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We prove that if $f$ belongs to the Besov class $B_{\\be,1}^1(\\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\\|f(A_1,B_1)-f(A_2,B_2)\\|_{\\bS_1}\\le\\const(\\|A_1-A_2\\|_{\\bS_1}+\\|B_1-B_2\\|_{\\bS_1})$. However, the condition $f\\in B_{\\be,1}^1(\\R^2)$ does not imply the Lipschitz type estimate in the operator norm."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.1815","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-11-07T03:20:33Z","cross_cats_sorted":["math.CA","math.CV","math.SP"],"title_canon_sha256":"0034ce5d07cc3c2195819ef53b68ac8b3cf738d0f64ca295f78cba4cd82a8c89","abstract_canon_sha256":"91bcda17345abbdc02612251af51acbfd0fc4e2bcc684e9ce1fc074db4ad140f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:25.939038Z","signature_b64":"anWJKbY2Xb4W97aC+Ww3wwOUEfZGw5iy7JBWlYmzWvS3TXynyINQFDr086HLBvy5kvLWBuSYHbkQrI9XBZc/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20daefb39fee4336228a2b826cf7ede1d3ad5e7e28f275daf3f42c2fb2ffcb10","last_reissued_at":"2026-05-18T02:38:25.938587Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:25.938587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functions of perturbed noncommuting self-adjoint 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, Fedor Nazarov, Vladimir Peller","submitted_at":"2014-11-07T03:20:33Z","abstract_excerpt":"We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\\be,1}^1(\\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\\|f(A_1,B_1)-f(A_2,B_2)\\|_{\\bS_1}\\le\\const(\\|A_1-A_2\\|_{\\bS_1}+\\|B_1-B_2\\|_{\\bS_1})$. However, the condition $f\\in B_{\\be,1}^1(\\R^2)$ does not imply the Lipschitz type estimate in the operator norm."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1815","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1411.1815","created_at":"2026-05-18T02:38:25.938653+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.1815v1","created_at":"2026-05-18T02:38:25.938653+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.1815","created_at":"2026-05-18T02:38:25.938653+00:00"},{"alias_kind":"pith_short_12","alias_value":"EDNO7M475ZBT","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"EDNO7M475ZBTMIUK","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"EDNO7M47","created_at":"2026-05-18T12:28:25.294606+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H","json":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H.json","graph_json":"https://pith.science/api/pith-number/EDNO7M475ZBTMIUKFOBGZ57N4H/graph.json","events_json":"https://pith.science/api/pith-number/EDNO7M475ZBTMIUKFOBGZ57N4H/events.json","paper":"https://pith.science/paper/EDNO7M47"},"agent_actions":{"view_html":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H","download_json":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H.json","view_paper":"https://pith.science/paper/EDNO7M47","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.1815&json=true","fetch_graph":"https://pith.science/api/pith-number/EDNO7M475ZBTMIUKFOBGZ57N4H/graph.json","fetch_events":"https://pith.science/api/pith-number/EDNO7M475ZBTMIUKFOBGZ57N4H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H/action/storage_attestation","attest_author":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H/action/author_attestation","sign_citation":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H/action/citation_signature","submit_replication":"https://pith.science/pith/EDNO7M475ZBTMIUKFOBGZ57N4H/action/replication_record"}},"created_at":"2026-05-18T02:38:25.938653+00:00","updated_at":"2026-05-18T02:38:25.938653+00:00"}