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We establish that a function $f$ defined on the unit circle $\\cir$ is $n$ times continuously Fr\\'echet $\\Sp^p$-differentiable at every point in $\\mathcal{U}(\\hilh)$ if and only if $f\\in C^n(\\cir)$. Take a function $U :\\R\\rightarrow\\mathcal{U}(\\hilh)$ such that the function $t\\in\\R\\mapsto U(t)-U(0)$ takes values in $\\Sp^{p}$ and is $n$ times continuously $\\Sp^{p}$-differentiable on $\\R$. Consequently, for $f\\in C^n(\\cir)$, we prove that $f$ is $n$ times con"},"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":"2404.08253","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2024-04-12T05:49:53Z","cross_cats_sorted":[],"title_canon_sha256":"53988439bf474248354c52d126d72674dba89a9d094e3d3b6d56ca0d2a2b71fd","abstract_canon_sha256":"60652b68c651c9d05fbb834d2e81926721f7399c69b333439fe73c1329fea5eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:21:03.572815Z","signature_b64":"jcwqNBVxu9ODWkBbZnSEM4Yl2YCCX6U5RRIAUHctgax0CitjKs+Ef2TK8OihdtIQNxyEaZGJotd4pH0zBdybDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c0df71169a69f096e007e1e8296a326dcc00d59c7e83ab54078f719e49a64dd","last_reissued_at":"2026-07-05T09:21:03.572194Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:21:03.572194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher order $\\mathcal{S}^{p}$-differentiability: The unitary case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Arup Chattopadhyay, Chandan Pradhan, Cl\\'ement Coine, Saikat Giri","submitted_at":"2024-04-12T05:49:53Z","abstract_excerpt":"Consider the set of unitary operators on a complex separable Hilbert space $\\hilh$, denoted as $\\mathcal{U}(\\hilh)$. Consider $1<p<\\infty$. We establish that a function $f$ defined on the unit circle $\\cir$ is $n$ times continuously Fr\\'echet $\\Sp^p$-differentiable at every point in $\\mathcal{U}(\\hilh)$ if and only if $f\\in C^n(\\cir)$. Take a function $U :\\R\\rightarrow\\mathcal{U}(\\hilh)$ such that the function $t\\in\\R\\mapsto U(t)-U(0)$ takes values in $\\Sp^{p}$ and is $n$ times continuously $\\Sp^{p}$-differentiable on $\\R$. Consequently, for $f\\in C^n(\\cir)$, we prove that $f$ is $n$ times con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.08253","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.08253/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2404.08253","created_at":"2026-07-05T09:21:03.572270+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.08253v2","created_at":"2026-07-05T09:21:03.572270+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.08253","created_at":"2026-07-05T09:21:03.572270+00:00"},{"alias_kind":"pith_short_12","alias_value":"LQG7OELJU2PQ","created_at":"2026-07-05T09:21:03.572270+00:00"},{"alias_kind":"pith_short_16","alias_value":"LQG7OELJU2PQS3QA","created_at":"2026-07-05T09:21:03.572270+00:00"},{"alias_kind":"pith_short_8","alias_value":"LQG7OELJ","created_at":"2026-07-05T09:21:03.572270+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2411.16426","citing_title":"Trace formulas for $\\mathcal{S}^p$-perturbations and extension of Koplienko-Neidhardt trace formulas","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3","json":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3.json","graph_json":"https://pith.science/api/pith-number/LQG7OELJU2PQS3QAPYPIFFVDE3/graph.json","events_json":"https://pith.science/api/pith-number/LQG7OELJU2PQS3QAPYPIFFVDE3/events.json","paper":"https://pith.science/paper/LQG7OELJ"},"agent_actions":{"view_html":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3","download_json":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3.json","view_paper":"https://pith.science/paper/LQG7OELJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.08253&json=true","fetch_graph":"https://pith.science/api/pith-number/LQG7OELJU2PQS3QAPYPIFFVDE3/graph.json","fetch_events":"https://pith.science/api/pith-number/LQG7OELJU2PQS3QAPYPIFFVDE3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3/action/storage_attestation","attest_author":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3/action/author_attestation","sign_citation":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3/action/citation_signature","submit_replication":"https://pith.science/pith/LQG7OELJU2PQS3QAPYPIFFVDE3/action/replication_record"}},"created_at":"2026-07-05T09:21:03.572270+00:00","updated_at":"2026-07-05T09:21:03.572270+00:00"}