{"paper":{"title":"On the extension of battys theorem on the semigroup asymptotic stability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS","math.FA"],"primary_cat":"math.OC","authors_text":"Bartosz Wasilewski, Grigory M. Sklyar, Piotr Polak","submitted_at":"2021-12-02T13:45:09Z","abstract_excerpt":"The well-known Batty's theorem states that if a $C_0$-semigroup $T(t)$ is bounded and the spectrum of the generator $A$ is contained in the open left-half plane of $\\mathbb{C}$, then $\\|T(t)A^{-1}\\|$ tends to $0$. This can be thought of as a particular case of a more general property that, for $\\omega_0>-\\infty$ and $(\\omega_0+i\\mathbb{R})\\cap \\sigma(A)=\\emptyset$ it holds $\\|T(t)(A-\\omega_0 I)^{-1}\\|/\\|T(t)\\|$ tends to 0. We show that it is true for $\\|T(t)\\|$ regular enough, however we give examples of unbounded semigroups, with the spectrum of the generator not contained in the open left-ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.01233","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.01233/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"}