{"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"}