{"paper":{"title":"On the Operator-valued $\\mu$-cosine functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","submitted_at":"2017-01-25T09:49:39Z","abstract_excerpt":"Let $(G,+)$ be a topological abelian group with a neutral element $e$ and let $\\mu : G\\longrightarrow\\mathbb{C}$ be a continuous character of $G$. Let $(\\mathcal{H}, \\langle \\cdot,\\cdot \\rangle)$ be a complex Hilbert space and let $\\mathbf{B}(\\mathcal{H})$ be the algebra of all linear continuous operators of $\\mathcal{H}$ into itself. A continuous mapping $ \\Phi: G\\longrightarrow \\mathbf{B}(\\mathcal{H})$ will be called an operator-valued $\\mu$-cosine function if it satisfies both the $\\mu$-cosine equation $$\\Phi(x+y)+\\mu(y)\\Phi(x-y)=2\\Phi(x)\\Phi(y),\\; x,y\\in G$$ and the condition $\\Phi(e)=I,$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07229","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"}