{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:X5RI7A6NTNMHGS4L27IZ4HD463","short_pith_number":"pith:X5RI7A6N","canonical_record":{"source":{"id":"1509.01510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T15:56:02Z","cross_cats_sorted":[],"title_canon_sha256":"ef5cb19ff3374f6dc233e1536793a094e8d8e470c3ef8767db981737180f464d","abstract_canon_sha256":"218428947bcfe5ef6214064bde28dd6ccddd66949a6aa0682111a2f27327dfd9"},"schema_version":"1.0"},"canonical_sha256":"bf628f83cd9b58734b8bd7d19e1c7cf6cf46cd612340f813ee7cf200649f8ab6","source":{"kind":"arxiv","id":"1509.01510","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01510","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01510v1","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01510","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"pith_short_12","alias_value":"X5RI7A6NTNMH","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5RI7A6NTNMHGS4L","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5RI7A6N","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:X5RI7A6NTNMHGS4L27IZ4HD463","target":"record","payload":{"canonical_record":{"source":{"id":"1509.01510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T15:56:02Z","cross_cats_sorted":[],"title_canon_sha256":"ef5cb19ff3374f6dc233e1536793a094e8d8e470c3ef8767db981737180f464d","abstract_canon_sha256":"218428947bcfe5ef6214064bde28dd6ccddd66949a6aa0682111a2f27327dfd9"},"schema_version":"1.0"},"canonical_sha256":"bf628f83cd9b58734b8bd7d19e1c7cf6cf46cd612340f813ee7cf200649f8ab6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:57.538563Z","signature_b64":"PvBwe4q86zIpg1hwkBfYVZE1YymP/4RBjdCt7jelKYqrX5sbjesC6VtSZ+bl1N1CNcBkrzPXr9rmPK7QZ27dDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf628f83cd9b58734b8bd7d19e1c7cf6cf46cd612340f813ee7cf200649f8ab6","last_reissued_at":"2026-05-18T01:33:57.538176Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:57.538176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.01510","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":"2026-05-18T01:33:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I/00WiD6rZOKmsLf0XBOqxd9XdVm48nMw7jEmPVFWccIOPep625khpl9TeUrFePiCh6dERoc2QzRF8rziOpxBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T16:23:59.022474Z"},"content_sha256":"5ea2d5759f81eb8ccade0e381688f8969ef52db4d08df5058c36c810e4884616","schema_version":"1.0","event_id":"sha256:5ea2d5759f81eb8ccade0e381688f8969ef52db4d08df5058c36c810e4884616"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:X5RI7A6NTNMHGS4L27IZ4HD463","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Adjoints of linear fractional composition operators on weighted Hardy spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Trieu Le, Zeljko Cuckovic","submitted_at":"2015-09-04T15:56:02Z","abstract_excerpt":"It is well known that on the Hardy space $H^2(\\mathbb{D})$ or weighted Bergman space $A^2_{\\alpha}(\\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01510","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":"2026-05-18T01:33:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gTo2d9OQBPLHR5DCj4SYNONO1TI43oGGYw3R0aj8uS3ofyOwhw4H2sFkqkJ/EcMP/VI6X+zNxXluonnGzKQmDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T16:23:59.022812Z"},"content_sha256":"cddefdeabc8c3082fb1a9e972de385ade0ff5ef3488cfdb5bf6a49189c941062","schema_version":"1.0","event_id":"sha256:cddefdeabc8c3082fb1a9e972de385ade0ff5ef3488cfdb5bf6a49189c941062"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X5RI7A6NTNMHGS4L27IZ4HD463/bundle.json","state_url":"https://pith.science/pith/X5RI7A6NTNMHGS4L27IZ4HD463/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X5RI7A6NTNMHGS4L27IZ4HD463/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":"2026-06-01T16:23:59Z","links":{"resolver":"https://pith.science/pith/X5RI7A6NTNMHGS4L27IZ4HD463","bundle":"https://pith.science/pith/X5RI7A6NTNMHGS4L27IZ4HD463/bundle.json","state":"https://pith.science/pith/X5RI7A6NTNMHGS4L27IZ4HD463/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X5RI7A6NTNMHGS4L27IZ4HD463/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:X5RI7A6NTNMHGS4L27IZ4HD463","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":"218428947bcfe5ef6214064bde28dd6ccddd66949a6aa0682111a2f27327dfd9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T15:56:02Z","title_canon_sha256":"ef5cb19ff3374f6dc233e1536793a094e8d8e470c3ef8767db981737180f464d"},"schema_version":"1.0","source":{"id":"1509.01510","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01510","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01510v1","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01510","created_at":"2026-05-18T01:33:57Z"},{"alias_kind":"pith_short_12","alias_value":"X5RI7A6NTNMH","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"X5RI7A6NTNMHGS4L","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"X5RI7A6N","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:cddefdeabc8c3082fb1a9e972de385ade0ff5ef3488cfdb5bf6a49189c941062","target":"graph","created_at":"2026-05-18T01:33:57Z","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":"It is well known that on the Hardy space $H^2(\\mathbb{D})$ or weighted Bergman space $A^2_{\\alpha}(\\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.","authors_text":"Trieu Le, Zeljko Cuckovic","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T15:56:02Z","title":"Adjoints of linear fractional composition operators on weighted Hardy spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01510","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":"sha256:5ea2d5759f81eb8ccade0e381688f8969ef52db4d08df5058c36c810e4884616","target":"record","created_at":"2026-05-18T01:33:57Z","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":"218428947bcfe5ef6214064bde28dd6ccddd66949a6aa0682111a2f27327dfd9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T15:56:02Z","title_canon_sha256":"ef5cb19ff3374f6dc233e1536793a094e8d8e470c3ef8767db981737180f464d"},"schema_version":"1.0","source":{"id":"1509.01510","kind":"arxiv","version":1}},"canonical_sha256":"bf628f83cd9b58734b8bd7d19e1c7cf6cf46cd612340f813ee7cf200649f8ab6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf628f83cd9b58734b8bd7d19e1c7cf6cf46cd612340f813ee7cf200649f8ab6","first_computed_at":"2026-05-18T01:33:57.538176Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:57.538176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PvBwe4q86zIpg1hwkBfYVZE1YymP/4RBjdCt7jelKYqrX5sbjesC6VtSZ+bl1N1CNcBkrzPXr9rmPK7QZ27dDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:57.538563Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.01510","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ea2d5759f81eb8ccade0e381688f8969ef52db4d08df5058c36c810e4884616","sha256:cddefdeabc8c3082fb1a9e972de385ade0ff5ef3488cfdb5bf6a49189c941062"],"state_sha256":"f138211dd3f1e30cef8c2168b80b5e500bbc46dde57fb56c1da957ffccd7a059"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FP5sR7i0zgIs20Zp8xBc0g9h9W8+XDaLNYgS69vt1X81Eq4/GUSOpql7xPyQINqlzAxqgiBGzysU9QJyT9lxBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T16:23:59.024707Z","bundle_sha256":"51b5244de77eb6fbaaf73bbace66969526fbe7feb604dc484030d590d6fb23aa"}}