{"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"}