{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:PVRTHVFW3YQMKFUSIYCL26R22M","short_pith_number":"pith:PVRTHVFW","canonical_record":{"source":{"id":"1108.4637","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-08-23T15:48:14Z","cross_cats_sorted":["math.CA","math.CV","math.SP"],"title_canon_sha256":"ad147fe9db399e8abf2204e6bf349941decd0408538b0e8708b7be2218624020","abstract_canon_sha256":"fc24272cb041cefa26be638ba1c3adf93620b72f0dc01630e8fe8c6e105525b5"},"schema_version":"1.0"},"canonical_sha256":"7d6333d4b6de20c516924604bd7a3ad30e432f1afc23f4f76ab8dccf894df7c6","source":{"kind":"arxiv","id":"1108.4637","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4637","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4637v1","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4637","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"pith_short_12","alias_value":"PVRTHVFW3YQM","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PVRTHVFW3YQMKFUS","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PVRTHVFW","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:PVRTHVFW3YQMKFUSIYCL26R22M","target":"record","payload":{"canonical_record":{"source":{"id":"1108.4637","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-08-23T15:48:14Z","cross_cats_sorted":["math.CA","math.CV","math.SP"],"title_canon_sha256":"ad147fe9db399e8abf2204e6bf349941decd0408538b0e8708b7be2218624020","abstract_canon_sha256":"fc24272cb041cefa26be638ba1c3adf93620b72f0dc01630e8fe8c6e105525b5"},"schema_version":"1.0"},"canonical_sha256":"7d6333d4b6de20c516924604bd7a3ad30e432f1afc23f4f76ab8dccf894df7c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:00.606428Z","signature_b64":"NSeG0Sph5Znrk5YqUwL4B639zmhD+rI8hXlMgfyBn54ChzI8WRdfSDnUSzW1sgq62u2U7eaMo4QtIU24y1IqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d6333d4b6de20c516924604bd7a3ad30e432f1afc23f4f76ab8dccf894df7c6","last_reissued_at":"2026-05-18T02:58:00.605885Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:00.605885Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.4637","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":"2026-05-18T02:58:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"evWP84jyaFqUWT8u/MsNS1Jd0sV5ckF3GSsoT67AXSVPyF/DZTW4cStavXhxDujw8Fk9NtXVpYWtbeeAr66jDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:10:44.380148Z"},"content_sha256":"372092ea4b9b96b2bf5f94eba99934d53994eaf2bdfa384caa2e9a585c4361a2","schema_version":"1.0","event_id":"sha256:372092ea4b9b96b2bf5f94eba99934d53994eaf2bdfa384caa2e9a585c4361a2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:PVRTHVFW3YQMKFUSIYCL26R22M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Operator and commutator moduli of continuity for normal operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV","math.SP"],"primary_cat":"math.FA","authors_text":"Aleksei Aleksandrov, Vladimir Peller","submitted_at":"2011-08-23T15:48:14Z","abstract_excerpt":"We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \\cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\\|N_1R- RN_2\\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\\a<1$ and $f$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4637","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":"2026-05-18T02:58:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tMkNOErfHZTGGXf/mgu8DTC7wldtqWA3r0CxGYEo6jhQEpG5exn5H5H4svUBt924ix+LIA9oL0fy+9mALQnbAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:10:44.380522Z"},"content_sha256":"1acb5b9afd8e8299afb7a4f4ee4a0866da022fefda101248eabdf2917e3f8f4b","schema_version":"1.0","event_id":"sha256:1acb5b9afd8e8299afb7a4f4ee4a0866da022fefda101248eabdf2917e3f8f4b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PVRTHVFW3YQMKFUSIYCL26R22M/bundle.json","state_url":"https://pith.science/pith/PVRTHVFW3YQMKFUSIYCL26R22M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PVRTHVFW3YQMKFUSIYCL26R22M/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":"2026-05-31T00:10:44Z","links":{"resolver":"https://pith.science/pith/PVRTHVFW3YQMKFUSIYCL26R22M","bundle":"https://pith.science/pith/PVRTHVFW3YQMKFUSIYCL26R22M/bundle.json","state":"https://pith.science/pith/PVRTHVFW3YQMKFUSIYCL26R22M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PVRTHVFW3YQMKFUSIYCL26R22M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:PVRTHVFW3YQMKFUSIYCL26R22M","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":"fc24272cb041cefa26be638ba1c3adf93620b72f0dc01630e8fe8c6e105525b5","cross_cats_sorted":["math.CA","math.CV","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-08-23T15:48:14Z","title_canon_sha256":"ad147fe9db399e8abf2204e6bf349941decd0408538b0e8708b7be2218624020"},"schema_version":"1.0","source":{"id":"1108.4637","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4637","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4637v1","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4637","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"pith_short_12","alias_value":"PVRTHVFW3YQM","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"PVRTHVFW3YQMKFUS","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"PVRTHVFW","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:1acb5b9afd8e8299afb7a4f4ee4a0866da022fefda101248eabdf2917e3f8f4b","target":"graph","created_at":"2026-05-18T02:58:00Z","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":"We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \\cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\\|N_1R- RN_2\\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\\a<1$ and $f$ is ","authors_text":"Aleksei Aleksandrov, Vladimir Peller","cross_cats":["math.CA","math.CV","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-08-23T15:48:14Z","title":"Operator and commutator moduli of continuity for normal operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4637","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":"sha256:372092ea4b9b96b2bf5f94eba99934d53994eaf2bdfa384caa2e9a585c4361a2","target":"record","created_at":"2026-05-18T02:58:00Z","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":"fc24272cb041cefa26be638ba1c3adf93620b72f0dc01630e8fe8c6e105525b5","cross_cats_sorted":["math.CA","math.CV","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-08-23T15:48:14Z","title_canon_sha256":"ad147fe9db399e8abf2204e6bf349941decd0408538b0e8708b7be2218624020"},"schema_version":"1.0","source":{"id":"1108.4637","kind":"arxiv","version":1}},"canonical_sha256":"7d6333d4b6de20c516924604bd7a3ad30e432f1afc23f4f76ab8dccf894df7c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d6333d4b6de20c516924604bd7a3ad30e432f1afc23f4f76ab8dccf894df7c6","first_computed_at":"2026-05-18T02:58:00.605885Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:00.605885Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NSeG0Sph5Znrk5YqUwL4B639zmhD+rI8hXlMgfyBn54ChzI8WRdfSDnUSzW1sgq62u2U7eaMo4QtIU24y1IqCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:00.606428Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.4637","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:372092ea4b9b96b2bf5f94eba99934d53994eaf2bdfa384caa2e9a585c4361a2","sha256:1acb5b9afd8e8299afb7a4f4ee4a0866da022fefda101248eabdf2917e3f8f4b"],"state_sha256":"c6f6c4a28f65a02542093bfb90614045dfba76604fc07b0299d0e1c72d155635"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"41VqCaHHERmcMByeTpXzR8ZfSUS3XfaAVN8QfrWEfyQYLilLKWarcpCpOacXoOozDWhW7j25gH2zzRE/isU6DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T00:10:44.382545Z","bundle_sha256":"851f0d2b366faccde897e78353986404af8a2f6f08725adfb846d5430647769d"}}