{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:EZD766XSP4QT2VCT5ZTRWATJCL","short_pith_number":"pith:EZD766XS","canonical_record":{"source":{"id":"0802.3921","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-02-26T22:08:10Z","cross_cats_sorted":[],"title_canon_sha256":"df384441768ebac6497a35aa64bb1219a3595fbefe2edbbf1cab76ca96947df6","abstract_canon_sha256":"f7f1ef6fb8e2447026cbdb61cfff44538c38d0d355b5eb4432d77fe894787001"},"schema_version":"1.0"},"canonical_sha256":"2647ff7af27f213d5453ee671b026912ee126468686087ddb99b99e6e034620b","source":{"kind":"arxiv","id":"0802.3921","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.3921","created_at":"2026-05-18T02:16:00Z"},{"alias_kind":"arxiv_version","alias_value":"0802.3921v1","created_at":"2026-05-18T02:16:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.3921","created_at":"2026-05-18T02:16:00Z"},{"alias_kind":"pith_short_12","alias_value":"EZD766XSP4QT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"EZD766XSP4QT2VCT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"EZD766XS","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:EZD766XSP4QT2VCT5ZTRWATJCL","target":"record","payload":{"canonical_record":{"source":{"id":"0802.3921","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-02-26T22:08:10Z","cross_cats_sorted":[],"title_canon_sha256":"df384441768ebac6497a35aa64bb1219a3595fbefe2edbbf1cab76ca96947df6","abstract_canon_sha256":"f7f1ef6fb8e2447026cbdb61cfff44538c38d0d355b5eb4432d77fe894787001"},"schema_version":"1.0"},"canonical_sha256":"2647ff7af27f213d5453ee671b026912ee126468686087ddb99b99e6e034620b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:00.828252Z","signature_b64":"wPh8JSytTLPwLfWJhKfRkJo2DXIzUMoi739u/VKSANXlT3jZ4IW6fkptrPBzLXcRGLDjrB1f+UewbPDveXJGCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2647ff7af27f213d5453ee671b026912ee126468686087ddb99b99e6e034620b","last_reissued_at":"2026-05-18T02:16:00.827869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:00.827869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0802.3921","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":"2026-05-18T02:16:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AxvK5rnpCW3q66B+MWETcTgOjHwt7HpJ8iI3H2vfdufoxasQAqDrmwOpgKLXS1v/VFHfXXdJf8qSqUV1mHNcBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T23:33:14.945434Z"},"content_sha256":"b8b9e6582dc27364e8f245ea60361f4a6343bb2968d02ca967b0397a87e795b6","schema_version":"1.0","event_id":"sha256:b8b9e6582dc27364e8f245ea60361f4a6343bb2968d02ca967b0397a87e795b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:EZD766XSP4QT2VCT5ZTRWATJCL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The commutants of certain Toeplitz operators on weighted Bergman spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Trieu Le","submitted_at":"2008-02-26T22:08:10Z","abstract_excerpt":"For $\\alpha>-1$, let $A^2_{\\alpha}$ be the corresponding weighted Bergman space of the unit ball in $\\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_{\\alpha}$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.3921","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":"2026-05-18T02:16:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KvxoOp//3QHG0GO6zGjlFqUlB5vyd6ivC9mRIGXxyCg4iYLHHRpY9vZ8Tf56/FX74wYSXdTjeeVv6pZ650kmAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T23:33:14.945766Z"},"content_sha256":"c350b362e046d2945027298e2e182c529fffdc7f2db9f2563443dc6cb78b839e","schema_version":"1.0","event_id":"sha256:c350b362e046d2945027298e2e182c529fffdc7f2db9f2563443dc6cb78b839e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EZD766XSP4QT2VCT5ZTRWATJCL/bundle.json","state_url":"https://pith.science/pith/EZD766XSP4QT2VCT5ZTRWATJCL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EZD766XSP4QT2VCT5ZTRWATJCL/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":"2026-06-01T23:33:14Z","links":{"resolver":"https://pith.science/pith/EZD766XSP4QT2VCT5ZTRWATJCL","bundle":"https://pith.science/pith/EZD766XSP4QT2VCT5ZTRWATJCL/bundle.json","state":"https://pith.science/pith/EZD766XSP4QT2VCT5ZTRWATJCL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EZD766XSP4QT2VCT5ZTRWATJCL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:EZD766XSP4QT2VCT5ZTRWATJCL","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":"f7f1ef6fb8e2447026cbdb61cfff44538c38d0d355b5eb4432d77fe894787001","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-02-26T22:08:10Z","title_canon_sha256":"df384441768ebac6497a35aa64bb1219a3595fbefe2edbbf1cab76ca96947df6"},"schema_version":"1.0","source":{"id":"0802.3921","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0802.3921","created_at":"2026-05-18T02:16:00Z"},{"alias_kind":"arxiv_version","alias_value":"0802.3921v1","created_at":"2026-05-18T02:16:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.3921","created_at":"2026-05-18T02:16:00Z"},{"alias_kind":"pith_short_12","alias_value":"EZD766XSP4QT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"EZD766XSP4QT2VCT","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"EZD766XS","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:c350b362e046d2945027298e2e182c529fffdc7f2db9f2563443dc6cb78b839e","target":"graph","created_at":"2026-05-18T02:16:00Z","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":"For $\\alpha>-1$, let $A^2_{\\alpha}$ be the corresponding weighted Bergman space of the unit ball in $\\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_{\\alpha}$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$.","authors_text":"Trieu Le","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-02-26T22:08:10Z","title":"The commutants of certain Toeplitz operators on weighted Bergman spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.3921","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:b8b9e6582dc27364e8f245ea60361f4a6343bb2968d02ca967b0397a87e795b6","target":"record","created_at":"2026-05-18T02:16:00Z","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":"f7f1ef6fb8e2447026cbdb61cfff44538c38d0d355b5eb4432d77fe894787001","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2008-02-26T22:08:10Z","title_canon_sha256":"df384441768ebac6497a35aa64bb1219a3595fbefe2edbbf1cab76ca96947df6"},"schema_version":"1.0","source":{"id":"0802.3921","kind":"arxiv","version":1}},"canonical_sha256":"2647ff7af27f213d5453ee671b026912ee126468686087ddb99b99e6e034620b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2647ff7af27f213d5453ee671b026912ee126468686087ddb99b99e6e034620b","first_computed_at":"2026-05-18T02:16:00.827869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:16:00.827869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wPh8JSytTLPwLfWJhKfRkJo2DXIzUMoi739u/VKSANXlT3jZ4IW6fkptrPBzLXcRGLDjrB1f+UewbPDveXJGCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:16:00.828252Z","signed_message":"canonical_sha256_bytes"},"source_id":"0802.3921","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8b9e6582dc27364e8f245ea60361f4a6343bb2968d02ca967b0397a87e795b6","sha256:c350b362e046d2945027298e2e182c529fffdc7f2db9f2563443dc6cb78b839e"],"state_sha256":"cfd2ef58cd9fb44ac8c81ea3adbaf6ab88fb6a0d6208a677829e1a8c88f7b6a0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UT9bmk3CjJY6Sz/J1bEuSr49TwgQsrTnxjh+bYMES0RmfP9z/z6cerVwMQ+aubIS/N+HjOQWMMyKAF8r0oEpBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T23:33:14.947524Z","bundle_sha256":"29c587bbe0f8c97c92ec4bbcee29ff88e59907fb872a0eb39d5d629ee59a275a"}}