{"paper":{"title":"On measuring unboundedness of the $H^\\infty$-calculus for generators of analytic semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Felix Schwenninger","submitted_at":"2015-02-05T13:27:48Z","abstract_excerpt":"We investigate the boundedness of the $H^\\infty$-calculus by estimating the bound $b(\\varepsilon)$ of the mapping $H^{\\infty}\\rightarrow \\mathcal{B}(X)$: $f\\mapsto f(A)T(\\varepsilon)$ for $\\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\\varepsilon)=\\mathcal{O}(|\\log\\varepsilon|)$ in general, whereas $b(\\varepsilon)=\\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01535","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":""},"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"}