df856796866ecd6abdeb8a672f35574a640c023525af2ab1eb4b8","first_computed_at":"2026-05-18T03:21:53.408920Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:53.408920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k/k+H6nBgZRcARCEEjNwE+Du1F4t9l/wyVdy/GiziSgOzjwBOSGu78apJUmIjJEHrYzr3b4cU282VkWef45sAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:53.409507Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.0724","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a286c4506844bbfdb80bcbca36ba6fbe4e9697d0f5a4be81f0b094768c03b2dd","sha256:0a72037d490d29ab1f11a6aa86d7705b32c46916efda1d3617fda3e332f4a280"],"state_sha256":"2474284397ec8c258e71d40ed90b387a8d7c50285de1277062941d5135ed92ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KrtpKBgt8L8BxUq9YOsaxftioK1M8ay2mqHfnMpaHjmCZGnoAOJCOF0b9BfdvvUObL2r3DmuaZB0svUyTm0oBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T00:43:42.616266Z","bundle_sha256":"fc197d5fbbf3a71b0e87766d69e1014be978ef5464781b9488610ba7164ce9ab"}}