{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:XF6NJ7ZIVWY66UUAQ53ECQ7773","short_pith_number":"pith:XF6NJ7ZI","schema_version":"1.0","canonical_sha256":"b97cd4ff28adb1ef528087764143fffec089f1b10840200a31f606d70fde1646","source":{"kind":"arxiv","id":"2112.01233","version":1},"attestation_state":"computed","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"},"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":"2112.01233","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2021-12-02T13:45:09Z","cross_cats_sorted":["math.DS","math.FA"],"title_canon_sha256":"6c6f660795619b1eee84cb10662081d020b4852cffd64f18e149d1ccff37eef9","abstract_canon_sha256":"113f41d607d591362eee60438b194a1ac564a3352436f4aac6447711c55f7981"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:37:06.808185Z","signature_b64":"tblwtQ9WiQNB24uUkPy8L3m+P+qex3Lg1+VfEKuaj3FnrF872wsZNPlXmMjLWBSMzCjR4UaJRWbptm9njZesAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b97cd4ff28adb1ef528087764143fffec089f1b10840200a31f606d70fde1646","last_reissued_at":"2026-07-05T03:37:06.807789Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:37:06.807789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2112.01233","created_at":"2026-07-05T03:37:06.807850+00:00"},{"alias_kind":"arxiv_version","alias_value":"2112.01233v1","created_at":"2026-07-05T03:37:06.807850+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.01233","created_at":"2026-07-05T03:37:06.807850+00:00"},{"alias_kind":"pith_short_12","alias_value":"XF6NJ7ZIVWY6","created_at":"2026-07-05T03:37:06.807850+00:00"},{"alias_kind":"pith_short_16","alias_value":"XF6NJ7ZIVWY66UUA","created_at":"2026-07-05T03:37:06.807850+00:00"},{"alias_kind":"pith_short_8","alias_value":"XF6NJ7ZI","created_at":"2026-07-05T03:37:06.807850+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773","json":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773.json","graph_json":"https://pith.science/api/pith-number/XF6NJ7ZIVWY66UUAQ53ECQ7773/graph.json","events_json":"https://pith.science/api/pith-number/XF6NJ7ZIVWY66UUAQ53ECQ7773/events.json","paper":"https://pith.science/paper/XF6NJ7ZI"},"agent_actions":{"view_html":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773","download_json":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773.json","view_paper":"https://pith.science/paper/XF6NJ7ZI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2112.01233&json=true","fetch_graph":"https://pith.science/api/pith-number/XF6NJ7ZIVWY66UUAQ53ECQ7773/graph.json","fetch_events":"https://pith.science/api/pith-number/XF6NJ7ZIVWY66UUAQ53ECQ7773/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773/action/storage_attestation","attest_author":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773/action/author_attestation","sign_citation":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773/action/citation_signature","submit_replication":"https://pith.science/pith/XF6NJ7ZIVWY66UUAQ53ECQ7773/action/replication_record"}},"created_at":"2026-07-05T03:37:06.807850+00:00","updated_at":"2026-07-05T03:37:06.807850+00:00"}