{"paper":{"title":"Compactness Properties of Weighted Summation Operators on Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mikhail Lifshits, Werner Linde","submitted_at":"2010-06-19T12:44:45Z","abstract_excerpt":"We investigate compactness properties of weighted summation operators $V_{\\alpha,\\sigma}$ as mapping from $\\ell_1(T)$ into $\\ell_q(T)$ for some $q\\in (1,\\infty)$. Those operators are defined by $$ (V_{\\alpha,\\sigma} x)(t) :=\\alpha(t)\\sum_{s\\succeq t}\\sigma(s) x(s)\\,,\\quad t\\in T\\;, $$ where $T$ is a tree with induced partial order $t \\preceq s$ (or $s \\succeq t$) for $t,s\\in T$. Here $\\alpha$ and $\\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\\alpha,\\sigma})$, the (dyadic) entropy numbers of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3867","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"}