{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:VR2X5AWNFCPEZDZUOVWMBBNNKY","short_pith_number":"pith:VR2X5AWN","canonical_record":{"source":{"id":"1008.1638","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-10T06:08:52Z","cross_cats_sorted":["math.CA","math.CV","math.SP"],"title_canon_sha256":"d66f012140646ab4f894ab6f37d79bfb64fa3fd26908ca77a7b76b1a20bf3153","abstract_canon_sha256":"19eae58393dd15b4b158cb71b8cddfb2aa4145309dae39fe684b7a0685f931cb"},"schema_version":"1.0"},"canonical_sha256":"ac757e82cd289e4c8f34756cc085ad5615e3f6879bff789669cb51c34c01e57d","source":{"kind":"arxiv","id":"1008.1638","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1638","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1638v1","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1638","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"VR2X5AWNFCPE","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"VR2X5AWNFCPEZDZU","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"VR2X5AWN","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:VR2X5AWNFCPEZDZUOVWMBBNNKY","target":"record","payload":{"canonical_record":{"source":{"id":"1008.1638","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-10T06:08:52Z","cross_cats_sorted":["math.CA","math.CV","math.SP"],"title_canon_sha256":"d66f012140646ab4f894ab6f37d79bfb64fa3fd26908ca77a7b76b1a20bf3153","abstract_canon_sha256":"19eae58393dd15b4b158cb71b8cddfb2aa4145309dae39fe684b7a0685f931cb"},"schema_version":"1.0"},"canonical_sha256":"ac757e82cd289e4c8f34756cc085ad5615e3f6879bff789669cb51c34c01e57d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:23.657981Z","signature_b64":"P17bSle3lyF7dVnt7J1N4ryScNNVVs41Co/tUtugfLGyrWf1QMgeIL6fJRQlb0H1KQX+gLLVQHlNcLPuDszMBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac757e82cd289e4c8f34756cc085ad5615e3f6879bff789669cb51c34c01e57d","last_reissued_at":"2026-05-18T04:42:23.657572Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:23.657572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.1638","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":"2026-05-18T04:42:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RB1Gre4TmSQKjnXx+EAlaiLckORuzVOnzBodZHoC3kcoaVcB0GS1yrKSxB9BQHBciQ24ibcMPQrih5+bTrYFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:11:02.090562Z"},"content_sha256":"067e00ae7359105677f81097aa79df47092af6f153e211e858a2b5a5ce6c545d","schema_version":"1.0","event_id":"sha256:067e00ae7359105677f81097aa79df47092af6f153e211e858a2b5a5ce6c545d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:VR2X5AWNFCPEZDZUOVWMBBNNKY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Functions of normal operators under perturbations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV","math.SP"],"primary_cat":"math.FA","authors_text":"Alexei Aleksandrov, Denis Potapov, Fedor Sukochev, Vladimir Peller","submitted_at":"2010-08-10T06:08:52Z","abstract_excerpt":"In \\cite{Pe1}, \\cite{Pe2}, \\cite{AP1}, \\cite{AP2}, and \\cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the H\\\"older class $\\L_\\a(\\R^2)$, $0<\\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\\|f(N_1)-f(N_2)\\|\\le\\const\\|f\\|_{\\L_\\a}\\|N_1-N_2\\|^\\a$. We obtain a more general result for functions in the space $\\L_\\o(\\R^2)=\\big\\{f:"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1638","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":"2026-05-18T04:42:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Pmdwyx1vxx+GgTXvNSrCzzHvIPfqxFur0ulMvjGDCHFKw6gBkf/8Sjeigg60f6mhs0j1BT4XTJ7LHcjyvryDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:11:02.091190Z"},"content_sha256":"c789f1569dfc2de308f529999a6058c599907469c79a78c69658564af46e2cde","schema_version":"1.0","event_id":"sha256:c789f1569dfc2de308f529999a6058c599907469c79a78c69658564af46e2cde"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY/bundle.json","state_url":"https://pith.science/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY/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":"2026-05-31T00:11:02Z","links":{"resolver":"https://pith.science/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY","bundle":"https://pith.science/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY/bundle.json","state":"https://pith.science/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VR2X5AWNFCPEZDZUOVWMBBNNKY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:VR2X5AWNFCPEZDZUOVWMBBNNKY","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":"19eae58393dd15b4b158cb71b8cddfb2aa4145309dae39fe684b7a0685f931cb","cross_cats_sorted":["math.CA","math.CV","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-10T06:08:52Z","title_canon_sha256":"d66f012140646ab4f894ab6f37d79bfb64fa3fd26908ca77a7b76b1a20bf3153"},"schema_version":"1.0","source":{"id":"1008.1638","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1638","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1638v1","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1638","created_at":"2026-05-18T04:42:23Z"},{"alias_kind":"pith_short_12","alias_value":"VR2X5AWNFCPE","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"VR2X5AWNFCPEZDZU","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"VR2X5AWN","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:c789f1569dfc2de308f529999a6058c599907469c79a78c69658564af46e2cde","target":"graph","created_at":"2026-05-18T04:42:23Z","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":"In \\cite{Pe1}, \\cite{Pe2}, \\cite{AP1}, \\cite{AP2}, and \\cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the H\\\"older class $\\L_\\a(\\R^2)$, $0<\\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\\|f(N_1)-f(N_2)\\|\\le\\const\\|f\\|_{\\L_\\a}\\|N_1-N_2\\|^\\a$. We obtain a more general result for functions in the space $\\L_\\o(\\R^2)=\\big\\{f:","authors_text":"Alexei Aleksandrov, Denis Potapov, Fedor Sukochev, 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":"2010-08-10T06:08:52Z","title":"Functions of normal operators under perturbations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1638","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":"sha256:067e00ae7359105677f81097aa79df47092af6f153e211e858a2b5a5ce6c545d","target":"record","created_at":"2026-05-18T04:42:23Z","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":"19eae58393dd15b4b158cb71b8cddfb2aa4145309dae39fe684b7a0685f931cb","cross_cats_sorted":["math.CA","math.CV","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-10T06:08:52Z","title_canon_sha256":"d66f012140646ab4f894ab6f37d79bfb64fa3fd26908ca77a7b76b1a20bf3153"},"schema_version":"1.0","source":{"id":"1008.1638","kind":"arxiv","version":1}},"canonical_sha256":"ac757e82cd289e4c8f34756cc085ad5615e3f6879bff789669cb51c34c01e57d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac757e82cd289e4c8f34756cc085ad5615e3f6879bff789669cb51c34c01e57d","first_computed_at":"2026-05-18T04:42:23.657572Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:42:23.657572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P17bSle3lyF7dVnt7J1N4ryScNNVVs41Co/tUtugfLGyrWf1QMgeIL6fJRQlb0H1KQX+gLLVQHlNcLPuDszMBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:42:23.657981Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.1638","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:067e00ae7359105677f81097aa79df47092af6f153e211e858a2b5a5ce6c545d","sha256:c789f1569dfc2de308f529999a6058c599907469c79a78c69658564af46e2cde"],"state_sha256":"4fa87111e59e3fc279e843baea08ddf6ac23a819b55eb88429fa47ee14733b46"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gT4V7meXnHFpRuKA2qxY+lCN/sxF4wIeY6dsKrfdEs4kFNR+dH+3kU0yTsiLykY5gzT5JpBIJT3lhKxV3HsJDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T00:11:02.093937Z","bundle_sha256":"c20833e5372d6e75c1ea8c63b4e23aad0221f8792cf9bd4c06ed2f0d465123a9"}}