{"paper":{"title":"Linear combination of composition operators on $H^\\infty$ and the Bloch space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Songxiao Li, Yecheng Shi","submitted_at":"2018-02-12T15:06:55Z","abstract_excerpt":"Let $\\lambda_i (i=1,...,k)$ be any nonzero complex scalars and $\\varphi_i (i=1,..,k)$ be any analytic self-maps of the unit disk $\\mathbb{D}$. We show that the operator $\\sum_{i=1}^k\\lambda_iC_{\\varphi_i}$ is compact on the Bloch space $\\mathcal{B}$ if and only if\n  $$\\lim_{n\\to\\infty}\\|\\lambda_1\\varphi_1^n+\\lambda_2\\varphi_2^n+...+\\lambda_k\\varphi_k^n\\|_{\\mathcal{B}}=0.$$ We also study the linear combination of composition operators on the Banach algebra of bounded analytic functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.04092","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"}