{"paper":{"title":"Zero-two law for cosine families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Felix Schwenninger, Hans Zwart","submitted_at":"2014-02-06T10:23:37Z","abstract_excerpt":"For $\\left(C(t)\\right)_{t \\geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\\limsup_{t\\to 0^{+}}\\|C(t) - I\\| <2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of $\\sup_{t\\geq0}\\|C(t)-I\\|<2$ yields that $C(t)=I$ for all $t\\geq0$. Additionally, we derive alternative proofs for similar results for $C_{0}$-semigroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1304","kind":"arxiv","version":3},"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"}