{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:SA4Y2SLDSTIYTRQKAIJR2YC5RA","short_pith_number":"pith:SA4Y2SLD","canonical_record":{"source":{"id":"1906.05585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-13T09:57:11Z","cross_cats_sorted":[],"title_canon_sha256":"6a71c91c1abdf57d81f9888a5ee2b7ec02654ad2ebeaa309713ace781bc8f074","abstract_canon_sha256":"862e49eb385a6083e80bdcac2931532ea637c8d87953dc261330361cc97ce9d8"},"schema_version":"1.0"},"canonical_sha256":"90398d496394d189c60a02131d605d8831111385532af1b81703c7ca08668c26","source":{"kind":"arxiv","id":"1906.05585","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.05585","created_at":"2026-05-17T23:43:24Z"},{"alias_kind":"arxiv_version","alias_value":"1906.05585v1","created_at":"2026-05-17T23:43:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.05585","created_at":"2026-05-17T23:43:24Z"},{"alias_kind":"pith_short_12","alias_value":"SA4Y2SLDSTIY","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SA4Y2SLDSTIYTRQK","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SA4Y2SLD","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:SA4Y2SLDSTIYTRQKAIJR2YC5RA","target":"record","payload":{"canonical_record":{"source":{"id":"1906.05585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-13T09:57:11Z","cross_cats_sorted":[],"title_canon_sha256":"6a71c91c1abdf57d81f9888a5ee2b7ec02654ad2ebeaa309713ace781bc8f074","abstract_canon_sha256":"862e49eb385a6083e80bdcac2931532ea637c8d87953dc261330361cc97ce9d8"},"schema_version":"1.0"},"canonical_sha256":"90398d496394d189c60a02131d605d8831111385532af1b81703c7ca08668c26","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:24.864473Z","signature_b64":"ncTnVEP/UoOCB55T2FYEsJ9Yesjtj+4RHZHFDX5z2LOJ+2XAlMI+ZypfGIQvY6hta2zYHO68etGUlNSkRKHEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90398d496394d189c60a02131d605d8831111385532af1b81703c7ca08668c26","last_reissued_at":"2026-05-17T23:43:24.863855Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:24.863855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.05585","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":"2026-05-17T23:43:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"npnrloJ0Wn+AveZSkXPAZ6D5OzD3TMXTY9wKCfzLwlG7mlW9IUDHTYCwHKB/I+eHmSDxJVZCKDlUsK86tNyGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:55:43.533813Z"},"content_sha256":"7d5e01a39ec01421926622f944fd76dd70d964cd4c78bb84558c077efd3b116a","schema_version":"1.0","event_id":"sha256:7d5e01a39ec01421926622f944fd76dd70d964cd4c78bb84558c077efd3b116a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:SA4Y2SLDSTIYTRQKAIJR2YC5RA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Perturbation theory and higher order $\\mathcal{S}^p$-differentiability of operator functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Cl\\'ement Coine","submitted_at":"2019-06-13T09:57:11Z","abstract_excerpt":"We establish, for $1 < p < \\infty$, higher order $\\mathcal{S}^p$-differentiability results of the function $\\varphi : t\\in \\mathbb{R} \\mapsto f(A+tK) - f(A)$ for selfadjoint operators $A$ and $K$ on a separable Hilbert space $\\mathcal{H}$ with $K$ element of the Schatten class $\\mathcal{S}^p(\\mathcal{H})$ and $f$ $n$-times differentiable on $\\mathbb{R}$. We prove that if either $A$ and $f^{(n)}$ are bounded or $f^{(i)}, 1 \\leq i \\leq n$ are bounded, $\\varphi$ is $n$-times differentiable on $\\mathbb{R}$ in the $\\mathcal{S}^p$-norm with bounded $n$th derivative. If $f\\in C^n(\\mathbb{R})$ with bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.05585","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":"2026-05-17T23:43:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0Jb/tHjatLPDcMHcv+ibCuF28/cAkEfKBsOF0f1Ye/aGUx76KMcwDCwSmG0ZICa1kR9jPEXR/wNcnN7iG4TmCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T12:55:43.534523Z"},"content_sha256":"9c7c9045011313860c047c09ff8fd405a9b57fd71c99b9f99af8ae4207f3f670","schema_version":"1.0","event_id":"sha256:9c7c9045011313860c047c09ff8fd405a9b57fd71c99b9f99af8ae4207f3f670"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA/bundle.json","state_url":"https://pith.science/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA/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":"2026-05-26T12:55:43Z","links":{"resolver":"https://pith.science/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA","bundle":"https://pith.science/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA/bundle.json","state":"https://pith.science/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SA4Y2SLDSTIYTRQKAIJR2YC5RA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:SA4Y2SLDSTIYTRQKAIJR2YC5RA","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":"862e49eb385a6083e80bdcac2931532ea637c8d87953dc261330361cc97ce9d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-13T09:57:11Z","title_canon_sha256":"6a71c91c1abdf57d81f9888a5ee2b7ec02654ad2ebeaa309713ace781bc8f074"},"schema_version":"1.0","source":{"id":"1906.05585","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.05585","created_at":"2026-05-17T23:43:24Z"},{"alias_kind":"arxiv_version","alias_value":"1906.05585v1","created_at":"2026-05-17T23:43:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.05585","created_at":"2026-05-17T23:43:24Z"},{"alias_kind":"pith_short_12","alias_value":"SA4Y2SLDSTIY","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SA4Y2SLDSTIYTRQK","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SA4Y2SLD","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:9c7c9045011313860c047c09ff8fd405a9b57fd71c99b9f99af8ae4207f3f670","target":"graph","created_at":"2026-05-17T23:43:24Z","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 establish, for $1 < p < \\infty$, higher order $\\mathcal{S}^p$-differentiability results of the function $\\varphi : t\\in \\mathbb{R} \\mapsto f(A+tK) - f(A)$ for selfadjoint operators $A$ and $K$ on a separable Hilbert space $\\mathcal{H}$ with $K$ element of the Schatten class $\\mathcal{S}^p(\\mathcal{H})$ and $f$ $n$-times differentiable on $\\mathbb{R}$. We prove that if either $A$ and $f^{(n)}$ are bounded or $f^{(i)}, 1 \\leq i \\leq n$ are bounded, $\\varphi$ is $n$-times differentiable on $\\mathbb{R}$ in the $\\mathcal{S}^p$-norm with bounded $n$th derivative. If $f\\in C^n(\\mathbb{R})$ with bo","authors_text":"Cl\\'ement Coine","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-13T09:57:11Z","title":"Perturbation theory and higher order $\\mathcal{S}^p$-differentiability of operator functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.05585","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":"sha256:7d5e01a39ec01421926622f944fd76dd70d964cd4c78bb84558c077efd3b116a","target":"record","created_at":"2026-05-17T23:43:24Z","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":"862e49eb385a6083e80bdcac2931532ea637c8d87953dc261330361cc97ce9d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-13T09:57:11Z","title_canon_sha256":"6a71c91c1abdf57d81f9888a5ee2b7ec02654ad2ebeaa309713ace781bc8f074"},"schema_version":"1.0","source":{"id":"1906.05585","kind":"arxiv","version":1}},"canonical_sha256":"90398d496394d189c60a02131d605d8831111385532af1b81703c7ca08668c26","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90398d496394d189c60a02131d605d8831111385532af1b81703c7ca08668c26","first_computed_at":"2026-05-17T23:43:24.863855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:24.863855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ncTnVEP/UoOCB55T2FYEsJ9Yesjtj+4RHZHFDX5z2LOJ+2XAlMI+ZypfGIQvY6hta2zYHO68etGUlNSkRKHEBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:24.864473Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.05585","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7d5e01a39ec01421926622f944fd76dd70d964cd4c78bb84558c077efd3b116a","sha256:9c7c9045011313860c047c09ff8fd405a9b57fd71c99b9f99af8ae4207f3f670"],"state_sha256":"2d035abce4d7373942ae51c214aad2807313ceb35ed204b8529c73b9426ccc12"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yW7NW7DjvAPoF4IE01p6bJBogx6GK+K+USlBEl/vF2uliKZHBSKxDOEFzUMp/0AwPj9eiyc7BuimGApIuqOMBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T12:55:43.538066Z","bundle_sha256":"c3acde71800e9c083fa30b4f39070018f3310723ed33d9b4605fe68c7453c34a"}}