{"paper":{"title":"A characterization of compact operators on $\\ell^p$-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mortaza Abtahi","submitted_at":"2025-06-07T09:18:13Z","abstract_excerpt":"Let $A$ be a Banach space, $p>1$, and $1/p+1/q=1$. If a sequence $a=(a_i)$ in $A$ has a finite $p$-sum, then the operator $\\Lambda_a:\\ell^q\\to A$, defined by $\\Lambda_a(\\beta)=\\sum_{i=1}^\\infty \\beta_i a_i, \\beta=(\\beta_i)\\in \\ell^q$, is compact. We present a characterization of compact operators $\\Lambda:\\ell^q\\to A$, and prove that $\\Lambda$ is compact if and only if $\\Lambda=\\Lambda_a$, for some sequence $a=(a_i)$ in $A$ with $\\{(\\phi(a_i)): \\phi\\in A^*, \\|\\phi\\|\\leq 1\\}$ being a totally bounded set in $\\ell^p$. For a sequence $(T_i)$ of bounded operators on a Hilbert space $H$, the corresp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.06726","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/2506.06726/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"}